# Robust mean field control: stochastic maximum principle and variational mean field games

Pierre Lavigne, François Delarue

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Pierre Lavigne, François Delarue. Robust mean field control: stochastic maximum principle and variational mean field games. 2026. hal-05531353

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# Robust mean field control: stochastic maximum principle and variational mean field games

François Delarue * Pierre Lavigne *

## Abstract

We introduce a class of robust control problems formulated in min-max form, in which the principal agent is viewed as a central planner facing Nature. The agent's cost is a nonlinear function of all its possible realizations, encompassing in particular the mean field regime where the cost depends on the distribution of the states. In parallel, Nature favors the occurrence of outcomes that are least favorable to the agent, at an entropic cost. We establish existence and uniqueness of solutions under appropriate assumptions, including suitable convexity-concavity conditions, and derive a related stochastic maximum principle. We further address a corresponding class of robust variational mean field games in which the interaction term is subject to ambiguity, and prove existence and uniqueness of solutions.

Keywords: Robust mean field control, Stochastic maximum principle, Risk-averse control, Quadratic backward stochastic differential equation, Entropic penalties.

MSC2020. Primary: 49N80, 91A16; Secondary: 93E20, 60H10.

## Contents

<table><tr><td>1 Introduction</td><td>2</td></tr><tr><td>2 Notations</td><td>9</td></tr><tr><td>3 Stochastic maximum principle</td><td>12</td></tr><tr><td>3.1 Main result</td><td>12</td></tr><tr><td>3.2 Examples of applications</td><td>21</td></tr><tr><td>4 Applications to mean field models</td><td>26</td></tr><tr><td>4.1 Robust mean field control</td><td>26</td></tr><tr><td>4.2 Examples</td><td>35</td></tr><tr><td>4.3 Variational mean field games</td><td>43</td></tr></table>

<footnote>* Université Côte d'Azur, CNRS, Laboratoire J.A. Dieudonné, 06108 Nice, France; Emails: francois.delarue@univ-cotedazur.fr, pierre.lavigne@univ-cotedazur.fr; F. Delarue and P. Lavigne acknowledge the financial support of the European Research Council (ERC) under the European Union's Horizon Europe research and innovation program (ELISA project, Grant agreement No. 101054746). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.</footnote>
<table>
  <tbody>
    <tr>
      <td>5</td>
      <td>Proof of Theorem 10</td>
      <td>45</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Existence of a saddle point to (P')</td>
      <td>45</td>
    </tr>
    <tr>
      <td>5.1.1</td>
      <td>Trajectories of the perspective problem and positivity of the optimal ones</td>
      <td>48</td>
    </tr>
    <tr>
      <td>5.1.2</td>
      <td>Solvability of the perspective min-max problem</td>
      <td>52</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Nature's control problem</td>
      <td>62</td>
    </tr>
    <tr>
      <td>5.2.1</td>
      <td>A priori estimate</td>
      <td>64</td>
    </tr>
    <tr>
      <td>5.2.2</td>
      <td>Necessary condition</td>
      <td>66</td>
    </tr>
    <tr>
      <td>5.2.3</td>
      <td>Sufficient condition</td>
      <td>82</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Central planner's control problem</td>
      <td>86</td>
    </tr>
    <tr>
      <td>5.3.1</td>
      <td>A priori estimate</td>
      <td>87</td>
    </tr>
    <tr>
      <td>5.3.2</td>
      <td>Necessary conditions</td>
      <td>88</td>
    </tr>
    <tr>
      <td>5.3.3</td>
      <td>Sufficient conditions</td>
      <td>95</td>
    </tr>
    <tr>
      <td>A</td>
      <td>Finite entropy and positive measures</td>
      <td>98</td>
    </tr>
    <tr>
      <td>B</td>
      <td>A priori estimates on a linear SDE</td>
      <td>100</td>
    </tr>
    <tr>
      <td>C</td>
      <td>Uniform integrability and convergence results</td>
      <td>104</td>
    </tr>
    <tr>
      <td>D</td>
      <td>Distance and differentiability on spaces of non-negative measures</td>
      <td>106</td>
    </tr>
    <tr>
      <td>D.1</td>
      <td>Generalized Wasserstein distance</td>
      <td>106</td>
    </tr>
    <tr>
      <td>D.2</td>
      <td>Differentiability</td>
      <td>108</td>
    </tr>
  </tbody>
</table>

# 1 Introduction

In this work we introduce a zero-sum non-local stochastic game in finite horizon between two players. Throughout, the first player is referred to as ‘Nature’ and the second one to as ‘the central planner’.

**Formulation of the problem.** The problem is defined on a finite interval $[0, T]$ and a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ equipped with a $d$-dimensional Brownian motion $W = (W_t)_{t \in [0, T]}$ and an independent $n$-dimensional random variable $\eta$ representing the initial condition of the central planner. Here, $n \in \mathbb{N}^*$ is the state dimension of the central planner and $d \in \mathbb{N}^*$ the noise dimension to which the central planner is subjected. The $\mathbb{P}$-complete filtration generated by $(\eta, W)$ is denoted by $\mathbb{F} = (\mathcal{F}_t)_{0 \le t \le T}$. On this probabilistic set-up, we consider the following inf-sup non-local (in the sense that the cost $\mathcal{G}$ below takes the entire random variables $q_T$ and $X_T^\psi$, and not only their realizations, as inputs) stochastic control problem:

$$ \sup_{q \in \mathcal{Q}} \inf_{\psi \in \mathcal{A}} \mathcal{J}(q, \psi), \quad \mathcal{J}(q, \psi) := \mathcal{R}(q, \psi) - \mathcal{S}(q), \qquad (P) $$

where

$$ \mathcal{R}(q, \psi) := \mathcal{G}(q_T, X_T^\psi) + \mathbb{E} \left[ \int_0^T q_s \ell(s, \psi_s) ds \right], \qquad (1) $$

$$ \mathcal{S}(q) := \mathbb{E} \left[ \int_0^T q_s f^*(s, Y_s^*, Z_s^*) ds \right]. \qquad (2) $$
In this formulation, equilibria are sought over open loop controls. Nature optimizes with respect to $q \in \mathcal{Q}$ and the central planner with respect to $\psi \in \mathcal{A}$, where the admissible sets $\mathcal{Q}$ and $\mathcal{A}$ can be roughly described as follows:

* The set $\mathcal{Q}$ is a class of $\mathbb{F}$-progressively measurable, positive-valued processes with finite entropy $\mathcal{S}$, accounting for changes in the historical measure $\mathbb{P}$ under uncertainty from Nature (here and throughout, 'positive' is understood in the sense of strictly positive). Precisely, a process $q$ belongs to $\mathcal{Q}$ if

$$ \mathcal{S}(q) < +\infty, \qquad (3) $$

$$ \text{and} \quad q_t = 1 + \int_0^t q_s Y_s^* \mathrm{d}s + \int_0^t q_s Z_s^* \cdot \mathrm{d}W_s, \quad t \in [0, T], \qquad (4) $$

where $Y^* = (Y_t^*)_{t \in [0,T]}$ and $Z^* = (Z_t^*)_{t \in [0,T]}$ are two $\mathbb{F}$-progressively measurable processes with values in $\mathbb{R}$ and $\mathbb{R}^d$, respectively. The process $q$ admits an explicit expression in terms of $Y^*$ and $Z^*$:

$$ q_t = e^{\int_0^t Y_s^* \mathrm{d}s} \mathcal{E}_t \left( \int_0^t Z_s^* \cdot \mathrm{d}W_s \right), \qquad (5) $$

where $(\mathcal{E}_t(\int_0^t Z_s^* \cdot \mathrm{d}W_s))_{t \in [0,T]}$ is the stochastic exponential associated to $Z^*$. From now on, we denote $q_T \mathbb{P}$ the equivalent (non-normalized) measure defined as $\int_A q_T \mathrm{d}\mathbb{P}$ for all $A \in \mathcal{F}$. When $|Y^*| = 0$, $q = (q_t)_{t \in [0,T]}$ is a Doléans-Dade exponential and defines an equivalent probability measure $q_T \mathbb{P}$. When $|Y^*| > 0$, $q$ defines a collection of equivalent non-normalized measures $(q_t \mathbb{P})_{t \in [0,T]}$, which we refer to as 'discounted measures'. We refer the reader to Appendix A for more details about the representation of $q$.

* The set $\mathcal{A}$ consists in a class of $\mathbb{F}$-progressively-measurable, $\mathbb{R}^n$-valued processes $\psi = (\psi_t)_{t \in [0,T]}$ such that

$$ \mathcal{S}^*(\psi) < +\infty, \quad \mathcal{S}^*(\psi) := \sup_{q \in \mathcal{Q}} \left\{ \mathbb{E} \left[ \int_0^T q_s |\psi_s|^2 \mathrm{d}s \right] - \gamma \mathcal{S}(q) \right\}. \qquad (6) $$

The coefficient $\gamma$ has to be fixed carefully and will be clearly defined in the Assumption A5 below, but we already mention that it should depend on the other data of the problem. For a given control $\psi \in \mathcal{A}$, the state $X^\psi = (X_t^\psi)_{t \in [0,T]}$ of the central planner is the solution to

$$ dX_t = b(t, X_t, \psi_t)dt + \sigma(t, \psi_t)dW_t, \quad X_0 = \eta, \qquad (7) $$

where the drift $b: \Omega \times [0, T] \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$ and the volatility $\sigma: \Omega \times [0, T] \times \mathbb{R}^n \to \mathbb{R}^{n \times d}$ are possibly random. Implicitly, $b$ and $\sigma$ are required to be $\mathbb{F}$-progressively measurable. The precise assumptions on the two of them will be clarified later in the article; see Subsection 3.1. In particular the state equation (7) will be assumed to be linear, but we keep it under general form for the exposition.

Returning to (1) and (2), $\ell: \Omega \times [0, T] \times \mathbb{R}^n \to \mathbb{R}$ is referred to as the running cost. The coefficient $f^*: \Omega \times [0, T] \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}$ is called the 'convex dual' driver (for reasons explained below). This function $f^*$ is typically viewed as (a perturbation
of) the square of its last argument. The function $\mathcal{G}$ represents the terminal cost. In its most general form, it is defined as a (measurable) real-valued mapping on $\Omega \times L^1(\Omega, \mathcal{F}_T, \mathbb{P}; \mathbb{R}_+) \times L^2(\Omega, \mathcal{F}_T, \mathbb{P}; \mathbb{R}^d)$. This formulation encompasses mean field functions with arguments such as $\mathbb{Q} \circ (X_T)^{-1}$ with $\mathbb{Q} = \mathcal{E}_T(\int_0^T Z_s^* \cdot dW_s)\mathbb{P}$; this example motivates the term central planner for the player optimizing over $\psi$. The problem is thus called non-local, since the functional $\mathcal{G}$ requires the full information on the terminal random variables $(q_T, X_T^\psi)$ to be evaluated. In principle, we could incorporate a running cost of a similar structure in (1), but for the sake of simplicity and clarity, we will omit this term from the remainder of the article.

The cost functions can be interpreted as follows: when the central planner chooses a strategy $\psi$, Nature tries to adjust the historical probability $\mathbb{P}$ by weighting it with $q$ in the worst possible way for the planner, thus maximizing the cost $\mathcal{R}(q, \psi)$. Conversely, once the weighting $q$ is chosen, the planner aims to select the best strategy $\psi$ to minimize $\mathcal{R}(q, \psi)$. This is an ‘almost classic’ stochastic control problem, depending on the form of the terminal cost $\mathcal{G}$. When $\mathcal{G}(q_T, X_T^\psi)$ is written as an expectation $\mathbb{E}[q_T g(X_T)]$, the planner solves a standard problem under the discounted measure $q\mathbb{P}$. When $\mathcal{G}(q_T, X_T^\psi)$ takes the form $\mathcal{G}((q_T\mathbb{P}) \circ (X_T^\psi)^{-1})$, with $\mathcal{G}$ being a cost function defined on the space $M_+(\mathbb{R}^n)$ of positive measures on $\mathbb{R}^n$, the planner solves a mean field control problem under the measures $q\mathbb{P}$. In both cases, the running cost $\ell$ can be chosen to be quadratic or to grow quadratically in $\psi$.

**A preview: risk averse control problem and BSDEs.** To better understand the problem (P), we focus in this paragraph on the first of the two cases above, namely, we assume that there exists a function $g: \mathbb{R}^n \to \mathbb{R}$ such that

$$ \mathcal{G}(q_T, X_T^\psi) := \mathbb{E} [q_T g(X_T^\psi)] = \mathbb{E}^{\mathbb{Q}} [e^{\int_0^T Y_s^* ds} g(X_T^\psi)], $$

where $\mathbb{Q}$ is the equivalent probability measure defined by $\mathbb{Q} = \mathcal{E}_T(\int_0^T Z_s^* \cdot dW_s)\mathbb{P}$. Here the term $Y^*$ can be understood as an actualization rate, which might be negative. In this framework, the problem (P) becomes

$$ \sup_{q \in \mathbb{Q}} \inf_{\psi \in \mathcal{A}} J(q, \psi), \quad J(q, \psi) := R(q, \psi) - \mathcal{S}(q), \qquad (\mathrm{P}_L) $$

where

$$ R(q, \psi) := \mathbb{E}^{\mathbb{Q}} \left[ e^{\int_0^T Y_s^* ds} g(X_T^\psi) + \int_0^T e^{\int_0^s Y_u^* du} \ell(s, \psi_s) ds \right]. $$

When $\psi \in \mathcal{A}$ is fixed, the penalty $\mathcal{S}(q)$ prevents Nature from choosing a singular measure (relative to the historical probability $\mathbb{P}$) that would only assign weight to the worst outcome for the central planner. In fact, the problem solved by Nature coincides with the risk-aversion problem presented in [86, Chapter 6.4], with the key difference being that the variable $Z^*$ is bounded in [86], which greatly simplifies the analysis. In particular, [86] provides a representation of the value of the problem (corresponding here to the problem solved by Nature) in the form of a Backward Stochastic Differential Equation (BSDE) driven by coefficients with at most linear growth. In our framework, this BSDE may become quadratic, as explained in the next paragraph.

To further fix the ideas about the ‘linear’ problem $(\mathrm{P}_L)$, assume that $f^*(t, y^*, z^*) = \frac{1}{2}|z^*|^2$, for all $(t, y^*, z^*) \in [0, T] \times \mathbb{R} \times \mathbb{R}^d$, and $Y^* \equiv 0$. Because the actualization
rate $Y^*$ is null, $q_T$ is the Radon-Nikodym derivative of $\mathbb{Q}$ with respect to $\mathbb{P}$, that is $d\mathbb{Q} = q_T d\mathbb{P}$. Due to the specific form of $f^*$, the penalty $S(q)$ is equal to $H(\mathbb{Q}|\mathbb{P})$ where $H(\mathbb{Q}|\mathbb{P}) := \mathbb{E}^{\mathbb{Q}}[\ln(d\mathbb{Q}/d\mathbb{P})]$ denotes the relative entropy of $\mathbb{Q}$ with respect to $\mathbb{P}$. Then, the cost simplifies to

$$J(q, \psi) = \mathbb{E}^{\mathbb{Q}} \left[ g(X_T^{\psi}) + \int_0^T \ell(s, \psi_s) ds \right] - H(\mathbb{Q}|\mathbb{P}). \quad (8)$$

Nature's problem then coincides with a maximization problem that frequently appears in large deviation theory. Indeed, the Donsker-Varadhan variational formula provides an interpretation of Nature's optimal value as the log-Laplace transform of a cost function defined on the Wiener space, as seen in works like [29, 55]. This problem has a long history in economic and finance literature [39, 63], and can be found under different names (ambiguity, robust or risk sensitive control problem, depending on the interpretation) We also refer to the recent contribution [20] for a systematic analysis of entropy-penalized stochastic optimal control problems.

When considering an optimizer $q \in \mathbb{Q}$ for Nature's problem in $(\mathbb{P}_L)$, the remaining central planner minimization problem over $\psi$ can be reformulated as a control problem over BSDEs:

$$\inf_{\psi \in \mathcal{A}} \mathbb{E} [Y_0^{\psi}],$$

where $(Y, Z)$ is the solution to,

$$-dY_t = (f(t, Y_t, Z_t) + \ell(t, \psi_t)) dt - Z_t \cdot dW_t, \quad Y_T = g(X_T^{\psi}), \quad (9)$$

and $f$ is the Fenchel transform of $f^*$ (see (2)). This connection is presented in [86, Chapter 6.4] in the particular case of linear growth drivers $f$. When $f$ is quadratic in the variable $z$, as considered throughout the remainder of the article, solving the BSDE in equation (9) becomes more challenging. The study of quadratic BSDEs began with the seminal work of [70] on equations driven by bounded terminal conditions. For a comprehensive presentation of the standard theory, see [99], which includes additional references. Subsequent research has extended the results on existence and uniqueness to unbounded terminal conditions, under the assumption of finite exponential order moments [24, 25, 50]. We will return to these references in the core of the article, as our analysis is typically conducted in the context where the terminal value $g$ is unbounded.

The BSDE in equation (9) can be interpreted as a nonlinear conditional expectation, specifically a $g$-expectation [85]. When the criterion $J$ is given by (8), that is, when $f(s, y, z) = \frac{1}{2}|z|^2$, the first component $Y$ of the BSDE is known in the literature as the entropic risk measure of the cost $g(X_T^\psi) + \int_0^T \ell(\psi_t)dt$. Entropic risk measures have been extensively studied in the $L^\infty$ case, i.e., for bounded costs, see [7].

**First contribution: From risk neutral to robust mean field control.** The main objective of our paper is twofold: first, from a technical perspective, to relax the growth conditions of the various cost functionals in the problem $(\mathbb{P}_L)$; and second, from a modeling perspective, to consider a mean field version, whose general form is given in (P). In this regard, the problem (P) encompasses not only mean field control problems with risk aversion but, more generally, problems in which the central planner is subject to uncertainty, here perceived as an adverse action of Nature. A series of examples are provided in Subsections 3.2 and 4.2 to illustrate these concepts.
In the risk-neutral case, stochastic mean field control problems are typically introduced as the limiting behavior of optimal control problems defined over large interacting particle systems. In these settings, a central planner seeks to optimize an objective function that depends on the collective dynamics of the particles. This class of problems has attracted significant attention in recent years [4, 18, 19, 27, 45, 46, 53, 73, 76]. For a comprehensive introduction to the subject, we refer to [11, 35, 36].

In this article, we establish the stochastic maximum principle for the problem (P). The stochastic maximum principle is a powerful tool for solving stochastic control problems, first introduced by [72] and further developed by [14], [65], [83], and [97]. It plays a central role in the stochastic mean field control and mean field game literature [28, 35, 36]. The standard theory of the stochastic maximum principle applies to risk-neutral control problems and is typically formulated in an $L^2$ framework, where both the state variables and the adjoint processes are assumed to belong to $L^2$. To establish the stochastic maximum principle, three key steps are typically followed: first, proving the existence of a solution to the control problem [64]; second, deriving the necessary conditions [84]; and third, demonstrating the sufficient conditions, which can be shown using a simple verification argument.

Here, we move beyond the scope of the standard theory for two main reasons, which align with the two primary objectives of our work. The first is to address a mean field problem with a risk-averse min-max structure. Extensions of the stochastic maximum principle to risk-averse problems have been studied in the context of optimal control of Forward-Backward Stochastic Differential Equations (FBSDEs). For example, see [82] for cases with linear growth drivers and jumps. The second objective is to allow the terminal condition $g$ to be unbounded. While bounded terminal conditions enable the use of the BMO theory for quadratic BSDEs [66], such assumptions are too restrictive for some applications. Moreover, they are rather incompatible with the convexity constraints typically required in the sufficient condition of the maximum principle. One natural approach to obtain stronger exponential integrability properties on $\psi$, compatible with those required in the theory of quadratic BSDEs, would be to follow the methodology of [41, 42] and work within an Orlicz space framework. Indeed, Orlicz spaces generalize $L^p$ spaces and, in particular, include random variables with finite exponential moments of arbitrary order, together with their dual space, which consists of random variables with finite entropy H. Such a dual space would be a natural candidate for carrying the variable $q$. That said, adopting this approach in our setting would require working with a quadratic driver of the form $f(t, y, z) = \frac{1}{\gamma}|z|^2$. For $\gamma$ large enough, this would provide the level of exponential integrability needed to apply the theory of quadratic BSDEs. However, for small values of $\gamma$, to the best of our knowledge, the stochastic maximum principle is not available even in this simpler setting. In contrast, our analysis goes one step further: the driver $f$ is only assumed to have at most quadratic growth, and may in fact exhibit subquadratic growth. Our strategy is to extend the duality inherent to Orlicz spaces of random variables to a setting involving dual spaces of stochastic processes. This perspective motivates the introduction of the mappings $S$ and $S^*$, which define the admissible sets $Q$ and $A$.

The first major contribution of this article is the proof of the stochastic maximum principle for the problem (P). Under appropriate concavity-convexity conditions, we show that this problem has a unique solution, where the minimizer is fully charac-
terized by the solution of a FBSDE. To establish this result, we begin by considering a constrained version of (P), for which we identify a topological structure ensuring semi-continuity, convexity/concavity, and compactness of the criterion $\mathcal{J}$ in each variable. This preliminary analysis allows us to apply Sion's min-max theorem and to deduce the existence of a saddle point for the constrained problem. We then relax the constraints by showing that there exists a level at which they are in fact non-binding, which in turn yields the existence of a saddle point for the original problem (P). The necessary and sufficient optimality conditions are obtained by coupling the first-order conditions associated with Nature's problem and the central planner's problem. It is worth emphasizing that the necessary conditions provided by the stochastic maximum principle require solving FBSDEs that go beyond the scope of the standard theory. The sufficient conditions ensure the uniqueness of these solutions. In the course of the analysis, we revisit the connection between entropy-type optimization problems and quadratic BSDEs with unbounded terminal conditions, a connection previously established for linear functionals $\mathcal{G}$ in [50].

**Second contribution: Robust mean field control and variational mean field games.** Mean field control (MFC) problems constitute a class of stochastic optimal control problems in which both the system dynamics and the associated cost functional may depend on the distribution of the controlled state process. Such problems naturally arise in the modeling of large populations of weakly interacting particles, where the influence of each individual is mediated through the empirical distribution of the population. Typical applications can be found in economics, statistical physics, and mathematical finance. In recent years, these problems have attracted significant attention; see, for instance, [28, 30, 34, 53, 73, 87], among many others. When treated from a probabilistic perspective, they are often addressed via the stochastic maximum principle.

In this article, we introduce a robust version of this problem, where the measure encoding the mean field interaction is biased by Nature. For a real-valued function $\mathcal{G}$, defined on the space of non-negative measures on $\mathbb{R}^n$, we thus consider the min-max problem

$$ \inf_{\psi \in A} \sup_{q \in \mathcal{Q}} \left\{ G(\mathbb{Q}_X^q) + \mathbb{E} \left[ \int_0^T q_s \ell(s, \psi_s) ds \right] - \mathcal{S}(q) \right\}, \quad (\text{MFC}) $$

where $(X_t^\psi)_{t \in [0,T]}$ is the solution to the controlled stochastic differential equation (7) and

$$ \mathbb{Q}_X^q := \mathbb{Q} \circ X^{-1}, \quad \mathbb{Q} = \exp \left( \int_0^T Y_s^* ds \right) \mathcal{E}_T \left( \int_0^T Z_s^* \cdot dW_s \right) \mathbb{P}. $$

This problem is a specification of the problem (P) when $\mathcal{G}(q, X) = G(\mathbb{Q}_X^q)$. Building upon the stochastic maximum principle established for (P), we derive the stochastic maximum principle for the problem (MFC) under the assumption that the mapping $\mathcal{G}$ is Lions differentiable, Lions convex and flat concave.

In addition, we also study a variational mean field game (MFG) problem. In contrast to MFC problems, which are cooperative in essence, MFGs are competitive problems. They are defined over a continuum of players whose interactions arise through a mean field functional. The theory of MFGs was introduced independently in [67] and [74, 75], and has since been extensively developed; see, for instance, [12, 16, 31, 32, 35, 36, 37]. MFGs have found numerous applications in economics
and finance [1, 33, 57, 80], environmental studies [69, 77], and electricity markets [3], to name just a few. We also refer to [35, 36] for a comprehensive monograph. The classical theory of MFGs typically considers risk-neutral agents. A natural extension is therefore to investigate models with risk-averse agents. Several approaches have been proposed in this direction, each relying on different ways of incorporating risk aversion into the representative agent's cost functional. Risk-sensitive MFGs [79, 95] introduce criteria depending on the variance of the state, while risk-averse MFGs [40, 56, 59] incorporate risk measures directly into the cost functional. MFGs in which agents optimize a worst-case criterion, using $H^\infty$ control techniques, were introduced in [8]. The theory of MFGs is closely related to that of MFC, particularly through variational (or potential) MFGs, which form a special class of MFGs. In brief, a variational MFG can be formulated as the first-order optimality conditions of a stochastic MFC problem [9, 10, 17, 26, 61, 62]. In particular, any solution to the MFC problem yields an equilibrium of the associated variational game. Moreover, when the MFC problem is strictly convex and coercive, the corresponding variational MFG admits a unique solution.

In this article, we study the following MFG problem. Given a non-negative measure $\mu$ on $\mathbb{R}^n$, representing the mean field coupling, a representative agent (in the continuum) minimizes a risk-averse objective functional

$$ \inf_{\psi \in \mathcal{A}} \sup_{q \in \mathcal{Q}} \mathcal{J}[\mu](q, \psi) := \mathbb{E} \left[ q_T \frac{\delta G}{\delta \mu}(X_T^\psi, \mu_T) + \int_0^T q_s \ell(s, \psi_s) ds \right] - \mathcal{S}(q), \quad (P_\mu) $$

where the controlled state process $(X_t^\psi)_{t \in [0, T]}$ satisfies the dynamics given in (7), and $\delta G/\delta \mu$ is the so-called flat derivative of $G$, see Section 4 for a reminder. For a saddle point $(\psi, q) \in \mathcal{A} \times \mathcal{Q}$, the mean field equilibrium condition is defined as follows: the measure $\mu$ is required to coincide with the law of the terminal state $X_T^\psi$ under the probability measure $\mathbb{Q}^q$ induced by Nature, that is,

$$ \mu = \mathbb{Q}_{X^\psi}^q := \mathbb{Q}^q \circ (X_T^\psi)^{-1}, \quad \mathbb{Q}^q = \mathcal{E} \left( \int_0^T Z_s^* \cdot dW_s \right). \quad (\text{MFG-eq}) $$

In other words, the MFG problem consists in finding a triple $(q, \psi, \mu)$, with $(q, \psi) \in \mathcal{Q} \times \mathcal{A}$ and $\mu$ being a non-negative measure, such that

$$ \mathcal{J}[\mu](q, \psi) = \inf_{\psi' \in \mathcal{A}} \sup_{q' \in \mathcal{Q}} \mathcal{J}[\mu](q', \psi'), \quad \mu = \mathbb{Q}^q \circ (X_T^\psi)^{-1}. \quad (\text{MFG}) $$

The optimization problem faced by the representative player can be interpreted as a risk-averse (non mean field) control problem. To find the optimal strategy, the representative agent solves a risk-averse stochastic control problem that falls within the scope of (non-mean field) control problems addressed in this work.

Under the same regularity and concavity-convexity assumptions on $G$ as those used in the analysis of the robust MFC (MFC), we establish the existence and uniqueness of an equilibrium, which ultimately coincides with the solution of (MFC). As such, this article is the first to identify a variational structure for risk-averse MFGs. The analysis of such robust MFGs is pursued further in our companion work [49], where we go beyond the variational setting.

**Organization of the article.** The article is organized as follows. In Section 2, we introduce the main notations and definitions used throughout the paper. Section
3 contains our main result, Theorem 10, which establishes a stochastic maximum principle for the problem (P), together with first examples of applications. Section 4 is devoted to the mean field setting. There, we establish the existence and uniqueness of solutions to a class of robust mean field control problems in Corollary 15, and we then consider a related class of robust variational mean field games, proving existence and uniqueness of equilibria in Corollary 16. Additional examples are provided in Subsection 4.2. Finally, Section 5 is dedicated to the proof of Theorem 10.

## 2 Notations

In this section, we introduce the main notations used in the article. Throughout, we work on the same filtered complete probability space $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ as in the definition of the problem (P).

**Spaces of random variables and random processes.** We begin by introducing the spaces of variables and stochastic processes on which our analysis relies. Unless otherwise stated, all notations are understood to be with respect to the probability measure $\mathbb{P}$. When a different measure, say $\mathbb{Q}$, is used, this will be made explicit. For example, in the context of the first example below, we will write $L^p(\dots, \mathbb{Q})$ to indicate the underlying measure. Moreover, for each of the spaces defined below, we will often omit the notation $\mathbb{R}^k$ when $k=1$.

*Usual random variable spaces.* For a given $k \in \mathbb{N}^*$ and for each $t \in [0, T]$, we denote by $L^0(\mathcal{F}_t, \mathbb{R}^k)$ the set of $\mathbb{R}^k$ valued and $\mathcal{F}_t$-measurable random variables (r.v.'s in short). And then, we define the sets

* $L^p(\mathcal{F}_t, \mathbb{R}^k)$ of r.v.'s $X \in L^0(\mathcal{F}_t, \mathbb{R}^d)$ s.t. $\|X\|_{L^p(\mathcal{F}_t, \mathbb{R}^k)} := \mathbb{E}[|X|^p] < +\infty$, for $p < +\infty$,

* $L^\infty(\mathcal{F}_t, \mathbb{R}^k)$ of r.v.'s $X \in L^0(\mathcal{F}_t, \mathbb{R}^k)$ s.t. $\|X\|_{L^\infty(\mathcal{F}_t, \mathbb{R}^k)} := \ess\sup_{\omega \in \Omega} \sup_{i \in \{1, \dots, d\}} |X^i(\omega)| < +\infty$.

*Usual random process spaces.* We denote by $L^0(\mathbb{F}, \mathbb{R}^k)$ the space of $\mathbb{F}$-progressively measurable random processes (r.p.'s in short) with values in $\mathbb{R}^k$, and by $S^0(\mathbb{F}, \mathbb{R}^k)$ the subset of $L^0(\mathbb{F}, \mathbb{R}^k)$ comprising processes with continuous trajectories. We define the sets

* $L^p(\mathbb{F}, \mathbb{R}^k)$ of r.p.'s $X \in L^0(\mathbb{F}, \mathbb{R}^k)$ s.t. $\|X\|_{L^p(\mathbb{F}, \mathbb{R}^k)} := \mathbb{E}\left[\left(\int_0^T |X_t|^p dt\right)^{1/p}\right] < +\infty$, for $p < +\infty$,

* $M^p(\mathbb{F}, \mathbb{R}^k)$ of r.p.'s $X \in L^0(\mathbb{F}, \mathbb{R}^k)$ s.t. $\|X\|_{M^p(\mathbb{F}, \mathbb{R}^k)} := \mathbb{E}\left[\left(\int_0^T |X_t|^2 dt\right)^{p/2}\right] < +\infty$,

* $L^\infty(\mathbb{F}, \mathbb{R}^d)$ of r.p.'s $X \in L^0(\mathbb{F}, \mathbb{R}^k)$ s.t. $\|X\|_{L^\infty(\mathbb{F}, \mathbb{R}^k)} := \sup_{t \in [0, T]} \|X_t\|_{L^\infty(\mathcal{F}_t, \mathbb{R}^k)} < +\infty$,

* $S^p(\mathbb{F}, \mathbb{R}^k)$ of r.p.'s $X \in S^0(\mathbb{F}, \mathbb{R}^k)$ s.t. $\|X\|_{S^p(\mathbb{F}, \mathbb{R}^k)} := \mathbb{E}\left[\sup_{t \in [0, T]} |X_t|^p\right] < +\infty$.
* $D(\mathbb{F}, \mathbb{R}^k)$ of r.p.'s $X \in S^0(\mathbb{F}, \mathbb{R}^k)$ such that the family $(|X_\tau|)_\tau$, with $\tau$ running over the set of $[0, T]$-valued $\mathbb{F}$-stopping times, is uniformly integrable.

The class $D(\mathbb{F}, \mathbb{R}^k)$, which is the least standard among the above classes, was introduced in [52, Definition 20].

Moreover, for a process $X = (X_t)_{t \in [0, T]}$, with continuous trajectories and with values in $\mathbb{R}^k$, we denote by $(X_t^* := \sup_{s \in [0, t]} |X_s|)_{t \in [0, T]}$ the running maximum of the norm of $X$.

Orlicz spaces. Following (3), we define the entropy function $h: \mathbb{R}_+ \to \mathbb{R}$:

$$h(x) := x(\ln(x) - 1), \qquad (10)$$

together with the two sets

* $L \log L(\mathcal{F}_T)$ of r.v.'s $X \in L^0(\mathcal{F}_T)$ s.t. $\mathbb{E}[h(X_T)] < +\infty$,

* $L \log L(\mathbb{F})$ of r.p.'s $X \in L^0(\mathcal{F}_T)$ s.t. $\sup_{t \in [0, T]} \mathbb{E}[h(X_t)] < +\infty$.

For any $X \in L^0(\mathcal{F}_T)$, with non-negative values, we call entropic risk measure of level $\vartheta > 0$ of $X$ the quantity

$$\rho_{\vartheta}[X] := \frac{1}{\vartheta} \ln \mathbb{E}[\exp(\vartheta X)], \qquad (11)$$

which makes it possible to define the sets

* $L_{\exp}^{p, \vartheta}(\mathcal{F}_t, \mathbb{R}^k)$ of r.v.'s $X \in L^0(\mathcal{F}_t, \mathbb{R}^d)$ s.t. $\rho_{\vartheta}[|X|^p] < +\infty$,

* $L_{\exp}^{p, \vartheta}(\mathbb{F}, \mathbb{R}^k)$ of r.p.'s $X \in L^0(\mathbb{F}, \mathbb{R}^k)$ s.t. $\rho_{\vartheta}[\int_0^T |X_s|^p ds] < +\infty$,

* $S_{\exp}^{p, \vartheta}(\mathbb{F}, \mathbb{R}^k)$ of r.p.'s $X \in L^0(\mathbb{F}, \mathbb{R}^k)$ s.t. $\rho_{\vartheta}[|X_T^*|^p] < +\infty$,

for $k \in \mathbb{N}^*$, $p > 0$ and $t \in [0, T]$. When $X \in L^0(\mathcal{F}_T)$ and $|X| \in L_{\exp}^{1, \vartheta}(\mathcal{F}_T)$, the right-hand side (11) still makes sense and we can define $\rho_{\vartheta}[X]$ accordingly. Moreover, for $p > 0$, we denote $L_{\exp}^p(\mathcal{F}_t, \mathbb{R}^k)$ the set of random variables $X \in L^0(\mathcal{F}_t, \mathbb{R}^k)$ such that $X \in L_{\exp}^{p, \vartheta}(\mathcal{F}_t, \mathbb{R}^k)$ for some $\vartheta > 0$. The sets $L_{\exp}^p(\mathbb{F}, \mathbb{R}^d)$ and $S_{\exp}^p(\mathbb{F}, \mathbb{R}^d)$ are defined in an analogous way.

**Spaces of measures.** For a metric space $(\mathcal{X}, d)$, we call $\mathcal{B}(\mathcal{X})$ its Borel $\sigma$-field, $\mathcal{P}(\mathcal{X})$ the set of probability measures on $\mathcal{X}$, and $\mathcal{M}(\mathcal{X})$ the set of finite non-negative measures on $\mathcal{X}$. Let $p \ge 1$ we define the sets

* $\mathcal{P}_p(\mathcal{X})$ of $\mu \in \mathcal{P}(\mathcal{X})$ s.t. $\int_{\mathcal{X}} d(x_0, x)^p d\mu(x) < +\infty$ for some $x_0 \in \mathcal{X}$,

* $\mathcal{M}_p(\mathcal{X})$ of $\mu \in \mathcal{M}(\mathcal{X})$ s.t. $\int_{\mathcal{X}} d(x_0, x)^p d\mu(x) < +\infty$ for some $x_0 \in \mathcal{X}$.

For any finite measure $\mathbb{Q}$ on $\Omega$ and measurable mapping $X: \Omega \to \mathcal{X}$, we denote $\mathbb{Q} \circ X^{-1}$, or $\mathbb{Q}_X$, the image measure of $\mathbb{Q}$ by $X$. When $f$ is a non-normalized non-negative measurable function on $\Omega$, we denote by $f\mathbb{P}$ the equivalent non-normalized measure $\mathbb{Q}: \mathcal{F} \ni A \mapsto \mathbb{Q}(A) := \int_A f d\mathbb{P}$. In particular, for $X$ as before, $(f\mathbb{P})_X$ stands for the image of $f\mathbb{P}$ by $X$.
Lastly, for $\mu^1, \mu^2 \in \mathcal{P}(\mathcal{X})$, we define the relative entropy by

$$H(\mu^1|\mu^2) := \int_{\mathcal{X}} \ln\left(\frac{d\mu^1}{d\mu^2}\right) d\mu^1,$$

if $\mu^1$ is absolutely continuous with respect to $\mu^2$, and we set $H(\mu^1|\mu^2) = +\infty$ otherwise. Additional material on the metric structures of $\mathcal{P}(\mathcal{X})$ and $\mathcal{M}(\mathcal{X})$ is introduced in Subsection 4.1.

**Duality.** We end this section with duality results.

*Fenchel transform.* The following duality is used repeatedly all along the article. By Fenchel duality, we have, for any $x^* \in \mathbb{R}$ and $x \in \mathbb{R}_+$ (recalling the definition of $h$ in (10)),

$$\exp(x^*) + h(x) \geq x^*x. \quad (12)$$

We often make use of the duality inequality (12), when reformulated in the form

$$\begin{aligned} x^*x &= (\vartheta x^*) \frac{x}{\vartheta} \leq h\left(\frac{x}{\vartheta}\right) + \exp(\vartheta x^*) \\ &= \frac{1}{\vartheta}h(x) - \ln(\vartheta)x + \exp(\vartheta x^*), \end{aligned} \quad (13)$$

for all $\vartheta > 0$ and any $x, x^* > 0$.

*Duality between S and S*. By definition of $S$ and $S^*$ in (2) and (6), we have for any F-progressively measurable processes $q$ and $\zeta$, valued in $\mathbb{R}$,

$$S(q) + S^*(\zeta) \geq \frac{1}{\gamma} \mathbb{E} \left[ \int_{0}^{T} q_s |\zeta_s|^2 ds \right], \quad (14)$$

where $S(q)$ and $S^*(\zeta)$ might take infinite values. This inequality is a direct consequence of the definition of $S^*$. Equality holds whenever

$$q \in \arg \max_{q' \in \mathcal{Q}} \left\{ \mathbb{E} \left[ \int_{0}^{T} q_s |\zeta_s|^2 ds \right] - \gamma S(q) \right\}.$$

*Dual Donsker-Varadhan variational formula.* Let $\mu \in \mathcal{P}(\mathbb{R}^n)$ and $\vartheta > 0$ be such that $\int_{\mathbb{R}^n} \exp(\alpha \cdot x) d\mu(x) < +\infty$ for all $|\alpha| \leq \vartheta$. If there is a finite constant $L > 0$ such that $|k(x)| \leq L(1 + |x|)$ for any $x \in \mathbb{R}^n$ then

$$- \ln \int_{\mathbb{R}^n} \exp(-k(x)) d\mu(x) = \inf_{m \in \mathcal{P}(\mathbb{R}^n): H(m|\mu) < \infty} \left\{ H(m|\mu) + \int_{\mathbb{R}^n} k(x) dm(x) \right\}. \quad (15)$$

This formula can be found in [29, Proposition 2.3]. It remains valid even when $\mu$ does not satisfy exponential integrability, provided that $k$ is bounded from above, with no assumption on its growth from below.

**Miscellaneous.** Throughout the article, we use a generic constant $C > 0$ that depends only on the data of the problem. The value of $C$ may change from line to line. As for the data of the problem themselves, they are introduced and specified in the assumptions section.

When $x$ and $y$ are vectors of finite dimension, $x \cdot y$ denotes the scalar product between $x$ and $y$.
## 3 Stochastic maximum principle

In this section, we establish the stochastic maximum principle for the (non-mean field) problem (P). The section is organized into two main subsections. The main result, Theorem 10, which presents the stochastic maximum principle for the problem (P), is stated in Subsection 3.1. Its proof relies on an application of Sion's min-max theorem (recalled below), together with the stochastic maximum principles for both Nature's problem and the central planner's problem. These two problems are treated independently in Section 5. Two application examples are discussed in Subsection 3.2.

**Theorem 1** (Sion [93]). Let $M$ be a compact convex subset of a linear topological space and $N$ a convex subset of a linear topological space. Let $v: M \times N \to \mathbb{R}$ be such that

1. $v(\cdot, y)$ is lower semi-continuous and convex on $M$ for each $y \in N$,

2. $v(x, \cdot)$ is upper semi-continuous and concave on $N$ for each $x \in M$.

Then we have

$$ \min_{x \in M} \sup_{y \in N} v(x, y) = \sup_{y \in N} \min_{x \in M} v(x, y) $$

and supremum is attained whenever $N$ is compact.

### 3.1 Main result

We first present the assumptions used throughout the paper, even though some intermediate results are stated under weaker conditions. Additional assumptions are introduced in Section 4 when discussing the mean field setting.

**Assumptions.** The assumptions are stated in terms of two constants, $L > 0$ and $r \in \{0, 1\}$. They also make use of the notion of progressively-measurable field: for a metric space $(\mathcal{X}, d)$ and an integer $k \in \mathbb{N}^*$, a random field $\mathcal{G}: \Omega \times [0, T] \times \mathcal{X} \to \mathbb{R}^k$ is said to be progressively-measurable if, for any $t \in [0, T]$, its restriction to $\Omega \times [0, t] \times \mathcal{X}$ is $\mathcal{F}_t \otimes \mathcal{B}([0, t]) \otimes \mathcal{B}(\mathcal{X})/\mathcal{B}(\mathbb{R}^k)$ measurable.

A1 *Initial condition and drift.* The initial condition $\eta$ in (7) belongs to $L^\infty(\mathcal{F}_0, \mathbb{R}^n)$, i.e.

$$ \|\eta\|_{L^\infty(\mathcal{F}_0, \mathbb{R}^n)} < +\infty. $$

The drift $b: \Omega \times [0, T] \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$ is linear and of separated form

$$ b(t, x, \psi) = a_t + b_t x + c_t \psi, $$

where $a$, $b$ and $c$ belong respectively to $L^\infty(\mathbb{F}, \mathbb{R}^n)$, $L^\infty(\mathbb{F}, \mathbb{R}^{n \times n})$ and $L^\infty(\mathbb{F}, \mathbb{R}^{n \times n})$, i.e. $\|a\|_{L^\infty(\mathbb{F}, \mathbb{R}^n)} + \|b\|_{L^\infty(\mathbb{F}, \mathbb{R}^{n \times n})} + \|c\|_{L^\infty(\mathbb{F}, \mathbb{R}^{n \times n})} < +\infty$. In particular, $b$ is a progressively-measurable random field.

A2 *Volatility.* The volatility $\sigma: \Omega \times [0, T] \times \mathbb{R}^n \to \mathbb{R}^{n \times d}$ is linear in the control variable. Precisely, the $n \times d$ entries of the matrix $\sigma$ are of the form

$$ (\sigma(t, \psi))_{i,j} = (\nu_t)_{i,j} + r(\sigma_t)_{i,j,k} \psi_k, $$
$(i, j, k) \in \{1, \dots, n\} \times \{1, \dots, d\} \times \{1, \dots, n\}$, with $\nu$ and $\sigma$ belonging respectively to $L^\infty(\mathbb{F}, \mathbb{R}^{n \times d})$ and $L^\infty(\mathbb{F}, \mathbb{R}^{n \times d})$ and satisfying $\|\nu\|_{L^\infty(\mathbb{F}, \mathbb{R}^{n \times d})} + \|\sigma\|_{L^\infty(\mathbb{F}, \mathbb{R}^{n \times d \times n})} \le L$.

A3 Driver. The driver $f: \Omega \times [0, T] \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}$ is progressively-measurable, convex and twice differentiable with respect to its last two variables, the corresponding derivatives of order 2 are bounded by $L$. Moreover, there exist two constants $\alpha, \beta \ge 0$ such that $t$,

$$f(t, y, z) \le |f_t^0| + \alpha|y| + \frac{\beta}{2}|z|^2, \quad (y, z) \in \mathbb{R} \times \mathbb{R}^d,$$

where $f^0 := f(0, 0) \in L^\infty(\mathbb{F})$.

A4 Running cost. The running cost $\ell: \Omega \times [0, T] \times \mathbb{R}^n \to \mathbb{R}$ is progressively-measurable and twice differentiable in the last variable. It satisfies

$$(\nabla_{\psi}\ell(t, \psi) - \nabla_{\psi}\ell(t, \psi')) \cdot (\psi - \psi') \ge \frac{1}{L}|\psi - \psi'|^2, \quad |D_{\psi}^2\ell(t, \psi)| \le L,$$

and $|\ell(t, 0)| \le L$ for any $t \in [0, T]$ and $\psi, \psi' \in \mathbb{R}^n$. In particular, $\ell$ is strongly convex in the last variable, with a quadratic growth, uniformly in the other variables.

A5 Coefficients. The coefficient $\gamma$ in (6) is chosen as

$$\gamma = 8\beta \max(1, L)e^{\alpha T} \|\Gamma\|_{L^\infty(\mathbb{F}, \mathbb{R}^{n \times n})} \|\Gamma^{-1}\|_{L^\infty(\mathbb{F}, \mathbb{R}^{n \times n})} \\ \times \left( \|\nu\|_{L^\infty(\mathbb{F}, \mathbb{R}^{n \times d})} + 12 \max(1, L)e^{\alpha T} \|\sigma\|_{L^\infty(\mathbb{F}, \mathbb{R}^{n \times d \times n})} \right),$$

where $\Gamma$ is the resolvent of the linear ODE driven by $b$, i.e. the solution to

$$\frac{d}{dt}\Gamma_t = b_t\Gamma_t, \quad t \in [0, T], \quad \Gamma_0 = I_n,$$

with $I_n$ standing for the $n \times n$ identity matrix.

Moreover, when $r = 0$, the following smallness condition is satisfied:

$$4\beta e^{\alpha T} L \|\Gamma\|_{L^\infty(\mathbb{F}, \mathbb{R}^{n \times n})}^2 \|\Gamma^{-1}\|_{L^\infty(\mathbb{F}, \mathbb{R}^{n \times n})}^2 \|\nu\|_{L^\infty(\mathbb{F}, \mathbb{R}^{n \times d})}^2 T < 1,$$

The choice of $\gamma$ is discussed in Remark 5 below, and the smallness condition in Remark 7.

A6 Growth of the mapping $\mathcal{G}$ and its derivatives. Denoting by $\mathcal{G}$ the set of pairs $(q, X)$ of $\mathcal{F}_T$-measurable random variables with values in $\mathbb{R}_+ \times \mathbb{R}^n$ such that $\mathbb{E}[q|X|^{2-r}] < +\infty$, the cost $\mathcal{G}$ is a real-valued function on $\mathcal{G}$. Together with some mappings $\delta_q \mathcal{G}: \mathcal{G} \to L^0(\mathcal{F}, \mathbb{R})$ and $\delta_X \mathcal{G}: \mathcal{G} \to L^0(\mathcal{F}, \mathbb{R}^d)$, which are interpreted below as derivatives of $\mathcal{G}$ in the directions $q$ and $X$ respectively, it satisfies the growth properties:

$$\begin{aligned} |\mathcal{G}(q, X)| &\le L (1 + \mathbb{E}[(1+q)|X|^{2-r}]), \\ -L(1 + |X| + \mathbb{E}[q|X|^{2-r}]) &\le \delta_q \mathcal{G}(q, X) \le L (1 + |X|^{2-r} + \mathbb{E}[q|X|^{2-r}]), \\ |\delta_X \mathcal{G}(q, X)| &\le Lq (1 + |X|^{1-r} + \mathbb{E}[q|X|^{2-r}]). \end{aligned}$$
A7 First order Taylor expansion of the mapping $\mathcal{G}$. With $\mathcal{G}$, $\delta_q\mathcal{G}$ and $\delta_X\mathcal{G}$ as in the previous condition, the mapping $\mathcal{G}$ admits the following two first order expansions in $X$ and $q$ respectively:

$$ \begin{aligned} \mathcal{G}(q, X') &= \mathcal{G}(q, X) + \mathbb{E} [\delta_X \mathcal{G}(q, X) \cdot (X' - X)] + O(\mathbb{E} [q | X' - X|^2]) \\ \mathcal{G}(q', X) &= \mathcal{G}(q, X) + \mathbb{E} [\delta_q \mathcal{G}(q, X)(q' - q)] + o(\mathbb{E} [(1 + |X|^{2-r}) |q' - q|]), \end{aligned} $$

where $|O(r)| \le cr$ for a constant $c$ that only depends on $(q, X', X)$ via (any bound for) $\mathbb{E}[q|X|^{2-r}]$ and $\mathbb{E}[q|X'|^{2-r}]$, and where $|o(r)| \le \eta(r)r$ for a function $\eta$ that tends 0 with $r$ and that only depends on $(q, q', X)$ via (any bound for) $\mathbb{E}[q|X|^{2-r}]$ and $\mathbb{E}[q|X'|^{2-r}]$.

A8 Concavity-convexity of the mapping $\mathcal{G}$. The mapping $\mathcal{G}$ is concave with respect to the variable $q$ and convex with respect to the variable $X$, i.e., for any $(q, X)$, $(q^1, X^1)$ and $(q^2, X^2)$ in $\mathcal{G}$, with $\mathcal{G}$ as in (A6), and for any $\theta \in [0, 1]$.

$$ \begin{aligned} \mathcal{G}(\theta q^1 + (1 - \theta)q^2, X) &\ge \theta \mathcal{G}(q^1, X) + (1 - \theta) \mathcal{G}(q^2, X), \\ \mathcal{G}(q, \theta X^1 + (1 - \theta)X^2) &\le \theta \mathcal{G}(q, X^1) + (1 - \theta) \mathcal{G}(q, X^2). \end{aligned} $$

**Comments and examples.** We provide several comments and examples to clarify the assumptions.

**Remark 2.** Lower bound on the dual driver. In (2), $f^*: \Omega \times [0, T] \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}$ is the Fenchel transform of the driver $f$ with respect to its variables $(y, z)$, i.e.,

$$ f^*(t, y^*, z^*) := \sup_{(y,z) \in \mathbb{R} \times \mathbb{R}^d} \{\langle (y^*, z^*), (y, z) \rangle - f(t, y, z)\}. $$

Since $f$ is continuous in $(y, z)$, the supremum in the definition of $f^*$ can be reduced to a supremum over a countable set. We easily deduce that $f^*$ is progressively-measurable.

Moreover, because $f$ is twice differentiable in $(y, z)$ with bounded second-order derivatives, see A3, $f^*$ is $c$-strongly convex with respect to its last two variables, for a constant $c > 0$, see for instance [49] for an explicit proof.

Assumption A3 also implies that, for any $(t, y^*, z^*) \in [0, T] \times \mathbb{R} \times \mathbb{R}^d$,

$$ f^*(t, y^*, z^*) \ge -|f_t^0| + \chi_B(y^*/\alpha) + \frac{1}{2\beta}|z^*|^2, \quad (16) $$

where $\chi_B$ denotes the indicator function of the unit ball $B := \{x \in \mathbb{R}, |x| \le 1\}$, i.e.,

$$ \chi_B(y^*) = \begin{cases} 0, & \text{if } y^* \in B, \\ +\infty, & \text{otherwise.} \end{cases} $$

Indeed by the growth condition in A3, we have, for any $t \in [0, T]$, $(y^*, z^*) \in \mathbb{R} \times \mathbb{R}^d$ and $(y, z) \in \mathbb{R} \times \mathbb{R}^d$,

$$ \begin{aligned} f^*(t, y^*, z^*) &\ge \langle (y^*, z^*), (y, z) \rangle - f(t, y, z) \\ &\ge \langle (y^*, z^*), (y, z) \rangle - |f_t^0| - \alpha|y| - \frac{\beta}{2}|z|^2. \end{aligned} $$

Taking, on both sides, the supremum with respect to $(y, z) \in \mathbb{R} \times \mathbb{R}^d$ and recalling that the absolute value $|\cdot|$ and the indicator function $\chi_B$ are in duality, we get (16).
**Example 3.** Mean field structure of $\mathcal{G}$. As we already mentioned, the problem addressed in this section is not of mean field type. It is only in Section 4 that we clarify our application to the mean field case, by considering cost functions $\mathcal{G}$ of the form

$$ \mathcal{G}(q, X) = G((q\mathbb{P})_X), $$

where $(q\mathbb{P})_X$ denotes the law of $X$ under $q\mathbb{P}$, assuming that $q$ is a non-negative random variable, and $G$ is a cost function defined on the space of non-negative measures.

**Remark 4.** Linearity of the state equation. The linearity of the state equation, as guaranteed by Assumptions A1 and A2, ensures the concavity of the mapping $Q \ni q \mapsto \mathcal{J}(q, \psi)$ and the convexity of the mapping $\mathcal{A} \ni \psi \mapsto \mathcal{J}(q, \psi)$, which are proved in Proposition 20 and Lemma 22, respectively. Additionally, the assumption that the volatility is independent of the state variable is crucial for guaranteeing the existence of a finite exponential moment of $L|X_T^{0,*}|$, where $X^0$ denotes the solution to the state equation when $\psi \equiv 0$. Such a property would generally fail if the volatility depended linearly on the state variable. See also Remark 7 for further comments on the exponential integrability of $X_T^{0,*}$.

**Remark 5.** On the constant $\gamma$. The choice of $\gamma$, as specified in A5, stems from Lemma 32. Roughly speaking, the latter provides an a priori bound on the component $\psi$ of any saddle point $(q, \psi)$ of (P). This bound is formulated in terms of a bound on $S^*(\psi)$ and therefore requires an appropriate choice of $\gamma$.

**Remark 6.** On the constant $r$. The volatility is controlled when $r = 1$ and uncontrolled when $r = 0$. As suggested in the previous remark, the state variable $X$ has finite exponential moments of sufficiently small order when the volatility is controlled ($r = 1$). When the volatility is uncontrolled ($r = 0$), stronger results can be established, showing that $X$ has finite quadratic exponential moments of small order, as detailed in Lemma 42. The fact that the integrability properties are stronger when $r = 0$ explains why the growth assumption A6 is more general in this case.

**Remark 7.** On the smallness condition on $\|\nu\|_{L^\infty(\mathbb{F}, \mathbb{R}^{n \times d})}^2$. Part of our analysis relies on an a priori bound for the component $q$ of an arbitrary saddle point $(q, \psi)$ of (P). This bound is established in Lemma 26. To make the proof work, we require the existence of some $\psi \in \mathcal{A}$ (in fact, for simplicity, we choose $\psi \equiv 0$) such that $L|X_T^{\psi,*}|^{2-r}$ admits an exponential moment of sufficiently large order $\nu$ (with $\nu$ depending explicitly on the other parameters in the assumptions). When $r = 1$, the random variable $|X_T^{0,*}|^{2-r} = |X_T^{0,*}|$ admits exponential moments of all orders. When $r = 0$, the random variable $L|X_T^{0,*}|^{2-r} = L|X_T^{0,*}|^2$ admits an exponential moment of order $\nu$ provided the smallness condition stated in A5 is in force (see Lemma 42).

It must be stressed that we require stronger integrability properties of $|X_T^{0,*}|$ in the case $r = 0$ than in the case $r = 1$, due to the growth conditions imposed on $\mathcal{G}$. If we were to work with the same growth conditions as in the case $r = 1$, the smallness condition would no longer be needed.

Finally, we note that the smallness condition imposed here is reminiscent of the integrability assumptions appearing in the analysis of quadratic BSDEs with unbounded terminal data; see, for instance, [25, 50]. This is not surprising, since the characterization of the saddle points of (P) resulting from our analysis (see Theorem 10) relies on a forward-backward SDE that may be quadratic (if $f$ is). In this respect, it is worth emphasizing that our smallness condition is not imposed at the
level of the saddle point itself, but rather at the level of a single controlled trajectory. As such, it is more explicit and easier to verify.

**Remark 8.** On the running cost $\ell$. In our analysis, the running cost $\ell$ is assumed to be independent of the state variable. This assumption may be restrictive for certain applications. However, our approach also allows one to consider a running cost $\ell'$: $[0, T] \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that depends on the state variable $x$ and is of separated form. More precisely, there exists a function $c'$: $[0, T] \times \mathbb{R}^n \to \mathbb{R}$ such that

$$ \ell'(t, x, \psi) = c'(t, x) + \ell(t, \psi). $$

One has to assume that $c$ is convex with respect to its second variable, and satisfies the growth condition

$$ |c'(t, x)| \le L(1 + |x|^{2-r}). $$

The latter implies that

$$ \mathbb{E} \left[ \int_{0}^{T} q_s \ell'(s, X_s^0, 0) ds \right] \le L \left( 1 + \mathbb{E} \left[ \int_{0}^{T} (1 + q_s) |X_s^0|^{2-r} ds \right] \right), $$

which is, in particular, enough to reproduce the proofs of the two key Lemmas 26 and 32 (up to an adaptation of the two conditions in A5).

Inequality (16) has an important consequence, which we formalize in the following statement:

**Lemma 9.** Let $q = (q_t)_{t \in [0,T]}$ be an $\mathbb{F}$-progressively measurable positive-valued continuous process such that $S(q) < +\infty$. Then,

$$ \mathbb{P} \otimes \text{Leb}_{[0,T]} (\{(\omega, t) \in \Omega \times [0, T], |Y_t^*| > \alpha\}) = 0. \quad (17) $$

Moreover,

$$ \mathbb{P} \left( \left\{ \int_{0}^{T} |Z_{s}^{*}|^{2} ds < +\infty \right\} \right) = 1, \quad (18) $$

and $(\mathcal{E}_t(\int_0^T Z_s^* \cdot dW_s))_{t \in [0,T]}$ in (5) is a 'true' martingale.

While the first claim, (17), is quite obvious, the second one, (18), is more subtle. Indeed, we deduce from (16) that

$$ \mathbb{P} \left( \left\{ \int_{0}^{T} q_s |Z_s^*|^2 ds < +\infty \right\} \right) = 1. $$

And then, it is by continuity and strict positivity of $q$ that (18) follows. In particular, it must be observed that the stochastic integral in (5) is necessarily well-defined. It is then clear that (4) and (5) are equivalent: starting from (4), one obtains (5) by applying Itô's formula, while the converse implication also follows from Itô's formula, applied to the process $(\ln(q_t))_{t \in [0,T]}$, which is well defined since $q$ takes strictly positive values. The fact that $(\mathcal{E}_t(\int_0^T Z_s^* \cdot dW_s))_{t \in [0,T]}$ is a true martingale is a follows from Lemma 39, proved in Appendix A.
**Main result.** The main result of this section is presented in Theorem 10 below. We introduce the pre-Hamiltonian of the system $\mathcal{H}: \Omega \times [0, T] \times \mathbb{R}_+ \times \mathbb{R} \times \mathbb{R}^d \times \mathbb{R} \times \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^{n \times d} \to \mathbb{R}$,

$$ \mathcal{H}(t, q, y^*, z^*, y, z, x, \psi, p, k) := q(\mathrm{yy}^* + z \cdot z^* - f^*(t, y^*, z^*) + \ell(t, \psi)) \quad (19) \\ +p \cdot b(t, x, \psi) + \mathrm{Tr}(k\sigma^\top(t, \psi)). $$

Although the pre-Hamiltonian $\mathcal{H}$ explicitly appears in Isaac's condition discussed in Remark 11 below, for the purpose of our analysis, it is more convenient to split it into two parts, each corresponding to the pre-Hamiltonian used by either the central planner or Nature:

$$ \begin{aligned} F(t, q, y^*, z^*, y, z, \psi) &:= q(\mathrm{yy}^* + z \cdot z^* - f^*(t, y^*, z^*) + \ell(t, \psi)), \quad (20) \\ H(t, x, \psi, p, k, q) &:= q\ell(t, \psi) + p \cdot b(t, x, \psi) + \mathrm{Tr}(k\sigma^\top(t, \psi)). \end{aligned} $$

Given $q \in \mathcal{Q}$, we say that a tuple $(\psi, p, k, X)$ satisfies the first order condition (OptC) for the central planner problem if $(\psi, p, k, X)$ is a solution to

$$ \left\{ \begin{array}{ll} -\mathrm{d}p_t = \nabla_x H(t, X_t, \psi_t, p_t, k_t, q_t)\mathrm{d}t - k_t\mathrm{d}W_t, & p_T = \delta_X \mathcal{G}(q_T, X_T^\psi), \\ \mathrm{d}X_t = b(t, X_t, \psi_t)\mathrm{d}t + \sigma(t, \psi_t)\mathrm{d}W_t, & X_0 = \eta, \quad (\text{OptC}) \\ \psi_t \in \arg\min_\alpha H(t, X_t, \alpha, p_t, k_t, q_t), & \mathrm{d}\mathbb{P} \otimes \mathrm{d}t\text{-a.s.} \end{array} \right. $$

The first equation is interpreted as the adjoint equation for the central planner, the second equation as the state equation, and the last equation as the optimality condition. Because the last equation couples the two preceding equations, the system above is an FBSDE. The wordings 'first order condition' and 'optimality condition' are fully justified by the statement of Theorem 10 below.

Given $\psi \in \mathcal{A}$, we say that a tuple $(Y, Z, q)$ satisfies the first order condition (OptN) for Nature problem if $(Y, Z, q)$ is a solution to

$$ \left\{ \begin{array}{ll} -\mathrm{d}Y_t = \partial_q F(t, q_t, Y_t^*, Z_t^*, Y_t, Z_t, \psi_t)\mathrm{d}t - Z_t \cdot \mathrm{d}W_t, & Y_T = \delta_q \mathcal{G}(q_T, X_T^\psi), \\ \mathrm{d}q_t = q_t Y_t^* \mathrm{d}t + q_t Z_t^* \cdot \mathrm{d}W_t, & q_0 = 1, \\ (Y_t^*, Z_t^*) \in \arg\max_{(Y^{*'}, Z^{*'})} F(t, q_t, Y^{*'}, Z^{*'}, Y_t, Z_t, \psi_t), & \mathrm{d}\mathbb{P} \otimes \mathrm{d}t\text{-a.s.} \end{array} \right. \quad (\text{OptN}) $$

The first equation is interpreted as the adjoint equation for Nature, the second equation describes the dynamics of the control variable, and the last equation is the optimality condition. Similar to the previous one, this system of equations is also an FBSDE.

The two systems (OptC) and (OptN) above are presented in an abstract form. To clarify the result, we now give an explicit formulation using the concrete expressions of the coefficients. We start with the system (OptC). Computing the gradient of the Hamiltonian $\nabla_x H$ and the optimality condition, we have

$$ \left\{ \begin{array}{ll} -\mathrm{d}p_t = b_t^\top p_t \mathrm{d}t - k_t \mathrm{d}W_t, & p_T = \delta_X \mathcal{G}(q_T, X_T^\psi), \\ \mathrm{d}X_t = b(t, X_t, \psi_t) \mathrm{d}t + \sigma(t, \psi_t) \mathrm{d}W_t, & X_0 = \eta, \quad (21) \\ 0 = q_t \nabla_\psi \ell(t, \psi_t) + c_t^\top p_t + r \mathrm{Tr}(\sigma_t^\top k_t), & \mathrm{d}\mathbb{P} \otimes \mathrm{d}t\text{-a.s.} \end{array} \right. $$
where we denote, by convention,

$$ \mathrm{Tr} (\sigma_t^\top k_t) = \left( \sum_{i=1}^{n} \sum_{j=1}^{d} (\sigma_t)_{i,j,l} (k_t)_{i,j} \right)_{\ell=1,\dots,d} . \qquad (22) $$

We now turn to the system (OptN). The optimality condition is given by

$$ (Y_t^*, Z_t^*) = (\partial_y f(t, Y_t, Z_t), \partial_z f(t, Y_t, Z_t)). \qquad (23) $$

Computing the derivative of the Hamiltonian $\partial_q F$ and plugging the optimality condition into the backward equation, the latter equation becomes a (possibly quadratic) BSDE by Fenchel's duality

$$ \left\{ \begin{array}{l} -dY_t = (f(t, Y_t, Z_t) + \ell(t, \psi_t))dt - Z_t \cdot dW_t, \quad Y_T = \delta_q \mathcal{G}(q_T, X_T^\psi), \\ \\ dq_t = q_t Y_t^* dt + q_t Z_t^* \cdot dW_t, \quad q_0 = 1. \end{array} \right. \qquad (24) $$

For the purpose of analyzing these two systems, we define the following two spaces. The first is the space of solutions to the system (OptC) given $\psi$ within a certain sub-level set of $\mathcal{A}$ (which will be specified when necessary), and the second is the space of solutions to (OptN) given $q \in \mathcal{Q}$:

$$ \mathcal{A} := \mathcal{A} \times D(\mathbb{F}) \times \left( \bigcap_{\beta \in (0,1)} M^\beta(\mathbb{F}, \mathbb{R}^d) \right) \times S^{2-r}(\mathbb{F}, \mathbb{R}^n, \mathcal{Q}), \qquad (25) $$

$$ \mathcal{Q} := \left\{ (q, Y, Z) \in \mathcal{Q} \times D(\mathbb{F}, \mathcal{Q}) \times \left( \bigcap_{\beta \in (0,1)} M^\beta(\mathbb{F}, \mathbb{R}^d, \mathcal{Q}) \right) \right\}, \qquad (26) $$

where $\mathcal{Q}$ in the first line is the measure $q_T \mathbb{P}$. Here is now our main statement regarding the inf-sup mean field stochastic control problem (P).

**Theorem 10.** There exists a unique saddle point $(\bar{q}, \bar{\psi}) \in \mathcal{Q} \times \mathcal{A}$ to the problem (P), i.e.

$$ \min_{\psi \in \mathcal{A}} \max_{q \in \mathcal{Q}} \mathcal{J}(q, \psi) = \max_{q \in \mathcal{Q}} \min_{\psi \in \mathcal{A}} \mathcal{J}(q, \psi) = \mathcal{J}(\bar{q}, \bar{\psi}). $$

Moreover, if a pair $(\psi, q) \in \mathcal{A} \times \mathcal{Q}$ is a solution to the problem (P), then the tuples $(\psi, p, k, X)$, obtained by solving in $\mathcal{A}$ the two decoupled equations in (OptN), and $(q, Y, Z)$, obtained by solving in $\mathcal{Q}$ the two decoupled equations in (OptC), satisfy the optimality conditions in (OptN) and (OptC) respectively. Conversely, if $(\psi, p, k, X, q, Y, Z) \in \mathcal{A} \times \mathcal{Q}$ is a solution to (OptC)-(OptN), then the pair $(\psi, q) \in \mathcal{A} \times \mathcal{Q}$ is the solution to the problem (P).

We provide a sketch of the proof based on the results established in the core of the article. We believe this presentation will help the reader gain a global overview of the structure of the arguments.

The strategy relies on introducing two truncation parameters. For $c_1, c_2 > 0$, we define the two following sets:

$$ \mathcal{Q}_{c_1} := \{q \in \mathcal{Q}, S(q) \le c_1\}, \qquad (27) $$

$$ \mathcal{A}_{c_2} := \{ \psi \in L^2(\mathbb{F}, \mathbb{R}^n), S^*(\psi) \le c_2 \}. \qquad (28) $$

Accordingly, we define the following min-max problem, analogous to the problem (P), but with the above two sets as restricted admissible sets:

$$ \sup_{q \in \mathcal{Q}_{c_1}} \inf_{\psi \in \mathcal{A}_{c_2}} \mathcal{J}(q, \psi). \qquad (P') $$
Proof. Step 1: Existence of a saddle point to (P'). The problem (P') is studied in Subsection 5.1. Existence of a saddle point is established in Lemma 17.

Step 2: Interior solutions. By definition, any saddle point $(q, \psi)$ to (P') satisfies

$$ \mathcal{J}(q, 0) \geq \mathcal{J}(q, \psi) \geq \mathcal{J}(q^0, \psi), \qquad (29) $$

where $q^0 = (q_t^0 = 1)_{t \in [0,T]} \in \mathcal{Q}_{c_1}$ denotes the solution to $q_t = 1 + \int_0^t q_s Y_s^* ds + \int_0^t q_s Z_s^* ds$. Then, by the two forthcoming Lemmas 26 and 32, there exist two constants $c'_1, c'_2 > 0$ only depending on the data and independent of $c_1$ and $c_2$ such that (more precisely $c'_2$ depends on $c'_1$, which is only depending on the data) such that

$$ (q, \psi) \in (\mathcal{Q}_{c_1} \cap \mathcal{Q}_{c'_1}) \times (\mathcal{A}_{c_2} \cap \mathcal{A}_{c'_2}). $$

Now choosing $c_1$ and $c_2$ such that $c_1 > c'_1$ and $c_2 > c'_2$ yields that $(q, \psi) \in \mathcal{Q}_{c'_1} \times \mathcal{A}_{c'_2}$ and thus $(q, \psi)$ is an interior solution to the problem (P'), in the sense that $\mathcal{S}(q)$ and $\mathcal{S}^*(\psi)$ are respectively strictly less than $c_1$ and $c_2$.

Step 3: Nature's problem. Let $(q, \psi) \in \mathcal{Q}_{c_1} \times \mathcal{A}_{c_2}$ be a saddle point to (P'), for $c_1 > c'_1$. By the previous step, $q$ lies in the interior of $\mathcal{Q}_{c_1}$. Theorem 25 (whose statement and proof are the main objectives of Subsection 5.2.1 below) says that $q$ is a maximizer of the problem

$$ \sup_{q' \in \mathcal{Q}_{c_1}} \mathcal{J}(q', \psi), \qquad (30) $$

if and only if the triple $(q, Y, Z) \in \mathcal{Q}$ obtained by solving the decoupled FBSDE in (OptN) satisfies the optimality condition in (OptN). Theorem 25 also guarantees that the maximizer of the problem (30) is unique.

Step 4: Central planner's problem. Let $(q, \psi) \in \mathcal{Q}_{c_1} \times \mathcal{A}_{c_2}$ be a saddle point to (P'), for $c_2 > c'_2$. By Step 2, $\psi$ lies in the interior of $\mathcal{A}_{c_2}$. Then Theorem 31 (which is the main result of Subsection 5.3 below) establishes that $\psi \in \mathcal{A}_{c_2}$ is a minimizer of the problem

$$ \inf_{\psi \in \mathcal{A}_{c_2}} \mathcal{J}(q, \psi), \qquad (31) $$

if and only if the tuple $(\psi, p, k, X) \in \mathcal{A}$ obtained by solving the decoupled FBSDE in (OptC) satisfies the optimality condition in (OptC). Theorem 31 also guarantees that the minimizer of the problem (31) is unique.

Step 5: Conclusion. To conclude the proof, we show that the problems (P') and (P) have the same set of solutions if $c_1 > c'_1$ and $c_2 > c'_2$. We first show that any solution to (P') is solution to (P). To do so, we consider the tuple $(q, Y, Z, \psi, p, k, X) \in \mathcal{Q} \times \mathcal{A}$, solution to the coupled system of FBSDEs (OptC)-(OptN) (which solution is given by the previous two steps). By the sufficiency property of the two first order conditions (OptC) and (OptN) (see again the previous two steps), $(q, Y, Z, \psi, p, k, X)$ satisfies the following two properties:

* for any $c''_1 > c'_1$, $q$ is a maximizer of (30), with $c_1$ being replaced by $c''_1$ therein, and thus $\mathcal{J}(q, \psi) \geq \mathcal{J}(q', \psi)$ for any $q' \in \mathcal{Q}$;

* for any $c''_2 > c'_2$, $\psi$ is a minimizer of (31), with $c_2$ being replaced by $c''_2$ therein, and thus $\mathcal{J}(q, \psi) \leq \mathcal{J}(q, \psi')$ for any $\psi' \in \mathcal{A}$.
Therefore, $(q, \psi)$ is also a solution to the problem (P), which proves in particular that the problem (P) admits at least a solution.

We now show that any solution to (P) is also a solution to (P'), when $c_1 > c'_1$ and $c_2 > c'_2$. Any solution $(q, \psi) \in Q \times A$ to (P) necessarily belongs to $Q_{c'_1} \times A_{c'_2}$ for some $c''_1 > 0$ and $c''_2 > 0$ (since $Q = \cup_{c''_1>0} Q_{c'_1}$ and $A^{\vartheta} = \cup_{c''_2>0} A_{c'_2}^{\vartheta}$). This implies that $(q, \psi)$ also lies in $Q_{c'_1} \times A_{c'_2}$ by the same argument as in Step 2. Then, by repeating the arguments of Step 3 and 4, we deduce that $(q, \psi)$ is a solution to (P'), concluding the proof.

Uniqueness follows readily. Suppose that there exist two distinct saddle points $(q, \psi)$ and $(q', \psi')$ in $Q \times A$, and hence in $Q_{c'_1} \times A_{c'_2}$ by the analysis above. Then at least one of the following holds: $q' \neq q$ or $\psi' \neq \psi$. If $q' \neq q$, we use the fact that the optimization problem (30) admits a unique maximizer to deduce that $\mathcal{J}(q, \psi) > \mathcal{J}(q', \psi)$. By the saddle-point property of $(q', \psi')$, this implies

$$ \mathcal{J}(q, \psi) > \mathcal{J}(q', \psi) \geq \mathcal{J}(q', \psi'). $$

This is a contradiction, since both extreme terms are equal to $\min_{\tilde{\psi} \in A} \max_{\tilde{q} \in Q} \mathcal{J}(\tilde{q}, \tilde{\psi})$. Similarly, if $\psi \neq \psi'$, then $\mathcal{J}(q, \psi) < \mathcal{J}(q, \psi') \leq \mathcal{J}(q', \psi')$, which again leads to a contradiction by the saddle-point property. This concludes the proof of uniqueness. $\square$

The system described by equations (OptC) and (OptN) is inherently coupled. Specifically, the control $q$ played by Nature appears both in the pre-Hamiltonian $H$ and in the terminal condition for the adjoint variables $(p, k)$ of the central planner, as shown in (OptC). Similarly, the control $\psi$ of the central planner is present in the driver and in the terminal condition for the Nature adjoint variables $(Y, Z)$. As a consequence of Theorem 10, this coupled system has a unique solution, which characterizes the (unique) saddle point to the problem (P).

**Remark 11.** *Isaac's condition.* At optimality, the following Isaac's condition holds at the optimum:

$$ \begin{aligned} & \min_{\psi} \max_{(Y^*, Z^*)} \mathcal{H}(t, q_t, Y^*, Z^*, Y_t, Z_t, X_t, \psi, p_t, k_t) \\ & = \max_{(Y^*, Z^*)} \min_{\psi} \mathcal{H}(t, q_t, Y^*, Z^*, Y_t, Z_t, X_t, \psi, p_t, k_t), \end{aligned} \quad (32) $$

$d\mathbb{P} \otimes dt$-almost surely. This follows from the combination of (OptN) and (OptC), which say that, $d\mathbb{P} \otimes dt$-almost surely,

$$ \begin{aligned} & (Y_t^*, Z_t^*) \in \arg\max_{(Y^{*'}, Z^{*'})} F(t, q_t, Y^{*'}, Z^{*'}, Y_t, Z_t, \psi_t), \\ & \psi_t \in \arg\min_{\alpha} H(t, X_t, \alpha, p_t, k_t, q_t). \end{aligned} $$

Returning back to the definition (19) of $\mathcal{H}$, these two lines can be rewritten as

$$ \begin{aligned} & (Y_t^*, Z_t^*) \in \arg\max_{(Y^{*'}, Z^{*'})} \mathcal{H}(t, q_t, Y^{*'}, Z^{*'}, Y_t, Z_t, X_t, \psi_t, p_t, k_t), \\ & \psi_t \in \arg\min_{\alpha} \mathcal{H}(t, q_t, Y_t^*, Z_t^*, Y_t, Z_t, X_t, \alpha, p_t, k_t), \end{aligned} $$

from which the bound $\min \max \leq \max \min$ in (32) indeed follows, the converse bound being always true.
## 3.2 Examples of applications

We provide two examples of applications of Theorem 10. On purpose, the presentation is informal and contains no mathematical statement. Further examples are given in Subsection 4.2.

**Risk averse portfolio management with trading costs.** The first example is inspired by [58] and considers the regime “without investment control constraints,” in the absence of a risk-free asset, and over a finite time horizon.

Consider a financial market consisting of $d \in \mathbb{N}^*$ stocks, whose prices per share are encoded in the form of an $d$-dimensional process $S = (S^1, S^2, \dots, S^d)$, satisfying the following SDE:

$$ \frac{\mathrm{d}S_t^i}{S_t^i} = c_t^i \mathrm{d}t + (\sigma_t \mathrm{d}W_t)^i, \quad S_0^i = \zeta^i, \quad \forall i \in \{1, \dots, d\}, $$

where $\zeta = (\zeta^1, \dots, \zeta^d) \in L^\infty(\mathcal{F}_0, \mathbb{R}^d)$ are the initial prices, $c = (c^1, \dots, c^d) \in L^\infty(\mathbb{F}; \mathbb{R}^d)$ is the vector of stock appreciation rates, $\sigma \in L^\infty(\mathbb{F}, \mathbb{R}^{d \times d})$ is the volatility matrix, and $W = (W^1, \dots, W^d)$ is a $d$-dimensional Brownian motion. For simplicity we thus assume that the number of assets is equal to the number of noise sources. We further assume that there is no bond available on the market. For a given vector of amounts (or allocation strategies) $\psi \in \mathcal{A}$, the dynamics of the self-financing portfolio $X^\psi$ is given by

$$ \mathrm{d}X_t = \psi_t \cdot \frac{\mathrm{d}S_t}{S_t}, \quad X_0 = 1, $$

where the dot appearing on the right-hand side stands for the inner product in $\mathbb{R}^d$, the initial condition is arbitrarily chosen to be unitary, and (consistently with the fact there is no bond) the interest rate of the market is assumed to be null for simplicity. The problem of the risk averse investor under a min-max form is given by

$$ \sup_{q \in \mathcal{Q}} \inf_{\psi \in \mathcal{A}} \mathcal{J}(q, \psi), \qquad (33) $$

where

$$ \mathcal{J}(q, \psi) = \mathbb{E}^{\mathbb{Q}} \left[ X_T^\psi + \frac{1}{2} \int_0^T |\psi_s|^2 \mathrm{d}s \right] - \lambda \mathrm{H}(\mathbb{Q}|\mathbb{P}), \quad \mathbb{Q} = q_T \mathbb{P} $$

and $q_T = \mathcal{E}_T(\int_0^T Z_s^* \cdot \mathrm{d}W_s)$. The problem can be interpreted as follows. Given a probability measure $\mathbb{Q}$ equivalent to $\mathbb{P}$, the investor optimizes the average return of the portfolio while incurring a trading cost. Given an investment strategy chosen by the investor, Nature then selects the worst-case probability measure $\mathbb{Q}$, while being penalized by an entropic cost. The parameter $\lambda > 0$ models the level of risk aversion of the investor.

This is a sub-case of our setting. The random processes $a$ and $b$ are null, $c$ is valued in $\mathbb{R}^d$ instead of $\mathbb{R}^{1 \times d}$ and $\sigma$ in $\mathbb{R}^{d \times d}$ instead of $\mathbb{R}^{d \times 1 \times d}$. The terminal cost is linear in the measure $\mathbb{Q} \circ (X_T^\psi)^{-1}$ and the actualization rate $Y^*$ is null. By Theorem 10, the problem (33) admits a unique solution $(\psi, q)$, characterized by $\nabla_\psi H(t, X_t, \psi_t, p_t, k_t, q_t) = 0$ and $Z_t^* = \partial_z f(Z_t)$, which, after computations (with $k_t$ being viewed as a vector of dimension $d$), gives

$$ \psi_t = -q_t^{-1} (p_t c_t + \sigma_t^\top k_t), \quad Z_t = \frac{1}{\lambda} Z_t^*, $$
where the tuple of state and adjoint processes $(Y, Z, X, p, k, q)$ is the solution to

$$ egin{cases} -dY_t = (\frac{1}{2\lambda}|Z_t|^2 + |\psi_t|^2)dt - Z_t \cdot dW_t, & Y_T = X_T, \\ dX_t = \psi_t \cdot c_t dt + \psi_t \cdot (\sigma_t dW_t), & X_0 = 1, \\ -dp_t = -k_t \cdot dW_t, & p_T = q_T, \\ dq_t = \lambda q_t Z_t \cdot dW_t, & q_0 = 1. \end{cases} $$

By the last two equations, we have that $p_t = q_t$ and $k_t = \lambda q_t Z_t$. Then the solution simplifies to

$$ \psi_t = -c_t - \frac{1}{\lambda} \sigma_t^\top Z_t, $$

and

$$ egin{cases} -dY_t = (\frac{1}{2\lambda}|Z_t|^2 + |\psi_t|^2)dt - Z_t \cdot dW_t, & Y_T = X_T, \\ dX_t = \psi_t \cdot c_t dt + \psi_t \cdot (\sigma_t dW_t), & X_0 = 1, \end{cases} \quad (34) $$

which reduces the problem to a quadratic FBSDE. It seems that, due to the unboundedness of the terminal condition, the latter system is out of the scope of the theory of FBSDEs with a quadratic driver (in the backward equation) [60, 78, 68]. Very briefly, existing results on the solvability of quadratic BSDEs require the terminal state variable $X_T$ and the cost $\int_0^T |\psi_t|^2 dt$ (with $\psi$ standing for the optimal control) to admit an exponential moment with a sufficiently large exponent, depending on the parameters of the problem. In the present setting, we are only able to establish exponential integrability for small exponents. This result is not stated explicitly in the article, as it holds only in the case where the function $f$ is genuinely quadratic. In that case, the functional $S$ coincides with the standard entropy and, by a Donsker-Varadhan-type duality (see (15)), bounds on the conjugate functional $S^*$ yield bounds on certain exponential moments of $X_T$.

Here, the solution $(Y, Z)$ to the quadratic BSDE is obtained in the rather weak space

$$ D(\mathbb{F}, \mathbb{Q}) \times \bigcap_{\beta \in (0,1)} M^\beta(\mathbb{F}, \mathbb{R}^d, \mathbb{Q}). $$

Because the process $Y$ does not enjoy strong integrability properties, we are not able to justify the following identity, which is frequently used to establish the connection between the min-max and the risk-averse formulations:

$$ \frac{1}{\lambda} \ln \mathbb{E}[\exp(\lambda Y_0)] = \rho_{\lambda} \left[ X_T + \frac{1}{2} \int_0^T |\psi_s|^2 \, ds \right], $$

where $\rho_\lambda$ is defined in (11). The standard proof of this identity relies on the Hopf-Cole transform for quadratic BSDEs. However, it would require the random variable

$$ X_T + \frac{1}{2} \int_0^T |\psi_s|^2 \, ds $$

to admit an exponential moment of exponent $\lambda$, a property which appears to be out of reach in our framework.
**Control of systemic risk measure.** Systemic risk measures are risk assessment tools that evaluate the macro-level risk of a system composed of multiple interacting agents. The concept was first introduced axiomatically in [38] and has since been extensively explored in both management science [5, 13] and mathematical finance literature [71].

Denoting by $N$ the number of agents in the system, we consider the product space $(\Omega^{\times N}, \mathcal{F}^{\times N}, \mathbb{P}^N := \mathbb{P}^{\times N})$, and we equip its $i$-th factor with an $\mathbb{R}^d$-valued Brownian motion $W^i = (W_t^i)_{t \in [0,T]}$. We denote by $\mathbb{F}^N = (\mathcal{F}_t^N)_{t \in [0,T]}$ the completion of the filtration generated by $(W^1, \dots, W^N)$, and by $\mathcal{Q}^{(N)}$ the analogue of $\mathcal{Q}$ but on the product space, i.e. $\mathcal{Q}^{(N)}$ is the set of $q^N \in L \log L(\mathbb{F}^N)$ such that

$$dq_t^N = q_t^N \left( \sum_{i=1}^N Z_t^{\star,i} \cdot dW_t^i \right), \quad q_0 = 1, \quad \mathrm{H}(q^N \mathbb{P}^N | \mathbb{P}^N) < +\infty. \quad (35)$$

Below, we write $\mathbb{Q}^N := q^N \mathbb{P}^N = q^N \mathbb{P}^{\times N}$.

The function $f^*$ only depends on its last variable and is given, for a certain $\lambda > 0$, by $f^*(z^{(N)}) = \frac{\lambda}{2} \sum_{i=1}^N |z^i|^2$ for any $z^{(N)} = (z^1, \dots, z^N) \in [\mathbb{R}^d]^N$. We then denote by $\mathcal{A}^{(N)}$ the analogue of $\mathcal{A}$ but on the product space.

To simplify the presentation, we assume that the states of the agents follow dynamics similar to the one presented in the first example, but with each driven by its own noise $W^i$. We also assume that the coefficients are deterministic, which avoids the need to track how each player's coefficients depend on the various sources of noise. For $i \in \{1, \dots, N\}$, the state of the $i$-th agent is thus given by the solution $X^{i, \psi^i}$ of the state equation:

$$dX_t^{i, \psi^i} = \psi_t^i \cdot c_t dt + \psi_t^i \cdot \sigma_t dW_t^i, \quad X_0 = 1, \quad (36)$$

where $\psi^i$ is the control to player $i$.

We now address the construction of a risk measure for the system formed by the $N$ agents. An initial approach would consist in summing individual risk measures associated to each of the agents. However, as emphasized in [13, Section 2], this approach may fail to capture systemic risk effects in financial systems. Motivated by the latter article, we propose an alternative construction in which individual states are first aggregated through an increasing, convex, and nonlinear function $g: \mathbb{R} \to \mathbb{R}$, then summed, and finally evaluated via an individual risk measure, the nonlinearity of the aggregation function being essential for practical relevance. A typical example for $g$ is the cost function $g(x) = \max\{x - x_0, 0\}$, where $x_0 \in \mathbb{R}$ is a finite threshold. That said, in order to fit within our framework, in which $g$ is typically required to be differentiable, we consider instead a smooth version of it, still convex, obtained for instance via regularization. We thus define the systemic risk measure $\mathfrak{p}_{\lambda}$, by letting

$$\mathfrak{p}_{\lambda}^{N}(\psi^{1}, \dots, \psi^{N}) = \rho_{\lambda} \left[ \sum_{i=1}^{N} \left( g(X_{T}^{i, \psi^{i}}) + \frac{1}{2} \int_{0}^{T} |\psi_{t}^{i}|^{2} dt \right) \right]. \quad (37)$$

*Choice of the normalization.* We emphasize that the sum inside $\rho_{\lambda}$ in the definition of $\mathfrak{p}_{\lambda}^N$ diverges as $N$ tends to $+\infty$. In contrast, if we normalize this sum (inside the risk measure) by an additional factor $1/N$, we obtain, in the limit $N \to +\infty$, a model in which risk aversion disappears. To see this, assume that the independence property of the noises $W^1, \dots, W^N$ is asymptotically transmitted to the optimal controls, as in
a standard MFC problem without risk aversion. Equivalently, restrict the definition of $\mathfrak{p}_{\lambda}^{N}$ to controls $\psi^{1}, \dots, \psi^{N}$ that are each constructed as a common progressively measurable function of $W^{i}$, for the corresponding index $i \in \{1, \dots, N\}$. Then, a purely formal application of the law of large numbers (without further justification) allows one to pass to the limit inside $\rho_{\lambda}$, yielding

$$ \lim_{N \to +\infty} \rho_{\lambda} \left[ \frac{1}{N} \sum_{i=1}^{N} \left( g(X_{T}^{i, \psi^{i}}) + \int_{0}^{T} |\psi_{t}^{i}|^{2} dt \right) \right] = \int g d\mu_{T}^{\psi} + \rho_{\lambda} \left[ \int_{0}^{T} |\psi_{t}|^{2} dt \right], \quad (38) $$

where $\mu_{T}^{\psi} = \mathcal{L}(X_{T}^{\psi})$ and the dynamics of $X^{\psi}$ is given by

$$ dX_t = \psi_t \cdot c_t dt + \psi_t \cdot (\sigma_t dW_t), \quad X_0 = 1. \quad (39) $$

As announced, the risk aversion has disappeared in the limit. Intuitively, the limiting problem (obtained by letting $N \to +\infty$) is a standard MFC problem. For this reason, we propose below an alternative construction of risk measures for $N$-particles system. *Solving the N-fixed problem.* Instead, we want to keep the sum over $g(X_{T}^{i, \psi})$ unnormalized in the definition (37) of $p_{\lambda}^{N}$. Accordingly, our objective is to explain, at least informally, what is the behaviour of $\frac{1}{N}p_{\lambda}^{N}$ as $N$ tends to $+\infty$.

The first step is to observe from the Donsker-Varadhan formula (15) that the minimization of $\mathfrak{p}_{\lambda}^{N}$ can be reformulated as a min-max problem, i.e.,

$$ \begin{aligned} & \inf_{\psi^{N} \in \mathcal{A}^{(N)}} \frac{1}{N} \mathfrak{p}_{\lambda}^{N}(\psi^{1}, \dots, \psi^{N}) \\ & = \inf_{\psi^{(N)} \in \mathcal{A}^{(N)}} \sup_{q^{N} \in \mathcal{Q}^{(N)}} \frac{1}{N} \left\{ \mathbb{E}^{\mathbb{Q}^{N}} \left[ \sum_{i=1}^{N} \left( g(X_{T}^{i, \psi^{i}}) + \int_{0}^{T} |\psi_{t}^{i}|^{2} dt \right) \right] - \lambda H(\mathbb{Q}^{N} | \mathbb{P}^{N}) \right\}. \end{aligned} $$

As $g$ is convex, we observe that the min-max problem appearing in the right-hand side enters the framework of Theorem 10, with $n = N$ and $d$ replaced by $d \times N$, with $q^{N}$ and $Z^{*(N)} = (Z^{*(i)})_{i \in \{1, \dots, N\}}$ satisfying (35) (implicitly, $\alpha = 0$ and $Y^{*(N)} = (Y^{*(i)})_{i \in \{1, \dots, N\}} \equiv 0$), with $\psi^{(N)} = (\psi^{i})_{i \in \{1, \dots, N\}}$ and $X^{(N)} = (X^{i, \psi^{i}})_{i \in \{1, \dots, N\}}$ solving (36), and with the terminal cost functions

$$ \begin{aligned} \mathcal{G}(q_T^N, X_T^{(N)}) &= \mathbb{E} \left[ q_T^N \sum_{i=1}^N g(X_T^{i, \psi^i}) \right], \\ \ell(t, \psi^{(N)}) &= \frac{1}{2} \sum_{i=1}^N |\psi_t^i|^2, \quad f^*(t, Z^{*(N)}) = \frac{\lambda}{2} \sum_{i=1}^N |Z_t^{*(i)}|^2. \end{aligned} $$

The saddle-point is characterized by

$$ \psi_t^i = -(q_t^N)^{-1} (p_t^i c_t + \sigma_t^\top k_t^i), \quad Z_t^i = \frac{1}{\lambda} Z_t^{*(i)}, $$

for each $i \in \{1, \dots, N\}$, where $p_t^i$ takes values in $\mathbb{R}$ and $k_t^i$ in $\mathbb{R}^d$, and where the tuple of state and adjoint processes $(Y^N, Z^{(N)} = (Z^1, \dots, Z^N), X^{(N)}, p^{(N)} = (p^1, \dots, p^N), k^{(N)} = (k^1, \dots, k^N), q^{(N)}$) is the solution to

$$ \left\{ \begin{aligned} -dY_t^N &= \left( \frac{1}{2\lambda} \sum_{j=1}^N |Z_t^j|^2 + \frac{1}{2} \sum_{j=1}^N |\psi_t^j|^2 \right) dt - \sum_{j=1}^N Z_t^j \cdot dW_t^j, \\ dX_t^i &= \psi_t^i \cdot c_t dt + \psi_t^i \cdot (\sigma_t dW_t^i), \\ -dp_t^i &= -\sum_{j=1}^N k_t^{i,j} \cdot dW_t^j, \\ dq_t^N &= \lambda q_t^N \sum_{j=1}^N Z_t^j \cdot dW_t^j, \end{aligned} \right. $$
with terminal conditions

$$Y_T^N = \sum_{j=1}^{N} g(X_T^{j, \psi^j}), \quad X_0^{N, i} = 1, \quad p_T^i = q_T^N g'(X_T^{i, \psi^i}), \quad q_0^N = 1,$$

for any $i \in \{1, \dots, N\}$.

**Towards a robust MFC problem.** We now provide a heuristic derivation of the limiting problem as $N \to +\infty$. The purpose of this discussion is solely to identify the structure of the limiting model; no claim of rigor is made at this stage. Proceeding as in (38), we assume that, at the saddle point, the controls $\psi^1, \dots, \psi^N$ are each constructed as a common progressively measurable function of the individual Brownian motion $W^i$, for the corresponding index $i \in \{1, \dots, N\}$, and similarly for the controls $Z^{\star,1}, \dots, Z^{\star,N}$. Strictly speaking, such an independence structure does not hold at the finite-$N$ saddle point. However, this assumption can be justified a posteriori by reverse engineering: starting from a solution to the limiting problem, one may construct an approximate optimizer for the finite-$N$ problem, which is a standard approach in mean field control theory. Under this assumption, and applying Girsanov's theorem (all similar changes of probability measures will be justified in the core of the article, but we prefer not to address such technical questions in this informal discussion), the vector $X^{(N)} = (X^1, \dots, X^N)$ satisfies the following dynamics under $\mathbb{Q}^N$:

$$dX_t^i = \psi_t^i \cdot (c_t + \sigma_t Z_t^{\star, i}) dt + \psi_t^i \cdot \sigma_t dW_t^{i, \mathbb{Q}^N},$$

where

$$(W_t^{i, \mathbb{Q}^N} = W_t^i - \int_0^t Z_s^{\star, i} ds)_{t \in [0, T], i=1, \dots, N}$$

is (expected to be) an $N$-dimensional Brownian motion under $\mathbb{Q}^N$.

And then, under $\mathbb{Q}^N$, the processes $(X^i, \psi^i)_{i=1, \dots, N}$ are independent and identically distributed, which makes it possible to derive, by a new application of the law of large numbers, the following approximation for the cost underpinning the min-max problem:

$$\frac{1}{N} \left\{ \mathbb{E}^{\mathbb{Q}^N} \left[ \sum_{i=1}^{N} \left( g \left( X_T^{i, \psi^N} \right) + \int_0^T |\psi_t^{N, i}|^2 dt \right) \right] - \lambda \mathrm{H} \left( \mathbb{Q}^N | \mathbb{P}^N \right) \right\} \\ \approx G \left( (q_T^1 \mathbb{P})_{X_T^{1, \psi^1}} \right) + \mathbb{E} \left[ q_T^1 \int_0^T |\psi_t^1|^2 dt \right] - \lambda \mathrm{H} \left( q_T^1 \mathbb{P} | \mathbb{P} \right).$$

as $N \to +\infty$, where $q_T^1 \mathbb{P})_{X_T^{1, \psi^1}}$ denotes the law of $X_T^{1, \psi^1}$, regarded as a random variable on $(\Omega, \mathcal{F})$ equipped with the probability measure $q_T^1 \mathbb{P}$.

We thus conjecture that, in the limit, we are faced with a robust control problem involving a single agent, but with a cost depending on the law of the agent under the distribution resulting from Nature's choice. In other words, we expect that the asymptotic problem (obtained by letting $N \to \infty$) consists of the following min-max problem:

$$\inf_{\psi \in \mathcal{A}} \sup_{q \in \mathbb{Q}} \left\{ G(\mathcal{L}^{\mathbb{Q}}(X_T^{\psi})) + \mathbb{E}^{\mathbb{Q}} \left[ \int_0^T |\psi_t|^2 dt \right] - \lambda \mathrm{H}(\mathbb{Q}|\mathbb{P}) \right\},$$
formulated on $(\Omega, \mathcal{F}, \mathbb{P})$, where $\mathbb{Q} := q_T \mathbb{P}$ denotes the probability measure induced by Nature's strategy. A more general treatment of this problem is provided in the next section, which is devoted to the mean field regime.

# 4 Applications to mean field models

In this section, we develop a robust formulation of classical mean field control problems and subsequently analyze an associated variational mean field game problem. Subsection 4.1 is devoted to the robust mean field control problem (MFC). Relying on Theorem 10 from the previous section, we establish in Corollary 15 the existence and uniqueness of a saddle point, together with the corresponding stochastic maximum principle. Subsection 4.2 provides two examples of applications in this context. Subsection 4.3 then turns to a class of variational mean field games. Building on Corollary 15, we prove in Corollary 16 the existence and uniqueness of a Nash equilibrium, which is fully characterized by a McKean-Vlasov forward-backward stochastic differential equation.

## 4.1 Robust mean field control

This subsection is dedicated to the study of the robust mean field control problem (MFC), which we recall here

$$ \inf_{\psi \in \mathcal{A}} \sup_{q \in \mathcal{Q}} \left\{ G \left( (q_T \mathbb{P})_{X_T^\psi} \right) + \mathbb{E} \left[ \int_0^T q_s \ell(s, \psi_s) \mathrm{d}s \right] - \mathcal{S}(q) \right\}, \quad (\text{MFC}) $$

where $G: \mathcal{M}_+(\mathbb{R}^n) \to \mathbb{R}$ is a mean field mapping of the positive measure $(q_T \mathbb{P})_{X_T^\psi} = (q_T \mathbb{P}) \circ X_T^{\psi,-1}$ and $(X_T^\psi)_{t \in [0,T]}$ denotes the solution to the controlled stochastic differential equation (7).

This problem can be recast as problem (P) assuming that the mapping $\mathcal{G}$ is specified as follows

$$ \mathcal{G}(q, X) = G((q\mathbb{P})_X). \quad (40) $$

When the actualization rate $Y^*$ is null, the measure $q_T \mathbb{P}$ is a probability measure, equal to $\mathcal{E}_T(\int_0^T Z_s^* \cdot dW_s)$, and the domain of definition of $G$ can be reduced to $\mathcal{P}(\mathbb{R}^n)$. In the latter case, $G$ is a true mean field function. By extension, we still call the model 'mean field' even if the mass of $q_T$ is unnormalized.

As the domain of definition of $G$ is larger than $\mathcal{P}_1(\mathbb{R}^n)$, we are led, under the assumptions below, to redefine implicitly the notion of flat and Lions derivatives. Since the objects thus redefined coincide, in the mean field case, with the true flat and Lions derivatives, we nevertheless use the same notations $\delta G/\delta \mu$ and $\partial_\mu G$ as in the introduction. This is the rational behind the introduction of the following distances.

**Spaces of positive measures** We introduce, for any $p \ge 1$, a variant of the total variation distance, adapted to elements of $\mathcal{M}_p(\mathbb{R}^n) := \{\mu \in \mathcal{M}_+(\mathbb{R}^n), M_p(\mu) < +\infty\}$, where here and throughout $M_p(\mu) := \int_{\mathbb{R}^n} |x|^p d\mu(x)$. For such a $p$, we let
(the proof of the fact that the right-hand side below defines a distance is left to the reader):

$$d_p(\mu, \nu) := \sup_{\varphi} \int_{\mathbb{R}^n} \varphi(x) d(\mu - \nu)(x), \quad \mu, \nu \in \mathcal{M}_p(\mathbb{R}^n), \qquad (41)$$

where the supremum is taken over measurable functions $\varphi: \mathbb{R}^n \to \mathbb{R}$ such that $|\varphi(x)| \le 1 + |x|^p$.

We also use the standard p-Wasserstein distance, when restricted to subsets of measures with equal mass. Below, we refer to these subsets as “isomass subsets”. For the same p as above, and for $\mu$ and $\nu$ in $\mathcal{M}_p(\mathbb{R}^n)$ such that $\mu(\mathbb{R}^n) = \nu(\mathbb{R}^n)$, we let

$$W_p(\mu, \nu) := \inf_{\substack{\pi \in \mathcal{M}_p(\mathbb{R}^n \times \mathbb{R}^n), \\ \pi \circ e_1^{-1} = \mu, \pi \circ e_2^{-1} = \nu}} \int_{\mathbb{R}^n \times \mathbb{R}^n} |x - x'|^p d\pi(x, x') \le \nu,$$

where $e_i: \mathbb{R}^n \times \mathbb{R}^n \ni (x_1, x_2) \mapsto x_i$, for $i = 1, 2$.

We combine the two distances $d_p$ and $W_p$ by considering functions, defined on $\mathcal{M}_p(\mathbb{R}^n)$, that are continuous with respect to $d_p$ on the entire $\mathcal{M}_p(\mathbb{R}^n)$, and that are continuous with respect to $W_p$ on any isomass subset of $\mathcal{M}_p(\mathbb{R}^n)$. We prove in Subsection D.1 of the Appendix that those functions are continuous with respect to the so-called generalized p-Wasserstein distance, and conversely. That said, we feel easier, in our specific framework, to use separately the two distances $d_p$ and $W_p$, instead of the single generalized Wasserstein distance.

**Assumptions** We now state the required assumptions on $G$. We still assume A1-A5 to hold. We recall that, the parameter $r$ used throughout, is defined in A2.

A9 We assume that there exist two functions

$$\frac{\delta G}{\delta \mu} : \mathcal{M}_{2-r}(\mathbb{R}^n) \times \mathbb{R}^n \to \mathbb{R}, \quad \partial_{\mu} G : \mathcal{M}_{2-r}(\mathbb{R}^n) \times \mathbb{R}^n \to \mathbb{R}^n,$$

with $\delta G/\delta \mu$ being differentiable in the second argument when the first one is fixed such that, for any $\mu \in \mathcal{M}_{2-r}(\mathbb{R}^n)$ and any $x \in \mathbb{R}^n$,

$$\frac{\delta G}{\delta \mu}(\mu, x) = \frac{d}{d\varepsilon}|_{\varepsilon=0+} G(\mu + \varepsilon \delta_x), \qquad (42)$$

where $\delta_x$ is the delta mass at point $x$, and

$$\partial_{\mu} G(\mu, x) = \nabla_{x} \frac{\delta}{\delta \mu} G(\mu, x). \qquad (43)$$

a) We assume that these three mappings satisfy the following growth conditions:

$$\begin{align}
& |G(\mu)| \le L (1 + M_{2-r}(\mu)), \nonumber \\
& -L (1 + M_{2-r}(\mu) + |x|) \le \frac{\delta G}{\delta \mu}(\mu)(x) \le L (1 + M_{2-r}(\mu) + |x|^{2-r}), \tag{44} \\
& |\partial_{\mu} G(\mu, x)| \le L (1 + M_{2-r}(\mu) + |x|^{1-r}). \nonumber
\end{align}$$

b) We assume that $G$ is continuous with respect to $d_{2-r}$ on the entire $\mathcal{M}_{2-r}(\mathbb{R}^n)$, and continuous with respect to $W_{2-r}$ on isomass subsets.
As for the derivative $\delta G/\delta \mu$ (which is already required to be locally Lipschitz, uniformly in $\mu$, thanks to (44)), we assume it to be continuous with respect to $d_{2-r}$, with a modulus of continuity that grows at most like $|x|^{2-r}$, and that is uniform in $\mu \in \mathcal{M}_{2-r}(\mathbb{R}^n)$ satisfying $M_{2-r}(\mu) \le C$ for some $C > 0$. More precisely, for any $\varepsilon > 0$ and $C > 0$, we assume that there exists $v > 0$, such that, for any $\mu, \mu' \in \mathcal{M}_{2-r}(\mathbb{R}^n)$ satisfying $M_{2-r}(\mu), M_{2-r}(\mu') \le C$ and $d_{2-r}(\mu, \mu') \le v$, we have

$$ \sup_{x \in \mathbb{R}^n} \left[ \frac{1}{1 + |x|^{2-r}} \left| \frac{\delta G}{\delta \mu}(\mu)(x) - \frac{\delta G}{\delta \mu}(\mu')(x) \right| \right] \le \varepsilon. \quad (45) $$

We also assume that, for any $x \in \mathbb{R}^n$, $\mu \mapsto [\delta G/\delta \mu](\mu, x)$ is continuous with respect to $W_{2-r}$ on isomass subsets.

At last, we require that, when the first argument is restricted to the isomass subset $\{\mu \in \mathcal{M}_{2-r}(\mathbb{R}^n), \mu(\mathbb{R}^n) = c\}$, for some $c \ge 0$, the function $\partial_\mu G$ is locally Lipschitz continuous in $(\mu, x)$ in the following sense: for any $C > 0$, there exists $L_C \ge 0$ such that, for any $\mu, \mu' \in \mathcal{M}_{2-r}(\mathbb{R}^n)$ with $\mu(\mathbb{R}^n) = \mu'(\mathbb{R}^n)$ and $M_{2-r}(\mu), M_{2-r}(\mu') \le C$, and any $x, x' \in \mathbb{R}^n$,

$$ \frac{1}{1 + |x|^{1-r}} \left| \partial_{\mu} G(\mu, x) - \partial_{\mu} G(\mu', x) \right| \le L_C W_{2-r}(\mu, \mu'), \quad (46) \\ \left| \partial_{\mu} G(\mu, x) - \partial_{\mu} G(\mu, x') \right| \le L_C |x - x'|. $$

c) We finally assume that $G$ is flat concave, i.e., for $\mu, \mu' \in \mathcal{M}_{2-r}(\mathbb{R}^n)$,

$$ \int_{\mathbb{R}^n} \left( \frac{\delta G}{\delta \mu}(\mu, x) - \frac{\delta G}{\delta \mu}(\mu', x) \right) d(\mu - \mu')(x) \le 0, \quad (47) $$

and $G$ is displacement convex on isomass subsets, i.e., for any $\mu, \mu' \in \mathcal{M}_{2-r}(\mathbb{R}^n)$ with $\mu(\mathbb{R}^n) = \mu'(\mathbb{R}^n)$, for any measure $\pi \in \mathcal{M}_{2-r}(\mathbb{R}^n \times \mathbb{R}^n)$ with $\mu$ and $\mu'$ as marginals,

$$ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \left( \partial_{\mu} G(\mu, x) - \partial_{\mu} G(\mu', x') \right) \cdot (x - x') d\pi(x, x') \ge 0. \quad (48) $$

*Remark 12.* For presentation purpose we only consider a mean field terminal cost. But one could also consider mean field running cost of the separated form

$$ \ell'(t, \psi_t, \mu_t) = \ell(t, \psi_t) + c(X_t, \mu_t), $$

where $\mu_t$ is the marginal law of $X_t$ under the probability measure $q_T \mathbb{P}$ induced by Nature. The assumptions on $c$ should be analogous to the assumptions required for $g$ above (growth, flat differentiable and Lions differentiable, with the appropriate regularity, flat concave and displacement convex).

*Remark 13.* The following comments are in order.

The first remark is that it suffices, for our purpose, to have all the above conditions satisfied for $\mu$ and $\mu'$ of mass less than $\exp(\alpha T)$. This follows from the fact that, in our applications, $\mathbb{E}[q_T] \le \exp(\alpha T)$.

The second observation is that the notion of displacement convexity, as mentioned in (48), is usually reserved to functions defined on the space of probability
measures. In (48), we can easily recover the case when $\mu$ and $\mu'$ are probability measures by normalizing them. Indeed, for a given $c > 0$ representing the common mass of $\mu$ and $\mu'$, we can consider the function $G^{(c)}: \mathcal{P}_{2-r}(\mathbb{R}^n) \ni \mu \mapsto G(c\mu)$. Obviously, the standard flat and Lions derivatives (according to their usual definitions for functionals defined on $\mathcal{P}_{2-r}(\mathbb{R}^n)$, the common construction of the Lions derivative being restricted to the case $r = 0$) are

$$ \frac{\delta G^{(c)}}{\delta \mu}(\mu, x) = c \frac{\delta G}{\delta \mu}(\mu, x), \quad \partial_{\mu} G^{(c)}(\mu, x) = c \partial_{\mu} G(\mu, x). $$

If $G^{(c)}$ has second-order derivatives in $\mu$ and $x$, then it satisfies (48) for any two probability measures $\mu$ and $\mu'$ if

$$ \int_{\mathbb{R}^n \times \mathbb{R}^n} \mathrm{Tr} \left[ \partial_{\mu}^2 G^{(c)}(\mu, x, x') \beta(x) \otimes \beta(x') \right] d\mu(x) d\mu(x') + \int_{\mathbb{R}^n} \mathrm{Tr} \left[ \partial_x \partial_{\mu} G^{(c)}(\mu, x) \beta(x) \otimes \beta(x) \right] d\mu(x) \geq 0, \quad (49) $$

for any bounded measurable function $\beta$ from $\mathbb{R}^n$ to itself. The above can be found in [35, Chapter 5], when $r = 0$. Returning back to unnormalized measures (i.e., changing $\mu$ into $c\mu$), it easy to see that, when $r = 0$, (48) is true (whethever the mass of $\mu$ and $\mu'$) if (49) is true with $G$ being substituted for $G^{(c)}$. In fact, (49) remains also a sufficient condition when $r = 1$: It implies (48) when $\mu$ and $\mu'$ therein have finite second-order moments; by a standard approximation argument, the inequality remains true when $\mu$ and $\mu'$ are just in $M_1(\mathbb{R}^n)$. Below, we thus call Hessian of $G$ in the direction $\beta$ the quantity

$$ \mathcal{H}_G(\beta) := \int_{\mathbb{R}^n \times \mathbb{R}^n} \mathrm{Tr} \left[ \partial_{\mu}^2 G(\mu, x, x') \beta(x) \otimes \beta(x') \right] d\mu(x) d\mu(x') + \int_{\mathbb{R}^n} \mathrm{Tr} \left[ \partial_x \partial_{\mu} G(\mu, x) \beta(x) \otimes \beta(x) \right] d\mu(x). $$

**Constructing flat concave and displacement convex functions** We first note that any linear functional of the form

$$ G_1(\mu) = \int_{\mathbb{R}^n} v_1(x) d\mu(x), \quad (50) $$

where $v_1: \mathbb{R}^n \to \mathbb{R}$ is smooth and convex, is an ideal candidate to satisfy Assumption A9. Indeed it is flat concave as it is linear in $\mu$ and displacement convex by convexity of $v_1$. The main point is to check that $v_1$ satisfies the required integrability properties, depending on whether $r = 0$ or $r = 1$, which prompts us to distinguish between these two cases below.

Regardless of the integrability properties, $G_1$ satisfies

$$ \frac{\delta G_1}{\delta \mu}(\mu, x) = v_1(x), $$

for any $x \in \mathbb{R}^n$, so that (47) is trivially satisfied, and

$$ \partial_{\mu} G_1(\mu, x) = \nabla_x v_1(x), $$
so that (48) is expected to be satisfied if $v_1$ is convex. In particular, the Hessian $\mathcal{H}_{G_1}(\beta)$ is equal to

$$ \mathcal{H}_{G_1}(\beta) = \int_{\mathbb{R}^n} \mathrm{Tr} \left[ \nabla_x^2 v_1(x) \beta(x) \otimes \beta(x) \right] \mathrm{d}\mu(x), \quad (51) $$

which is obviously non-negative when $v_1$ is convex.

Before we discuss more in depth the integrability properties, we notice, as $G_1$ is linear in $\mu$, that any composition of $G_1$ by a (smooth) concave function $\varphi : \mathbb{R} \to \mathbb{R}$ is expected to be flat concave. Such an example can be written as

$$ G_2(\mu) = \varphi \left( \int_{\mathbb{R}^n} v_2(x) \mathrm{d}\mu(x) \right), \quad (52) $$

where $v_2$ satisfies the required integrability constraints (similar to $v_1$, as discussed below), and $\varphi$ is smooth and concave. In this situation, we have (at least formally),

$$ \partial_{\mu} G_2(\mu, x) = \varphi' \left( \int_{\mathbb{R}^n} v_2(x) \mathrm{d}\mu(x) \right) \nabla_x v_2(x), $$

and then,

$$ \partial_{\mu}^2 G_2(\mu, x, x') = \varphi'' \left( \int_{\mathbb{R}^n} v_2(x) \mathrm{d}\mu(x) \right) \nabla_x v_2(x) \otimes \nabla_x v_2(x'), $$

$$ \nabla_x \partial_{\mu} G_2(\mu, x) = \varphi' \left( \int_{\mathbb{R}^n} v_2(x) \mathrm{d}\mu(x) \right) \nabla_x^2 v_2(x). $$

In particular, by Cauchy-Schwarz inequality, it is quite easy to see that, for $\mu(\mathbb{R}^n) = \mu'(\mathbb{R}^n) \le \exp(\alpha T)$,

$$ |\mathcal{H}_{G_2}(\beta)| \le C(\varphi, v_2, \mu) \int_{\mathbb{R}^n} \left( |\nabla_x v_2(x)|^2 + |\nabla_x^2 v_2(x)| \right) |\beta(x)|^2 \mathrm{d}\mu(x), \quad (53) $$

where $C(\varphi, v_2, \mu) := \max(|\varphi'|(|\int_{\mathbb{R}^n} v_2 \mathrm{d}\mu|, \exp(\alpha T)|\varphi''|(|\int_{\mathbb{R}^n} v_2 \mathrm{d}\mu|))$.

To produce a wider class of functions that are flat concave and displacement convex, we can sum $G_1$ and $G_2$. Indeed, we observe that $G_1 + G_2$ is always flat concave. To obtain that the sum is displacement convex, we only need to ensure that

$$ \mathcal{H}_{G_1}(\beta) + \mathcal{H}_{G_2}(\beta) \ge 0. $$

Combining (51) and (53), the latter inequality holds true if

$$ \forall x, y \in \mathbb{R}^n, \quad \mathrm{Tr} \left[ \nabla_x^2 v_1(x) y \otimes y \right] \ge C(\varphi, v_2, \mu) \left( |\nabla_x v_2(x)|^2 + |\nabla_x^2 v_2(x)| \right) |y|^2. \quad (54) $$

We stress that the inequality must be true for any $\mu$ with a mass less than $\exp(\alpha T)$. This puts an additional constraint due to the dependence of the constant $C(\varphi, v_2, \mu)$ on $\mu$. That said, when $v_2$ is bounded, the constant $C(\varphi, v_2, \mu)$ can be bounded independently of $\mu$, since $\mu(\mathbb{R}^n) \le \exp(\alpha T)$; in that case, we can substitute $C(\varphi, v_2)$ for $C(\varphi, v_2, \mu)$ and then get a condition that is independent of $\mu$.

In order to give more explicit examples, we need to take into account the integrability conditions of $\mu$, as the latter dictate the growth properties of the derivatives of $v_1$ and $v_2$.
Case $r=0$. When $r=0$, a prototypal example is $v_1(x) = \lambda|x|^2/2$, for $\lambda > 0$. Then, (54) holds if

$$C(\varphi, v_2, \mu) (|\nabla_x v_2|^2 + |\nabla_x^2 v_2(x)|) \leq \lambda.$$

An interesting example is $v_2 = x$ and $\varphi(u) = -u^2/2$, in which case $G = G_1 + G_2$ writes

$$G(\mu) = \frac{\lambda}{2} \int_{\mathbb{R}^n} |x|^2 d\mu(x) - \frac{1}{2} \left| \int_{\mathbb{R}^n} x d\mu(x) \right|^2.$$

Here,

$$\frac{\delta G}{\delta \mu}(\mu, x) = \frac{\lambda}{2} |x|^2 - x \cdot \left( \int_{\mathbb{R}^n} x' d\mu(x') \right), \quad \partial_\mu G(\mu, x) = \lambda x - \int_{\mathbb{R}^n} x' d\mu(x'),$$

and it is easy to check (44), (45) and (46). Moreover, for any measure $\pi \in \mathcal{M}_2(\mathbb{R}^n \times \mathbb{R}^n)$ with $\mu$ and $\mu'$ as marginal measures,

$$\begin{align*}
& \int_{\mathbb{R}^n \times \mathbb{R}^n} (\partial_\mu G(\mu, x) - \partial_\mu G(\mu', x')) \cdot (x - x') d\pi(x, x') \\
&= \lambda \int_{\mathbb{R}^n \times \mathbb{R}^n} |x - x'|^2 d\pi(x, x') - \left| \int_{\mathbb{R}^n \times \mathbb{R}^n} (x - x') d\pi(x, x') \right|^2 \\
&\geq (\lambda - \mu(\mathbb{R}^n)) \int_{\mathbb{R}^n \times \mathbb{R}^n} |x - x'|^2 d\pi(x, x'),
\end{align*}$$

with the last line following from Cauchy-Schwarz inequality, and from the fact that $\pi(\mathbb{R}^n \times \mathbb{R}^n) = \mu(\mathbb{R}^n)$. This shows that, for $\lambda \ge \exp(\alpha T)$, (48) is satisfied for any $\mu, \mu'$ such that $\mu(\mathbb{R}^n) = \mu'(\mathbb{R}^n) \le \exp(\alpha T)$. At the threshold $\lambda = \mu(\mathbb{R}^n)(= \mu'(\mathbb{R}^n)),$

$$G(\mu) = \frac{1}{4} \int_{\mathbb{R}^n \times \mathbb{R}^n} |x - x'|^2 d\mu(x) d\mu(x').$$

Still for $v_1(x) = \lambda|x|^2/2$, we can choose $v_2$ bounded, with bounded derivatives of order 1 and 2. In that case, it is easy to check (44), (45) and (46). Moreover, (54) holds true if

$$\lambda \ge \sup_{|c| \le \exp(\alpha T) \|v_2\|_{\infty}} \max (|\varphi|(c), \exp(\alpha T) |\varphi''|(c)) (\|\nabla_x v_2\|_{\infty}^2 + \|\nabla_x^2 v_2\|_{\infty}).$$

Case $r=1$. When $r=1$, we can no longer choose $v_1$ of quadratic growth (since $v_1$ must have a finite integral with respect to elements of $\mathcal{M}_1(\mathbb{R}^n)$). Instead, we can work with

$$v_1(x) = \lambda v_1^0(x), \quad \text{with} \quad v_1^0(x) := (1 + |x|^2)^{1/2},$$

for some $\lambda > 0$. Then, for any coordinates $i, j \in \{1, \dots, n\}^2$,

$$\partial_{x_i} v_1^0(x) = \frac{x_i}{v_1^0(x)}, \quad \partial_{x_i x_j}^2 v_1^0(x) = \frac{\delta_{i,j}}{v_1^0(x)} - \frac{x_i x_j}{v_1^0(x)^3},$$

where $\delta$ is the Kronecker delta here, which gives for any $y \in \mathbb{R}^n$,

$$\mathrm{Tr} [\nabla_x^2 v_1(x) y \otimes y] = \lambda \left[ \frac{|y|^2}{v_1^0(x)} - \frac{(x \cdot y)^2}{v_1^0(x)^3} \right] \geq \lambda \frac{|y|^2}{v_1^0(x)^3}.$$
If we assume that $v_2$ is bounded, with bounded derivatives of order 1 and 2, it is easy to check (44), (45) and (46). Moreover, (54) holds true if

$$ \frac{\lambda}{v_1^0(x)^3} \geq \sup_{|c| \leq \exp(\alpha T) \|v_2\|_{\infty}} \max(|\varphi'(c), \exp(\alpha T)|\varphi''|(c)) (|\nabla_x v_2(x)|^2 + |\nabla_x^2 v_2(x)|). $$

For instance, the above holds true if $v_2$ is compactly supported and $\lambda$ is large enough.

**Main result** The following result is standard in the literature (see for instance [35, 89]). For completeness, the proof is given in the Appendix, see Subsection D.2.

**Lemma 14.** Let $G$ satisfy A9. Then, on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$ as before, for any two random variables $q, q'$ with values in $\mathbb{R}_+$, such that $\mathbb{E}[q], \mathbb{E}[q'] < +\infty$, and any random variable $X$ with values in $\mathbb{R}^n$, such that $\mathbb{E}[q|X|^{2-r}], \mathbb{E}[q'|X|^{2-r}] < +\infty$,

$$ G((q'\mathbb{P})_X) - G((q\mathbb{P})_X) = \int_0^1 \left[ \int_{\mathbb{R}^n} \frac{\delta G}{\delta \mu} ((\theta q' + (1-\theta)q)\mathbb{P})_X, x) d[(q'\mathbb{P})_X - (q\mathbb{P})_X](x) \right] d\theta, \quad (55) $$

and, for any random variable $X'$ with values in $\mathbb{R}^n$, such that $\mathbb{E}[q|X'|^{2-r}] < +\infty$,

$$ G((q\mathbb{P})_{X'}) - G((q\mathbb{P})_X) = \int_0^1 \mathbb{E} \left[ \partial_{\mu} G \left( (q\mathbb{P})_{\theta X' + (1-\theta)X}, \theta X' + (1-\theta)X \right) \cdot (X' - X) \right] d\theta. \quad (56) $$

Here is now the main result of this section:

**Corollary 15.** Let Assumptions A1-A5 and A9 be satisfied. Then, there exists a unique saddle point $(\bar{q}, \bar{\psi}) \in \mathcal{Q} \times \mathcal{A}$ to the problem (MFC). Moreover, if a pair $(q, \psi) \in \mathcal{Q} \times \mathcal{A}$ is a solution to the problem (P), then the tuples $(\psi, p, k, X)$, obtained by solving in $\mathcal{A}$ the two decoupled equations in (Opt$_N$) with the terminal condition being specified by

$$ p_T = q_T \partial_{\mu} G((q_T \mathbb{P})_{X_T}, X_T), $$

and $(q, Y, Z)$, obtained by solving in $\mathcal{Q}$ the two decoupled equations in (Opt$_C$) with the terminal condition being specified by

$$ Y_T = \frac{\delta G}{\delta \mu}((q_T \mathbb{P})_{X_T}, X_T), $$

satisfy the optimality conditions in (Opt$_N$) and (Opt$_C$) respectively. Conversely, if $(\psi, p, k, X, q, Y, Z) \in \mathcal{A} \times \mathcal{Q}$ is a solution to (Opt$_C$)-(Opt$_N$) with the terminal condition specified above, then the pair $(\psi, q) \in \mathcal{A} \times \mathcal{Q}$ is the unique solution to the problem (MFC).

*Proof.* As the result is a direct application of Theorem 10, we just need to check that the mapping $G$ defined in (40) satisfies the Assumptions A6-A8 of the previous section.
Step 1: $\mathcal{G}$ verifies A6-A7. Let $(q, X), (q', X') \in \mathcal{G}$ (the definition of $\mathcal{G}$ can be found in Assumption A6), satisfying $\mathbb{E}[q|X'|^{2-r}]$ and $\mathbb{E}[q'|X|^{2-r}] < +\infty$. By (44), we can easily check A6 with

$$ \mathcal{G}(q, X) = G((q\mathbb{P})_X), \quad \delta_\mu\mathcal{G}(q, X) = \frac{\delta G}{\delta\mu}((q\mathbb{P})_X, X), \quad \delta_X\mathcal{G}(q, X) = \partial_\mu G((q\mathbb{P})_X, X). $$

In fact, the main point is to check that $\delta_\mu\mathcal{G}$ and $\delta_X\mathcal{G}$ are the derivatives of $\mathcal{G}$, in the directions $q$ and $X$ respectively, as required in A7. By Lemma 14, we know that

$$ G((q'\mathbb{P})_X) = G((q\mathbb{P})_X) + \int_0^1 \left[ \int_{\mathbb{R}^n} \frac{\delta G}{\delta \mu}((q^\theta\mathbb{P})_X, x) d[(q'\mathbb{P})_X - (q\mathbb{P})_X](x) \right] d\theta, \quad (57) $$

$$ G((q\mathbb{P})_{X'}) = G((q\mathbb{P})_X) + \int_0^1 \mathbb{E} \left[ q \partial_\mu G \left( (q\mathbb{P})_{X^\theta}, X^\theta \right) \cdot (X' - X) \right] d\theta, $$

with the convenient notation $X^\theta := \theta X' + (1-\theta)X$ and $q^\theta := \theta q' + (1-\theta)q$. Let us first prove the first line in A7. For a constant $C \ge 0$ and for $q, X$ and $X'$ satisfying $\mathbb{E}[q|X|^{2-r}], \mathbb{E}[q|X'|^{2-r}] \le C$, we rewrite the second line in (57) as

$$ G((q\mathbb{P})_{X'}) = G((q\mathbb{P})_X) + \mathbb{E} \left[ q \partial_\mu G \left( (q\mathbb{P})_X, X \right) \cdot (X' - X) \right] \quad (58) \\ + \int_0^1 \mathbb{E} \left[ q \left( \partial_\mu G \left( (q\mathbb{P})_{X^\theta}, X^\theta \right) - \partial_\mu G \left( (q\mathbb{P})_X, X \right) \right) \cdot (X' - X) \right] d\theta. $$

By (46) we have

$$ \begin{aligned} & \left| \partial_\mu G \left( (q\mathbb{P})_{X^\theta}, X^\theta \right) - \partial_\mu G \left( (q\mathbb{P})_X, X \right) \right| \\ & \le L_C (1 + |X|^{1-r}) \mathbb{E} \left[ q |X - X'|^{2-r} \right]^{1/(2-r)} + L_C |X' - X|, \end{aligned} $$

from which we deduce, by Cauchy-Schwarz inequality, that

$$ \begin{aligned} & \left| \int_0^1 \mathbb{E} \left[ q \left( \partial_\mu G \left( (q\mathbb{P})_{X^\theta}, X^\theta \right) - \partial_\mu G \left( (q\mathbb{P})_X, X \right) \right) \cdot (X' - X) \right] d\theta \right| \\ & \le L'_C \mathbb{E} [q|X' - X|^2], \end{aligned} $$

for a constant $L'_C$ depending on $L_C$ and $C$. Inserting the above display in (58), this proves the first line in A7. We now establish the second line in A7. where the modulus of continuity $\varpi_L$ might increase. We rewrite the first line in (57) as

$$ G((q'\mathbb{P})_X) = G((q\mathbb{P})_X) + \int_{\mathbb{R}^n} \frac{\delta G}{\delta \mu}((q\mathbb{P})_X, x) d[(q'\mathbb{P})_X - (q\mathbb{P})_X](x) \\ + \int_0^1 \left[ \int_{\mathbb{R}^n} \left( \frac{\delta G}{\delta \mu}((q^\theta\mathbb{P})_X, x) - \frac{\delta G}{\delta \mu}((q\mathbb{P})_X, x) \right) d[(q'\mathbb{P})_X - (q\mathbb{P})_X](x) \right] d\theta. \quad (59) $$

By assumption (45),

$$ \begin{aligned} & \left| \frac{\delta G}{\delta \mu}((q^\theta \mathbb{P})_X, x) - \frac{\delta G}{\delta \mu}((q\mathbb{P})_X, x) \right| \\ & \le (1 + |x|^{2-r}) \varpi (d_{2-r}((q^\theta \mathbb{P})_X, (q\mathbb{P})_X)) \\ & \le (1 + |x|^{2-r}) \varpi \left( \sup_{\varphi} \mathbb{E} [\varphi(X)(q - q')] \right) \\ & \le (1 + |x|^{2-r}) \varpi (\mathbb{E} [(1 + |X|^{2-r})|q - q'|]), \end{aligned} $$
where $\varpi$ in the first line is the modulus of continuity of $\delta G/\delta \mu$ in the first argument, and is (here) independent of $x$ but depends on $q, q'$ and $X$ via $C$. As for $\varphi$ on the second line, it satisfies $|\varphi(x)| \le 1 + |x|^{2-r}$.

Combining the last two displays, we obtain

$$ G((q'\mathbb{P})_X) = G((q\mathbb{P})_X) + \int_{\mathbb{R}^n} \frac{\delta G}{\delta \mu}((q\mathbb{P})_X, x) d[(q'\mathbb{P})_X - (q\mathbb{P})_X](x) \\ + o(\mathbb{E}[(1 + |X|^{2-r})|q - q'|]), $$

where $o(r)/|r| \to 0$ as $r$ tends to 0, uniformly in $q, q'$ and $X$ satisfying the two bounds $\mathbb{E}[|q|X|^{2-r}|], \mathbb{E}[|q'|X|^{2-r}|] \le C$. This proves the second line in A7.

**Step 2:** $G$ verifies A8. Consider again $(q, X), (q', X') \in \mathcal{G}$ such that $\mathbb{E}[|q|X'|^{2-r}|]$ and $\mathbb{E}[|q'|X|^{2-r}|] < +\infty$. By (58) and then (48), we have

$$ G((q\mathbb{P})_{X'}) = G((q\mathbb{P})_X) + \mathbb{E}[q\partial_\mu G((q\mathbb{P})_X, X) \cdot (X' - X)] \\ + \int_0^1 \frac{1}{\theta} \mathbb{E}[q(\partial_\mu G((q^\theta\mathbb{P})_X, X^\theta) - \partial_\mu G((q\mathbb{P})_X, X)) \cdot (X^\theta - X)] d\theta \\ \ge G((q\mathbb{P})_X) + \mathbb{E}[q\partial_\mu G((q\mathbb{P})_X, X) \cdot (X' - X)], $$

which proves the second condition in A8. In order to establish the first condition in A8, we notice that, in (59),

$$ (q^\theta \mathbb{P})_X = \theta(q' \mathbb{P})_X + (1 - \theta)(q \mathbb{P})_X = (q \mathbb{P})_X + \theta((q' \mathbb{P})_X - (q \mathbb{P})_X), $$

and then,

$$ G((q'\mathbb{P})_X) = G((q\mathbb{P})_X) + \int_{\mathbb{R}^n} \frac{\delta G}{\delta \mu}((q\mathbb{P})_X, x) d[(q'\mathbb{P})_X - (q\mathbb{P})_X](x) \\ + \int_0^1 \int_{\mathbb{R}^n} \frac{1}{\theta} \left[ \frac{\delta G}{\delta \mu}((q^\theta\mathbb{P})_X, x) - \frac{\delta G}{\delta \mu}((q\mathbb{P})_X, x) \right] d[(q^\theta\mathbb{P})_X - (q\mathbb{P})_X](x) d\theta \\ \le G((q\mathbb{P})_X) + \int_{\mathbb{R}^n} \frac{\delta G}{\delta \mu}((q\mathbb{P})_X, x) d[(q'\mathbb{P})_X - (q\mathbb{P})_X](x), $$

where the last line follows by the monotonicity assumption (47). $\square$

<figure><img src="image_3.png" /></figure>

**Perspectives** *Common noise.* Our approach, based on the stochastic maximum principle, would allow us to introduce a common noise into the model in a direct manner. Similar to [31], we can think of an additive white noise manifesting in the form of an extra term $\sigma^0 dW^0$ in the dynamics of $X$, where $W^0$ is a Brownian motion independent of $(W, \eta)$. Alternatively, we could randomize the coefficients independently of $(W, \eta)$. In any case, this additional source of randomness could be represented by tensorizing the space $\Omega$ (which carries the idiosyncratic noises) with a new space $\Omega^0$ (which carries the common noise). This approach is used in [36].

To incorporate this, the following changes would be necessary:

* The terminal cost would read

$$ \mathbb{E}^0[\mathcal{G}(q_T(\omega^0, \cdot), X_T(\omega^0, \cdot))] = \mathbb{E}^0[G((q_T(\omega^0, \cdot)\mathbb{P})_{X_T(\omega^0, \cdot)})], $$

for any element $\omega^0 \in \Omega^0$. This accounts for the fact that the common noise induces a conditioning.
* Assuming, without significant loss of generality, that the filtration on $\Omega^0$ is generated by a Brownian motion (denoted $W^0$), all the backward equations would include an additional penalization term in the form of a stochastic integral with respect to $W^0$, i.e., $\int_0^T Z_s^0 \cdot dW_s^0$. If the filtration were not Brownian, the penalization could instead be written as a (possibly discontinuous) martingale, which would make the model more complex to study.

* If the model were extended to incorporate risk aversion with respect to the common noise, the dynamics of $q^0$ would include an additional term of the form $q_s^0 Z_s^{0,*} \cdot dW_s^0$. As a consequence, the driver of the BSDE for $Y$ would also depend on the additional variable $Z^0$, where $Z^0$ arises from the martingale representation above. Accordingly, both the adjoint process and the Hamiltonian would have to be modified to account for this additional dependence.

*N-particles system.* A natural question is how the robust mean field model arises as the limit of an *N*-particle control problem. A thorough and rigorous analysis of this convergence process is beyond the scope of the present article and is left for future work. Nevertheless, we hope that the formal arguments provided in the second example of Subsection 3.2, as well as in the forthcoming examples presented in Subsection 4.2, will help the reader to identify, at least at an intuitive level, the underlying mechanisms from which the mean field model can be expected to emerge.

## 4.2 Examples

In this paragraph, we provide two examples that lead to a robust mean field control problem.

**Feynman-Kac path particle models.** Inspired by the monograph [48], we consider a large system of *N* weakly interacting *d*-dimensional particles, with Gibbs distributions on the path space $\mathcal{C}([0, T], \mathbb{R}^n)^N$:

$$ \exp\left(\beta\left[\frac{1}{2N}\sum_{i,j=1}^{N} G\left(X_{T}^{i, \psi^{i}}, X_{T}^{j, \psi^{j}}\right) + \frac{1}{2N}\sum_{i,j=1}^{N} \int_{0}^{T} F\left(X_{t}^{i, \psi^{i}}, X_{t}^{j, \psi^{j}}\right) dt + \frac{1}{2}\sum_{i=1}^{N} \int_{0}^{T} |\psi_{t}^{i}|^{2} dt\right]\right) \cdot \mathbb{P}^{\times N}, $$

where $\beta > 0$ and $\mathbb{P}$ is the Wiener measure on $\Omega := \mathcal{C}([0, T], \mathbb{R}^n)$, $\psi^i : \mathcal{C}([0, T], \mathbb{R}^n)^N \to \mathbb{R}^d$ is a progressively-measurable control for each $i \in \{1, \dots, N\}$ and

$$ X_t^{i, \psi^i}(\omega^1, \dots, \omega^N) = \omega_t^i + \int_0^t \psi_s^i(\omega^1, \dots, \omega^N) ds, \quad t \in [0, T]. $$

The goal is then to minimize, with respect to $(\psi^1, \dots, \psi^N)$, the free energy given (up to a logarithmic transformation) by

$$ \mathbb{E}^{\times N}\left[\exp\left(\beta\left[\frac{1}{2N}\sum_{i,j=1}^{N} G\left(X_{T}^{i, \psi^{i}}, X_{T}^{j, \psi^{j}}\right) + \frac{1}{2N}\sum_{i,j=1}^{N} \int_{0}^{T} F\left(X_{t}^{i, \psi^{i}}, X_{t}^{j, \psi^{j}}\right) dt + \frac{1}{2}\sum_{i=1}^{N} \int_{0}^{T} |\psi_{t}^{i}|^{2} dt\right]\right)\right]. $$
In order to simplify, we assume below that the running cost $F$ is equal to 0, but the analysis would be the same if $F$ were not trivial.

Thanks to Donsker-Varadhan's formula (see (15)) for a remainder, the free energy can be rewritten in the form

$$ \sup_{q^{(N)}} \left\{ \beta \mathbb{E}^{\times N} \left[ q_T^{(N)} \left( \frac{1}{2N} \sum_{i,j=1}^{N} G(X_T^{i, \psi^i}, X_T^{j, \psi^j}) + \frac{1}{2} \sum_{i=1}^{N} \int_0^T |\psi_t^i|^2 dt \right) \right] - H(q^{(N)}|\mathbb{P}^{\times N}|\mathbb{P}^{\times N}) \right\}, \quad (60) $$

where $q^{(N)}$ is taken in the space of densities on $\Omega^{\times N}$ with a finite entropy. We observe that the normalization in the potential is consistent with that used in the paragraph on risk measures in Subsection 3.2. This therefore constitutes a nonlinear version (in the sense that the potential now depends on the empirical measure through a second-order functional) of the previous example; for simplicity, this example is also presented in the case of uncontrolled volatility.

*Characterization of the saddle-point.* If $G$, viewed as a real-valued function on $\mathbb{R}^n \times \mathbb{R}^n$, is smooth, convex and at most of quadratic growth, then Theorem 10 applies to the minimization of the above quantity. The saddle point of the min-max problem (over $q^{(N)}$ and $(\psi^1, \dots, \psi^N)$) can be characterized via a 6-tuple

$$ (Y^{(N)}, Z^{(N)}, X^{(N)}, p^{(N)}, k^{(N)}, q^{(N)}), $$

with

$$ \begin{aligned} Z^{(N)} &= (Z^{(N),i})_{i=1,\dots,N}, & X^{(N)} &= (X^{(N),i})_{i=1,\dots,N}, \\ p^{(N)} &= (p^{(N),i})_{i=1,\dots,N}, & k^{(N)} &= (k^{(N),i,j})_{i,j=1,\dots,N}, \end{aligned} $$

solution of (using the notation $B_t^j(\omega^1, \dots, w^N) := \omega_t^j$, for $j = 1, \dots, N$ and $t \in [0, T]$)

$$ \left\{ \begin{aligned} -dY_t^{(N)} &= \left( \frac{1}{2} \sum_{j=1}^{N} |Z^{(N),j}|^2 + \frac{\beta}{2} \sum_{j=1}^{N} |\psi_t^j|^2 \right) dt - \sum_{j=1}^{N} Z_t^{(N),j} \cdot dB_t^j, \\ dX_t^{(N),i} &= \psi_t^i dt + dB_t^i, \\ -dp_t^{(N),i} &= -\sum_{i=1}^{N} k_t^{(N),i,j} \cdot dB_t^j, \\ dq_t^{(N)} &= q_t^{(N)} \sum_{i=1}^{N} Z_t^{(N),i} \cdot dB_t^i, \end{aligned} \right. \quad (61) $$

for $t \in [0, T]$, with the optimality condition $\psi_t^i = -[\beta q_t^{(N)}]^{-1} p_t^{(N),i}$ and the boundary conditions

$$ \left\{ \begin{aligned} Y_T^{(N)} &= \frac{\beta}{2N} \sum_{i,j=1}^{N} G(X_T^{(N),i}, X_T^{(N),j}) \\ p_T^{(N),i} &= q_T^{(N)} \frac{\beta}{2N} \sum_{j=1}^{N} \left[ \partial_x G(X_T^{(N),i}, X_T^{(N),j}) + \partial_y G(X_T^{(N),j}, X_T^{(N),i}) \right]. \end{aligned} \right. \quad (62) $$

Similar to the discussion initiated in Subsection 3.2, the question here is to understand, at least informally, how the above system is connected to the mean field control problem described in Subsection 4.1. To better appreciate the intuitive arguments that we present, it is worth mentioning from the analysis carried out in Subsection 5.2 (see in particular Lemma 27) that the solution of the BSDE (24) is understood via the product $qY = (q_t Y_t)_{t \in [0,T]}$. This prompts us to consider, here, the product
$q^{(N)}Y^{(N)}$. The aforementioned Lemma 27 says that $q^{(N)}Y^{(N)}$ is a semi-martingale under the probability measure $q_T^{(N)}\mathbb{P}$, satisfying

$$
\left\{
\begin{aligned}
-d \left[ q_t^{(N)} Y_t^{(N)} \right] &= \left( -\frac{1}{2} \sum_{j=1}^{N} q_t^{(N)} |Z^{(N),j}|^2 + \frac{\beta}{2} \sum_{j=1}^{N} q_t^{(N)} |\psi_t^j|^2 \right) dt \\
&\quad - q_t^{(N)} (1 + Y_t^{(N)}) \sum_{j=1}^{N} Z_t^{(N),j} \cdot dB_t^j, \\
q_T^{(N)} Y_T^{(N)} &= \frac{\beta}{2N} q_T^N \sum_{i,j=1}^{N} G \left( X_T^{(N),i}, X_T^{(N),j} \right).
\end{aligned}
\right.
\quad (63)
$$

*Corresponding robust MFC problem.* In parallel, consider the robust mean field con-
trol problem (MFC), which we recall below for convenience:

$$
\inf_{\psi} \sup_q \left\{ \mathbb{E} \left[ \frac{\beta}{2} \int_{\mathbb{R}^n} G(x, y) (q_T \mathbb{P}_{X_T^\psi})^{\otimes 2} (dx, dy) + \frac{\beta}{2} \int_0^T q_s |\psi_s|^2 ds \right] - H(q_T \mathbb{P} | \mathbb{P}) \right\},
$$

where for simplicity we do not specify the sets of admissibility in which $q$ and $\psi$
are taken. Generally speaking, this problem is defined on the original probability
space $(\Omega, \mathcal{F}, \mathbb{P})$ equipped with the Brownian motion $W$, but we can consider, for each
$i = 1, \dots, N$, the same problem but on the $i$th factor $\Omega^N$ and thus with respect to the
Brownian motion $B^i$ instead of $W$. For each $i = 1, \dots, N$, we then call $(\tilde{q}^i, \tilde{X}^i, \tilde{\psi}^i)$
the saddle point of the corresponding problem, which exists and is unique under
the assumptions of Corollary 15. At this stage, these assumptions are taken for
granted, but we will discuss its meaning in more depth at the end of this paragraph.
Importantly, we observe that $(\tilde{q}^i, \tilde{X}^i, \tilde{\psi}^i)$ is a function of the sole $\omega^i$. The Pontryagin
system characterizing $(\tilde{q}^i, \tilde{X}^i, \tilde{\psi}^i)$ reads

$$
\left\{
\begin{aligned}
-d \tilde{Y}_t^i &= \left( \frac{1}{2} |\tilde{Z}_t^i|^2 + \frac{\beta}{2} |\tilde{\psi}_t^i|^2 \right) dt - \tilde{Z}_t^i \cdot dB_t^i, \\
d \tilde{X}_t^i &= \tilde{\psi}_t^i dt + dB_t^i, \\
-d \tilde{p}_t^i &= - \tilde{k}_t^i \cdot dB_t^i, \\
d \tilde{q}_t^i &= \tilde{q}_t^i \tilde{Z}_t^i \cdot dB_t^i,
\end{aligned}
\right.
\qquad (64)
$$

for $t \in [0, T]$, with the optimality condition $\tilde{\psi}_t^i = -[\beta \tilde{q}_t^i]^{-1} \tilde{p}_t^i$ and the boundary conditions

$$
\left\{
\begin{aligned}
\tilde{Y}_T^i &= \frac{\beta}{2} \int_{\mathbb{R}^n} \left[ G(\tilde{X}_T^i, z) + G(z, \tilde{X}_T^i) \right] d(\tilde{q}_T^i \mathbb{P}) \tilde{X}_T^i(z) \\
\tilde{p}_T^i &= \frac{\beta}{2} \tilde{q}_T^i \int_{\mathbb{R}^n} \left[ \partial_x G(\tilde{X}_T^i, z) + \partial_y G(z, \tilde{X}_T^i) \right] d(\tilde{q}_T^i \mathbb{P}) \tilde{X}_T^i(z).
\end{aligned}
\right.
\qquad (65)
$$

We then let

$$
\tilde{Y}_t^{(N)} = \sum_{i=1}^{N} \tilde{Y}_t^i, \quad \tilde{q}_t^{(N)} = \prod_{i=1}^{N} q_t^i, \quad t \in [0, T],
$$

and, following (63), we consider the process $\tilde{q}^{(N)}\tilde{Y}^{(N)}$. It satisfies

$$
\left\{
\begin{aligned}
-d \left[ \tilde{q}_t^{(N)} \tilde{Y}_t^{(N)} \right] &= \left( -\frac{1}{2} \tilde{q}_t^{(N)} \sum_{i=1}^{N} |\tilde{Z}_t^i|^2 + \frac{\beta}{2} \tilde{q}_t^{(N)} \sum_{i=1}^{N} |\tilde{\psi}_t^i|^2 \right) dt \\
&\quad - \tilde{q}_t^{(N)} (1 + Y_t^{(N)}) \sum_{i=1}^{N} \tilde{Z}_t^i \cdot dB_t^i, \\
\tilde{q}_T^{(N)} \tilde{Y}_T^{(N)} &= \frac{\beta}{2} \tilde{q}_T^{(N)} \sum_{i=1}^{N} \int_{\mathbb{R}^n} \left[ G(\tilde{X}_T^i, z) + G(z, \tilde{X}_T^i) \right] d(\tilde{q}_T^i \mathbb{P}) \tilde{X}_T^i(z).
\end{aligned}
\right.
\quad (66)
$$
Connecting the two problems. Of course, in the above right-hand side, $(\tilde{q}_T^i \mathbb{P})_{\tilde{X}_T^i}$ is independent of $i$ and can be replaced by $(\tilde{q}_T^1 \mathbb{P})_{\tilde{X}_T^1}$. For simplicity, we remove below the index 1 and merely write $(\tilde{q}_T \mathbb{P})_{\tilde{X}_T}$. The connection between the form of the boundary condition for $\tilde{q}_T^{(N)} \tilde{Y}_T^{(N)}$ in (66) and the form of the boundary condition for $q_T^{(N)} Y_T^{(N)}$ in (63) can be better understood by applying the weak law of large numbers under the probability $\tilde{q}_T^{(N)} \mathbb{P}$. Indeed, since the random variables $\tilde{X}_T^1, \dots, \tilde{X}_T^N$ are independent under $\tilde{q}_T^{(N)}$, with $(\tilde{q} \mathbb{P})_{\tilde{X}_T}$ as common distribution, and because $\int_{\mathbb{R}^n \times \mathbb{R}^n} G(x, y) (\tilde{q}_T \mathbb{P}_{\tilde{X}_T})^{\otimes 2}(dx, dy) < +\infty$ (as a consequence of Lemma 40), we have

$$ \lim_{N \to +\infty} \mathbb{E} \left[ \tilde{q}_T^{(N)} \left| \frac{1}{N^2} \sum_{i,j=1}^{N} G(\tilde{X}_T^i, \tilde{X}_T^j) - \frac{1}{2N} \sum_{i=1}^{N} \int_{\mathbb{R}^n} \left( G(\tilde{X}_T^i, z) + G(z, \tilde{X}_T^i) \right) (\tilde{q}_T \mathbb{P})_{\tilde{X}_T} (dz) \right| \right] = 0. \quad (67) $$

Pay attention to the fact that the boundary conditions for $\tilde{q}_T^{(N)} \tilde{Y}_T^{(N)}$ in (66) and $q_T^{(N)} Y_T^{(N)}$ in (63) are of order $N$, whereas the two terms in the above difference are of order 1. That said, the above display shows that $\tilde{q}_T^{(N)} \tilde{Y}_T^{(N)}$ satisfies a boundary condition similar to the one satisfied by $q_T^{(N)} Y_T^{(N)}$ in (63), up to a remainder of order $o(N) = \varepsilon_N N$ with $\varepsilon_N$ converging to 0 in $L^1$ under $\mathbb{P}$. This makes it possible to view the process $\tilde{q}^{(N)} \tilde{Y}^{(N)}$ as a ‘nearly solution’ of the equation satisfied by $q^{(N)} Y^{(N)}$, but with $\psi^i$ replaced by $\tilde{\psi}^i$ in the generator of the backward component, and $(X_T^{(N),1}, \dots, X_T^{(N),N})$ replaced by $(\tilde{X}_T^1, \dots, \tilde{X}_T^N)$ in the terminal condition.

By the same argument, one can multiply each $\tilde{p}^i$ in (64), for $i \in \{1, \dots, N\}$, by $\tilde{q}^{(N)}(\tilde{q}^i)^{-1}$. The resulting process $\tilde{q}^{(N)}(\tilde{q}^i)^{-1}\tilde{p}^i$ remains a local martingale, and its boundary condition satisfies, up to a new remainder of order $o(N)$, a boundary condition similar to the one satisfied by $p^{(N),i}$ in (62). This shows that the process $\tilde{q}^{(N)}(\tilde{q}^i)^{-1}\tilde{p}^i$ is a ‘nearly solution’ of the equation satisfied by $p^{(N),i}$, but with $(X_T^{(N),1}, \dots, X_T^{(N),N})$ replaced by $(\tilde{X}_T^1, \dots, \tilde{X}_T^N)$. Next, rewriting the identity

$$ \tilde{\psi}_t^i = -(\beta \tilde{q}_t^i)^{-1} \tilde{p}_t^i $$

in the form

$$ \tilde{\psi}_t^i = -(\beta \tilde{q}_t^{(N)})^{-1} \tilde{q}_t^{(N)} (\tilde{q}_t^i)^{-1} \tilde{p}_t^i, $$

we observe that $\tilde{\psi}^i$ can be expressed in terms of $\tilde{q}^{(N)}$ and $\tilde{q}^{(N)}(\tilde{q}^i)^{-1}\tilde{p}^i$ via the same function that allows one to express $\psi^i$ in terms of $q^{(N)}$ and $p^{(N),i}$.

Altogether, this shows that the tuple

$$ (\tilde{Y}^{(N)}, \tilde{Z}^1, \dots, \tilde{Z}^N, \tilde{X}^1, \dots, \tilde{X}^N, \tilde{q}^{(N)}, \tilde{q}^{(N)}(\tilde{q}^1)^{-1}\tilde{p}^1, \dots, \tilde{q}^{(N)}(\tilde{q}^N)^{-1}\tilde{p}^N) $$

is a nearly solution of the forward-backward system solved by the tuple

$$ (Y^{(N)}, Z^{(N),1}, \dots, Z^{(N),N}, X^{(N),1}, \dots, X^{(N),N}, p^{(N),1}, \dots, p^{(N),N}, q^{(N)}), $$

which makes the connection between (60) and the robust MFC problem.

Assumptions on G. We now comment on the assumptions needed to apply Corollary 15 in the analysis of the robust MFC problem.
Quite surprisingly, although the convexity of $G$ suffices to apply Theorem 10 in order to characterize the saddle points of (60) –because the map $(q, x_1, \dots, x_n) \mapsto q \sum_{i,j=1} G(x_i, x_j)$ is linear in $q$ and convex in $(x_1, \dots, x_n)^-$, it does not suffice to apply Corollary 15. Indeed, the map

$$ \mu \in \mathcal{P}_2(\mathbb{R}^n) \mapsto \int_{(\mathbb{R}^n)^2} G(x, y) \, d\mu^{\times 2}(x, y) $$

is displacement convex –as a consequence of the convexity of $G$– but may fail to be flat concave. For instance, if $G(x, y) = \varphi(x)\varphi(y)$, for some non-negative convex function $\varphi: \mathbb{R}^n \to \mathbb{R}$, then $G$ is convex. However, for any $\mu \in \mathcal{P}_2(\mathbb{R}^n)$,

$$ \Gamma(\mu) := \int_{(\mathbb{R}^n)^2} G(x, y) \, d\mu^{\times 2}(x, y) = \left( \int_{\mathbb{R}^n} \varphi(x) \, d\mu(x) \right)^2, $$

which shows that the function $\Gamma: \mu \in \mathcal{P}_2(\mathbb{R}^n) \mapsto \Gamma(\mu)$ is flat convex.

Additional conditions are therefore required to apply Corollary 15. Although, in the previous paragraph, we did not provide a complete proof but only some intuition to justify the passage from (61)–(62) to (64)–(65), we believe that the need for extra assumptions to ensure existence and uniqueness of a solution to the mean field problem reflects the price to pay for passing to the limit (as $N \to \infty$) in the original problem (60).

Following the discussion in the previous subsection, we now provide an example of a class of convex functions $G$ for which $\mu \in \mathcal{P}_2(\mathbb{R}^n) \mapsto \Gamma(\mu)$ is flat concave. If $G$ itself is not convex but $\Gamma$ is flat concave, one may replace $G$ by the function $(x, y) \mapsto G(x, y) + a(|x|^2 + |y|^2)$, for $a > 0$ large enough, in order to enforce displacement convexity while preserving flat concavity. Thus, the remaining task is to provide an example of a function $G$ for which $\Gamma$ is concave. One such example is given by any function of the form

$$ (x, y) \mapsto - \sum_{i=1}^{k} \lambda_i h_i(x) h_i(y), $$

where $k \ge 0$, $\lambda_i > 0$, and $h_i$ is a smooth function with bounded derivative, for each $i = 1, \dots, k$.

**Robust approximation of a Gibbs measure on the path space.** We now present another example, building on the previous one, but which corresponds to the robustification, with respect to the central planner's strategy $\psi$, of a control problem defined on Nature's state $q$. It is inspired by recent works on stochastic algorithms (a more precise list of references is given below).

Given a potential $\mathcal{W}$ defined on the Wiener path space $\Omega := C([0, T], \mathbb{R}^d)$, one wants to approximate the normalized Gibbs probability measure

$$ \mathbb{P}^{\text{target}} := \frac{1}{\mathcal{Z}} \exp(-\mathcal{W}) \mathbb{P}, \quad \text{with} \quad \mathcal{Z} := \mathbb{E}[\exp(-\mathcal{W})], \qquad (68) $$

by the law of a controlled diffusion process of the form (say to simplify that $X_0^\phi = 0$)

$$ dX_t^\phi = \phi_t dt + dB_t, \quad t \in [0, T]. \qquad (69) $$

(Here, we use the notation $B$ instead of $\mathcal{W}$ for the canonical process, with is a Brownian motion under the Wiener measure $\mathbb{P}$; this to avoid confusion with the
potential W.) Typically, W(ω), where ω = (ω_t)_{t∈[0,T]} denotes the generic element of the space Ω, is chosen as

$$ W(\omega) = G(\omega_{\tau}) + \int_{0}^{\tau} F(\omega_{t})dt, \qquad (70) $$

where τ is the realization, at ω, of a stopping time, usually chosen as the first exit time of ω from a given domain. Obviously, the structure of W described above is especially adapted to Markovian dynamics, which leads us to choose, in this situation, the control (φ_t)_{t∈[0,T]} in a Markov feedback form (φ_t = Φ(t, X_t))_{t∈[0,T]}.

**Exact solution to the targeting problem.** In fact, under standard assumptions covering the Markovian framework, one can find a control ¯ψ such that the law P ◦ (X ¯ψ)⁻¹ of (X_t ¯ψ)_{t∈[0,T]} under P perfectly matches the target distribution P<sup>target</sup>, i.e.

$$ P \circ (X^{\bar{\psi}})^{-1} = P^{\text{target}}. \qquad (71) $$

Assume indeed that one can solve the FBSDE system (for simplicity, we do not specify the spaces in which solutions are taken because this would be useless for the rest of the paragraph)

$$ \left\{ \begin{array}{l} -dY_t = \frac{1}{2}|Z_t|^2 dt - Z_t \cdot dB_t, \quad Y_T = W(X), \\ dX_t = -Z_t dt + dB_t, \quad X_0 = 0. \end{array} \right. \qquad (72) $$

Then, the backward equation can be reformulated as

$$ \exp(-W(X)) \mathcal{E}_T \left( \int_0^t Z_r \cdot dB_r \right) = \exp(-Y_0), \qquad (73) $$

where we recall that $Y_0$ is deterministic (as it is the initial value of the BSDE in (71)). We deduce that, for any bounded and measurable function $\Phi: C([0, T], \mathbb{R}^d) \to \mathbb{R}$,

$$ \mathbb{E} \left[ \exp \left( Y_0 - W(X) \right) \mathcal{E}_T \left( \int_0^t Z_r \cdot dB_r \right) \Phi(X) \right] = \mathbb{E} [\Phi(X)]. $$

Thanks to the forward equation in (72) and provided that the Girsanov transformation can be rigorously applied, we observe that the left-hand side is equal to $\mathbb{E}[e^{Y_0-W}\Phi]$, because the law of $X$ under $\mathcal{E}_T(\int_0^t Z_r \cdot dB_r)P$ is the same as the law of $B$ under $P$. Since the function $\Phi$ is arbitrary, this proves that the law of $X$ under $P$ is the Gibbs measure $P^{\text{target}}$, as required.

The analysis of the FBSDE (72) is standard in the Markovian setting. In this case, there exists a function $\Psi$, given as the solution of an auxiliary nonlinear parabolic PDE (see [35, Chapter 3]), such that $(Z_t = -\Psi(t, X_t))_{t \in [0,T]}$. In particular, one can express $P \circ X^{-1}$ as $\mathcal{E}_T(\int_0^t \psi_t \cdot dB_t)P$, where $\psi_t(\omega) = \Psi(t, \omega_t)$ (the latter is different from $-Z_t(\omega) = \Psi(t, X_t(\omega))$).

**Reformulation as a Nature optimization problem.** Interestingly, this targeting problem can be recast as a minimization problem in the space of probability measures. Indeed, using Donsker-Varadhan's lemma, it holds, for any control $(\phi_t)_{t \in [0,T]}$ such that the measure $P^\phi := \mathcal{E}_T(\int_0^t \phi_t \cdot dB_t)P$ has a relative finite entropy $H(P^\phi|P)$,

$$ -\ln(\mathbb{E}[\exp(-W)]) \leq \mathbb{E}\left[\frac{dP^\phi}{dP}W\right] + H(P^\phi|P). \qquad (74) $$
When $\phi$ is equal to $\psi$, the right-hand side becomes

$$ 
\begin{aligned}
\mathbb{E} \left[ \frac{d\mathbb{P}^{\psi}}{d\mathbb{P}} \mathcal{W} \right] + \mathrm{H} \left( \mathbb{P}^{\psi} \middle| \mathbb{P} \right) &= \mathbb{E} \left[ \mathcal{W} \left( X^{\psi} \right) \right] + \mathrm{H} \left( \mathbb{P} \circ (X^{\psi})^{-1} \middle| \mathbb{P} \right) \\
&= \mathbb{E} \left[ \mathcal{W} (X) + \frac{1}{2} \int_{0}^{T} |Z_{t}|^{2} dt \right],
\end{aligned}
$$

which is equal (thanks to (72)) to $\mathbb{E}[Y_0] = \mathbb{E}[\exp(-\mathcal{W})]$ (with the latter following from (73) and a new application of Girsanov's formula). Therefore, $\psi$ solves the minimization problem

$$ \inf_{\phi} \left\{ \mathbb{E} \left[ \frac{d\mathbb{P}^{\phi}}{d\mathbb{P}} \mathcal{W} \right] + \mathrm{H} \left( \mathbb{P}^{\phi} \middle| \mathbb{P} \right) \right\}, \quad (75) $$

hence connecting the targeting problem (71) and the minimization problem (75). For example, these two problems are tackled in control based importance sampling methods for diffusion processes (see for instance [81, 90] and [92, Chapter 6], from which we borrowed part of the presentation) and in diffusion based models for generative adversarial networks (see for instance the fine tuning analysis provided in [94, 96] and the MFC interpretation of score matching approaches [98]).

In fact, the connection between the targeting problem (71) and the minimization problem (75) can be better understood by reformulating the latter, and then by observing that $\psi$ solves

$$ \inf_{\phi} \mathrm{H} \left( \mathbb{P}^{\phi} \middle| \mathcal{Z} e^{-\mathcal{W}} \mathbb{P} \right), \quad (76) $$

the optimal value being equal to 0. Above, we recall that $\mathcal{Z} = \mathbb{E}[\exp(-\mathcal{W})]$. Rephrased in our framework, the density $d\mathbb{P}^{\phi}/d\mathbb{P}$ appearing in both (75) and (76) must be identified with Nature's state $q_T$ at terminal time. Therefore, the two problems can be regarded as optimal control problem for Nature (even though the original problem (71)) is formulated as a control problem for the player).

*Robust version.* Now, consistently with the robust approach introduced in this work, one can think of a situation where there is some uncertainty on the precise form of the potential $\mathcal{W}$ in the targeting measure $e^{-\mathcal{W}} \cdot \mathbb{P}$ in (68). We thus change $\mathcal{W}$ into $\mathcal{W}(X^{\psi})$, with $X^{\psi}$ as in (69). Intuitively, this says that there is some uncertainty on the ‘observed values’ of $\mathcal{W}$, say for instance because the potential is computed along an approximation of the canonical process (as in the stochastic algorithms cited above).

Next, we introduce two related min-max problems. The first problem is a robust version of (75):

$$ \inf_{q \in \mathcal{Q}} \sup_{\psi \in \mathcal{A}} \left\{ \mathrm{H}(\mathbb{Q}^q | \mathbb{P}) + \mathbb{E} \left[ q_T \mathcal{W}(X^{\psi}) - \frac{1}{2} q_T \int_0^T |\psi_t|^2 dt \right] \right\}, \quad \mathbb{Q}^q = q_T \mathbb{P}, \quad (77) $$

which is quite similar to the second example in Subsection 3.2 and to the first example in this subsection. The second one is a robust version of (76):

$$ \inf_{q \in \mathcal{Q}} \sup_{\psi \in \mathcal{A}} \left\{ \mathrm{H} \left( \mathbb{Q}^q \middle| \mathbb{G}^\psi \right) - \frac{1}{2} \mathbb{E} \left[ q_T \int_0^T |\psi_t|^2 dt \right] \right\}, \quad \mathbb{G}^\psi = \frac{1}{\mathcal{Z}^\psi} e^{-\mathcal{W}(X^\psi)} \mathbb{P}, \quad (78) $$

with $\mathcal{Z}^\psi := \mathbb{E}[\exp(-\mathcal{W}(X^\psi))]$.
The two problems are not the same because of the presence of the normalization constant in the second one. In both situations, the penalty term $-\frac{1}{2}q_T \int_0^T |\psi_t|^2 dt$ should be regarded as a regularization of the problem that just amounts in replacing the original potential $W(X^\psi)$ by the effective one $W(X^\psi) - \frac{1}{2}q_T \int_0^T |\psi_t|^2 dt$. Also, the term $W(X^\psi)$ itself can be chosen as a function of the realization of the path but also of its statistical distribution under $\mathbb{P}$. For instance, a typical choice, consistent with the previous example on Feynman-Kac models, is

$$W(X_T^\psi) = W(X_T^\psi, \mathbb{P} \circ (X_T^\psi)^{-1}).$$

*Explicit computation.* When $\mathcal{W}$ is (say) a concave function of the terminal state, the first problem (77) satisfies our concavity-convexity conditions and there exists a (unique) saddle point to the min-max problem. To better illustrate the result, we just focus on the case when $d=1$ and $\mathcal{W}(\omega) = -\beta\omega_T$, for some parameter $\beta \in \mathbb{R}$. Then, very similar to the risk averse portfolio management problem addressed in Subsection 3.2, the unique saddle point $(\bar{q}, \bar{\psi})$ can be found explicitly. Here,

$$\bar{\psi}_t = -\beta, \qquad (79)$$

and

$$\bar{q}_T = \frac{1}{\bar{\mathcal{Z}}} \exp \left( -\mathcal{W}(X^{\bar{\psi}}) - \frac{1}{2} \int_0^T |\bar{\psi}_t|^2 dt \right) = \frac{1}{\bar{\mathcal{Z}}} \exp \left( \beta(B_T - T\beta) - \frac{1}{2}T\beta^2 \right), \quad (80)$$

where

$$\bar{\mathcal{Z}} = \mathbb{E} \left[ \exp \left( \beta(B_T - T\beta) - \frac{1}{2}T\beta^2 \right) \right] = \exp(-T\beta^2). \qquad (81)$$

The second problem (78) is more difficult to handle. Using the explicit form of $\mathcal{Z}^\psi$ (and expanding the various logarithms inside the definition of the entropy), it can be rewritten as

$$\inf_{q \in \mathcal{Q}} \sup_{\psi \in \mathcal{A}} \left\{ \mathbb{E} \left[ q_T \mathcal{W}(X^\psi) - \frac{1}{2} q_T \int_0^T |\psi_t|^2 dt \right] + \ln \left( \mathbb{E} \left[ \exp \left( -\mathcal{W}(X^\psi) \right) \right] \right) + \mathrm{H}(\mathbb{Q}^q | \mathbb{P}) \right\}.$$

Here we recall from (74) that the cumulant generating function appearing on the second line of the right-hand can be reformulated as the supremum (over $\phi$) of $\mathbb{E}[-(d\mathbb{P}^\phi/d\mathbb{P})\mathcal{W}(X^\psi)] - \mathrm{H}(\mathbb{P}^\phi|\mathbb{P})$. In particular, if $\mathcal{W}$ is concave, then $-\mathcal{W}$ is convex (in $X_T^\psi$) and the supremum (over $\phi$) is also convex in $X_T^\psi$. As a result, it is not clear whether the cost is concave in $\psi$, which prevents any application of the results obtained in the article.

Nevertheless, one can use the saddle point $(\bar{q}, \bar{\psi})$ obtained for the problem (77) in order to gain some insight into the problem (78). Indeed, by the saddle point property, we have

$$\mathbb{E} \left[ \bar{q}_T \mathcal{W} (X^{\bar{\psi}}) - \frac{1}{2} \bar{q}_T \int_0^T |\bar{\psi}_t|^2 dt \right] = \sup_{\psi} \mathbb{E} \left[ \bar{q}_T \mathcal{W} (X^\psi) - \frac{1}{2} \bar{q}_T \int_0^T |\psi_t|^2 dt \right].$$
By concavity of the cost function (with respect to $\psi$), one deduces that, for any $\psi$

$$ \mathbb{E} \left[ \bar{q}_T \mathcal{W} (X^{\bar{\psi}}) - \frac{1}{2} \bar{q}_T \int_0^T |\bar{\psi}_t|^2 dt \right] \geq \mathbb{E} \left[ \bar{q}_T \mathcal{W} (X^{\psi}) - \frac{1}{2} \bar{q}_T \int_0^T |\psi_t|^2 dt \right] \\ + \frac{1}{2} \mathbb{E} \left[ \bar{q}_T \int_0^T |\psi_t - \bar{\psi}_t|^2 dt \right]. $$

And then, by expanding the logarithm inside the definition of the entropy,

$$ \begin{aligned} & H(\mathbb{Q}^{\bar{q}} | \mathbb{G}^{\bar{\psi}}) - \frac{1}{2} \mathbb{E} \left[ \bar{q}_T \int_0^T |\bar{\psi}_t|^2 dt \right] \\ &= \mathbb{E} \left[ \bar{q}_T \mathcal{W} (X^{\bar{\psi}}) - \frac{1}{2} \bar{q}_T \int_0^T |\bar{\psi}_t|^2 dt \right] + \ln \left( \mathbb{E} \left[ \exp \left( -\mathcal{W}(X^{\bar{\psi}}) \right) \right] \right) + H(\mathbb{Q}^{\bar{q}} | \mathbb{P}) \\ &\geq H(\mathbb{Q}^{\bar{q}} | \mathbb{G}^{\psi}) - \frac{1}{2} \mathbb{E} \left[ \bar{q}_T \int_0^T |\psi_t|^2 dt \right] + \Delta(\psi, \bar{\psi}), \end{aligned} $$

with

$$ \Delta(\psi, \bar{\psi}) := - \ln \left( \frac{\mathbb{E}[\exp(-\mathcal{W}(X^{\psi}))]}{\mathbb{E}[\exp(-\mathcal{W}(X^{\bar{\psi}}))]} \right) + \frac{1}{2} \mathbb{E} \left[ \bar{q}_T \int_0^T |\psi_t - \bar{\psi}_t|^2 dt \right], $$

which gives a way to control the variation $\Delta(\psi, \bar{\psi})$ of the cost when $\psi$ is deviating from $\bar{\psi}$. We can illustrate this idea in this example, by means of in (79)-(80)-(81). We have

$$ \begin{aligned} \Delta(\psi, \bar{\psi}) &= - \ln \left( \frac{\mathbb{E}[\exp(\beta X^{\psi})]}{\mathbb{E}[\exp(\beta X^{\bar{\psi}})]} \right) + \frac{1}{2} \mathbb{E} \left[ \bar{q}_T \int_0^T |\psi_t - \bar{\psi}_t|^2 dt \right] \\ &= - \ln \left\{ \mathbb{E} \left[ \exp \left( \beta B_T - \frac{1}{2} \beta^2 T \right) \exp \left( \beta \int_0^T [\psi_t - \bar{\psi}_t] dt \right) \right] \right\} \\ &\quad + \frac{1}{2} \mathbb{E} \left[ \exp \left( \beta B_T - \frac{1}{2} \beta^2 T \right) \int_0^T |\psi_t - \bar{\psi}_t|^2 dt \right]. \end{aligned} $$

This gives a way to control the output performance in terms of the disturbance, which principle is underpinning the theory of $H^\infty$-control (see [6]).

## 4.3 Variational mean field games

In this section, we formulate a mean field game problem that is closely related to the robust mean field control problem introduced in the previous section, and that even derives from it for some specific choice of the coefficients. The latter situation is an extension, to the robust setting, of the connection that exists between mean field control problems and potential mean field games.

Generally speaking, a mean field game is defined as a fixed point problem on the distribution of a control problem (with the latter being solved by a so-called representative agent in a continuum of agents). In our case, the fixed point problem is set on a generic non-negative measure $\mu \in \mathcal{M}_{2-r}(\mathbb{R}^n)$; given $\mu$, the representative agent minimizes a risk-averse objective functional

$$ \inf_{\psi \in \mathcal{A}} \sup_{q \in \mathcal{Q}} \mathcal{J}[\mu](q, \psi), \quad \mathcal{J}[\mu](q, \psi) := \mathbb{E} \left[ q_T g(\mu, X_T^{\psi}) + \int_0^T q_s \ell(s, \psi_s) ds \right] - \mathcal{S}(q), \quad (P_{\mu}) $$
where the controlled state process $(X_t^\psi)_{t \in [0,T]}$ satisfies the dynamics given in (7). Assuming that, for each $\mu \in \mathcal{M}^{2-r}(\mathbb{R}^n)$, the function $(q, X) \mapsto qg(\mu, X)$ satisfies the assumption of Theorem 10 (we clarify the choice of $g$ right below), we can denote by $\psi^\mu$ and $q^\mu$ the optimal controls of the representative agent and of Nature, respectively. The fixed point condition requires that the measure $\mu$ coincides with the law of the terminal state $X_T^{\psi^\mu}$ under the measure $q_T^\mu \mathbb{P}$ induced by Nature, that is,

$$ \mu = (q_T^\mu \mathbb{P}) \circ (X_T^{\psi^\mu})^{-1}. \qquad (\text{MFG-eq}) $$

The mean field game problem thus consists in finding a triple $(q, \psi, \mu) \in \mathcal{Q} \times \mathcal{A} \times \mathcal{M}_{2-r}(\mathbb{R}^n)$ such that

$$ \mathcal{J}[\mu](q, \psi) = \inf_{\psi' \in \mathcal{A}} \sup_{q' \in \mathcal{Q}} \mathcal{J}[\mu](q', \psi'), \quad \mu = (q_T \mathbb{P}) \circ (X_T^\psi)^{-1}. \qquad (\text{MFG}) $$

Here are the assumptions required on $g$.

A10 We assume that there exists a function $G: \mathcal{M}_{2-r}(\mathbb{R}^n) \to \mathbb{R}$, satisfying Assumption A9, such that the mapping $g: \mathcal{M}_{2-r}(\mathbb{R}^n) \times \mathbb{R}^n \to \mathbb{R}$ satisfies $g(\mu, x) = \delta G/\delta \mu(\mu, x)$ and, thus, $\nabla_x g(\mu, x) = \partial_\mu G(\mu, x)$.

Generally speaking, a mean field game problem is said to be variational if the associated mean field game system can be interpreted as the first-order optimality condition of a variational problem. Usually (i.e., in standard mean field games), the criterion of the variational problem involves a potential functional whose derivative-understood in a suitable sense-coincides with the interaction cost of the game (see, for instance, [26] when the mean field game is formulated as a system of PDEs, and [35, Chapter 6] for the probabilistic counterpart). Here, the mean field mapping $G: \mathcal{M}_{2-r}(\mathbb{R}^n) \to \mathbb{R}$ introduced in the above assumption plays the role of the potential, with the derivative understood in the flat sense for Nature and in the Lions sense for the representative player.

**Corollary 16.** Let Assumptions A1-A5 and A10 be satisfied. Then, there exists a unique mean field game equilibrium $(\psi, q, \mu) \in \mathcal{A} \times \mathcal{Q} \times \mathcal{M}_{2-r}(\mathbb{R}^n)$, to the problem (MFG), where we recall that $r$ is defined in Assumption A2. The equilibrium is fully characterized as the solution to the system formed by (OptN)-(OptC) with the terminal conditions in the first two systems being replaced by

$$ p_T = q_T \nabla_x g(X_T^\psi, \mu), \quad Y_T = g(X_T^\psi, \mu), \qquad (82) $$

complemented by the equilibrium condition (MFG-eq), namely $\mu = (q_T \mathbb{P})_{X_T^\psi}$.

*Proof. Step 1: Necessary and sufficient condition for equilibrium.* Let $\mu \in \mathcal{M}_{2-r}(\mathbb{R}^n)$. Applying Corollary 15 to the parametrized mean field mapping $G[\mu]: \mathcal{M}_{2-r}(\mathbb{R}^n) \to \mathbb{R}$ defined as follows

$$ G[\mu](\nu) = \int_{\mathbb{R}^n} g(\mu, x) d\nu(x), \quad \nu \in \mathcal{M}_{2-r}(\mathbb{R}^n), $$

we deduce that the system (OptN)-(OptC), with the terminal conditions (82), is a necessary and sufficient condition for equilibrium when the interaction term $\mu$ is frozen.
When complemented by the equilibrium condition $\mu = (q_T\mathbb{P})_{X_T^\psi}$, they provide a characterization of the solutions to the mean field game (MFG).

*Step 2: Uniqueness.* By Corollary 15, there exists a unique solution $(\psi, q) \in \mathcal{A} \times \mathcal{Q}$ to the system $(\text{Opt}_N)-(\text{Opt}_C)$ with the terminal conditions

$$p_T = q_T \nabla_x g \left( (q_T \mathbb{P})_{X_T^\psi}, X_T^\psi \right), \quad Y_T = g \left( (q_T \mathbb{P})_{X_T^\psi}, X_T^\psi \right).$$

This system coincides with the necessary and sufficient condition identified in the first step, which proves that there exists a unique solution to (MFG) in the space mentioned in the statement. □

**Perspectives.** We conclude this section with a brief discussion about mean field game model beyond the variational case. A natural question arises as to how one might treat mean field games that lack an underlying variational structure. The monotonicity assumptions imposed on the flat and Lions derivatives of $G$ in A9 (and thus on $g$ in A10) in the MFC problem already suggest the type of conditions that can be imposed on the interaction terms to ensure uniqueness of solutions, in the spirit of the classical Lasry-Lions monotonicity condition for standard mean field games. We refer to our companion work [49] for complete results in this direction.

# 5 Proof of Theorem 10

In this section, we establish all the intermediate results used in the proof of Theorem 10. The presentation is organized into three subsections. In Subsection 5.1, we establish the existence of a min-max solution to the problem (P'), corresponding to Step 1 and Step 2 in the proof of Theorem 10. Subsection 5.2 provides the necessary and sufficient conditions for the control problem solved by Nature, thus covering the arguments developed in Step 3 of the proof. Finally, Subsection 5.3 focuses on the central planner and forms the basis of Step 4 in the proof of Theorem 10.

## 5.1 Existence of a saddle point to (P')

This subsection is dedicated to the proof of the existence of a saddle point to the problem (P'), for given values of $c_1, c_2 > 0$. This corresponds to the first step in the proof of Theorem 10. Without any loss of generality, we can assume that

$$c_1 > \mathbb{E} \int_0^T q_t^0 f(t, 0, 0)dt, \qquad (83)$$

where $\bar{q}^0$ denotes the solution of

$$d q_t^0 = q_t^0 \partial_y f(t, 0, 0) dt + q_t^0 \partial_z f(t, 0, 0) \cdot dW_t, \quad t \in [0, T]. \qquad (84)$$

We notice that the right-hand side on (83) is equal to $S(q^0)$. Indeed

$$S(q^0) = \mathbb{E} \int_0^T q_t^0 f^*(t, \partial_y f(t, 0, 0), \partial_z f(t, 0, 0)) dt = -\mathbb{E} \int_0^T q_t^0 f(t, 0, 0) dt.$$

The purpose is thus to establish the following statement:
**Lemma 17.** There exists a solution $(\psi, q) \in \mathcal{A}_{c_2} \times \mathcal{Q}_{c_1}$ to (P').

Before we provide a sketch of the proof of this result, we introduce a variant of the Nature optimization problem. Existence of a saddle point is proven by means of weak compactness arguments (in $L^p$ spaces), which are developed in this subsection. In this regard, the nonlinear form of the state equation (4) causes additional difficulties, as the product form of the coefficients is not appropriate for weak convergence arguments. For this reason, it is easier to consider weak limits of the two products $(q_t Y_t^*)_{t \in [0,T]}$ and $(q_t Z_t^*)_{t \in [0,T]}$, each being viewed as a single process. However, this makes more difficult the identification of the limit points as solutions of an equation of the form (4) (because weak limits of the products must be shown to have a product form, which writing may be difficult to establish if the weak limit of $(q_t)_{t \in [0,T]}$ vanishes). This prompts us to introduce a variant of the problem (P'), and in particular to define the perspective function $q f^*(\omega, t, y^*/q, z^*/q)$ for any $(\omega, t, q, y^*, z^*) \in \Omega \times [0, T] \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}$ of $f^*$ (see [15, 44] for a presentation) with respect to its last two variables. We further introduce its lower semi-continuous envelope (or its bidual) $\tilde{f}^*: \Omega \times [0, T] \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}$,

$$ \tilde{f}^*(\omega, t, q, y^*, z^*) = \begin{cases} qf^*\left(\omega, t, \frac{y^*}{q}, \frac{z^*}{q}\right), & q > 0, \\ \mathrm{rec}f^*(\omega, t, q, y^*, z^*), & q = 0, \\ +\infty, & q < 0. \end{cases} \quad (85) $$

Here $\mathrm{rec}f^*(\omega, t, q, \cdot, \cdot)$ denotes the recession function of $f^*(\omega, t, q, \cdot, \cdot)$ (with respect to the last two variables of $f^*$). By [15, Lemma 1.156], it coincides with the support function of $f^*(\omega, t, 0, \cdot, \cdot)$, i.e.,

$$ \mathrm{rec}f^*(\omega, t, 0, y^*, z^*) = \sup_{(y,z) \in \mathbb{R} \times \mathbb{R}^d} \{\langle y, y^* \rangle + \langle z, z^* \rangle, f(\omega, t, 0, y, z) < +\infty\}. $$

Because $f(\omega, t, q, \cdot, \cdot)$ has full support, the recession function at $q=0$ is given by

$$ \mathrm{rec}f^*(\omega, t, q, y^*, z^*) = \begin{cases} 0, & (y^*, z^*) = 0, \\ +\infty, & \text{otherwise.} \end{cases} $$

Finally, the function $\tilde{f}^*$ is equal to

$$ \tilde{f}^*(\omega, t, q, y^*, z^*) = \begin{cases} qf^*\left(\omega, t, \frac{y^*}{q}, \frac{z^*}{q}\right), & q > 0, \\ 0, & (q, y^*, z^*) = 0, \\ +\infty, & \text{otherwise.} \end{cases} \quad (86) $$

For simplicity, we call $\tilde{f}^*$ the perspective function of $f^*$ when there is no ambiguity. Because $f^*(\omega, t, \cdot, \cdot)$ is convex and lower semi-continuous, its perspective function $\tilde{f}^*$ is convex with respect to its three last variables and lower semi-continuous. For any non-negative valued Itô process $q = (q_t)_{t \in [0,T]}$, satisfying $\mathbb{E}[q_T^*] < +\infty$, and admitting the expansion

$$ dq_t = \tilde{Y}_t^* dt + \tilde{Z}_t^* \cdot dW_t, \quad t \in [0, T], \quad (87) $$

for some (uniquely defined) $\mathbb{F}$-progressively measurable process $\tilde{Y}^* = (\tilde{Y}_t^*)_{t \in [0,T]}$ and $\tilde{Z}^* = (\tilde{Z}_t^*)_{t \in [0,T]}$, with values in $\mathbb{R}$ and $\mathbb{R}^d$ respectively and satisfying

$$ \mathbb{P}\left(\left\{\int_{0}^{T} (|\tilde{Y}_{t}^{*}| + |\tilde{Z}_{t}^{*}|^{2}) dt < +\infty\right\} = 1, \quad (88) \right. $$
we define the perspective generalized entropy of $q$ by letting

$$ \tilde{\mathcal{S}}(q) := \mathbb{E} \left[ \int_{0}^{T} \tilde{f}^{*}(t, q_t, \tilde{Y}_{t}^{*}, \tilde{Z}_{t}^{*}) dt \right]. \qquad (89) $$

Recalling the lower bound (16) and using the fact that $\mathbb{E}[q_{T}^{*}] < +\infty$, we notice that the expectation right above is well-defined; it belongs to $(-\infty, +\infty]$. And then, we introduce the perspective min-max problem

$$ \sup_{q \in \tilde{\mathcal{Q}}_{c_1}} \inf_{\psi \in \mathcal{A}_{c_2}} \tilde{\mathcal{J}}(q, \psi), \qquad (\tilde{\mathbf{P}}') $$

where the mapping $\tilde{\mathcal{J}}$ is given by

$$ \tilde{\mathcal{J}}(q, \psi) := \mathcal{R}(q, \psi) - \tilde{\mathcal{S}}(q); \qquad (90) $$

recall (1) for the definition of $\mathcal{R}$. Above, the set $\tilde{\mathcal{Q}}_{c_1}$ is defined as the collection of $q \in \tilde{\mathcal{Q}}$ such that $\tilde{\mathcal{S}}(q) \le c_1$, where $\tilde{\mathcal{Q}}$ is the set of non-negative valued measurable Itô processes $q = (q_t)_{t \in [0,T]}$ satisfying (87), such that $q_0 = 1$, $\mathbb{E}[q_{T}^{*}] \le \exp(\alpha T)$, and $\tilde{\mathcal{S}}(q) < +\infty$.

Since we restricted the controlled dynamics (4) to processes $(q_t)_{t \in [0,T]}$ that do not vanish, it is easy to see that any $q \in \mathcal{Q}_{c_1}$ belongs to $\tilde{\mathcal{Q}}_{c_1}$. Indeed, $\mathcal{S}(q)$ and $\tilde{\mathcal{S}}(q)$ coincide in this setting. Moreover, the bound $\mathbb{E}[q_{T}^{*}] \le \exp(\alpha T)$ follows from the facts that $Y^{*}$ is bounded by $\alpha$ and $(\mathcal{E}_{t}(\int_{0}^{T} Z_{s}^{*} \cdot dW_{s}))_{t \in [0,T]}$ is a martingale, see Lemma 9.

Existence of a saddle point to $(\tilde{\mathbf{P}}')$ is established in the next subsection; see Lemma 23. Taking the latter for granted, Lemma 17 can be derived as follows:

*Proof of Lemma 17.* Since existence of a saddle point to $(\tilde{\mathbf{P}}')$ is provided by Lemma 23, it suffices to show that any solution to $(\tilde{\mathbf{P}}')$ is a solution to $(\mathbf{P}')$. Let $(\bar{q}, \bar{\psi})$ be a solution to $(\tilde{\mathbf{P}}')$, that is to say

$$ \tilde{\mathcal{J}}(q, \bar{\psi}) \le \tilde{\mathcal{J}}(\bar{q}, \bar{\psi}) \le \tilde{\mathcal{J}}(\bar{q}, \psi), \quad \forall (q, \psi) \in \tilde{\mathcal{Q}}_{c_1} \times \mathcal{A}_{c_2}. \qquad (91) $$

By Lemma 18 (which is stated and proven in Subsection 5.1.1 below), the process $\bar{q}$ is positive in the sense that $\mathbb{P}(\inf_{t \in [0,T]} \bar{q}_t > 0) = 1$. This makes it possible to let $(Y_t^* := \tilde{Y}_t^* / \bar{q}_t)_{t \in [0,T]}$ and $(Z_t^* = \tilde{Z}_t^* / \bar{q}_t)_{t \in [0,T]}$, from which we deduce

$$ d\bar{q}_t = \bar{q}_t Y_t^* dt + \bar{q}_t Z_t^* \cdot dW_t, \quad t \in [0, T]; \quad \bar{q}_0 = 1. $$

By definition of the perspective generalized entropy

$$ \mathcal{S}(\bar{q}) = \tilde{\mathcal{S}}(\bar{q}) \le c_1, $$

which proves that $\bar{q}$ belongs to $\mathcal{Q}_{c_1}$. Then, by the definition (90) of $\tilde{\mathcal{J}}$, by the optimality condition (91) and since $\tilde{\mathcal{Q}}_{c_1}$ contains $\mathcal{Q}_{c_1}$, we have

$$ \mathcal{J}(q, \bar{\psi}) \le \mathcal{J}(\bar{q}, \bar{\psi}) \le \mathcal{J}(\bar{q}, \psi), \quad \forall (q, \psi) \in \mathcal{Q}_{c_1} \times \mathcal{A}_{c_2}, $$

concluding the proof. $\square$

<figure><img src="image_4.png" /></figure>
### 5.1.1 Trajectories of the perspective problem and positivity of the optimal ones

The third item in the following lemma was used in the proof of Lemma 17. The first two items are also used in the proof of Lemma 23.

**Lemma 18.** Let $q \in \tilde{Q}_{c_1}$ and $(\tilde{Y}^*, \tilde{Z}^*)$ be as in the representation (87).

i. Letting

$$Y_t^* = 1_{\{q_t>0\}} \frac{\tilde{Y}_t^*}{q_t}, \quad Z_t^* = 1_{\{q_t>0\}} \frac{\tilde{Z}_t^*}{q_t}, \quad t \in [0, T], \qquad (92)$$

it holds

$$\mathbb{P} \otimes \text{Leb}_{[0,T]} (\{(\omega, t) \in \Omega \times [0, T], |Y_t^*| > \alpha\}) = 0, \qquad (93)$$

and $q$ can be expanded as

$$d q_t = q_t Y_t^* d t + q_t Z_t^* \cdot d W_t, \quad t \in [0, T]. \qquad (94)$$

ii. Moreover, letting $\tau := \inf\{t \in [0, T], q_t = 0\}$ (with $\inf\emptyset = +\infty$), it also holds $\mathbb{P}(\{\sup_{t \in [\tau, T]} q_t > 0\} \cap \{\tau < T\}) = 0$ (i.e., 0 is an absorbing state). And then,

$$\tilde{S}(q) = \mathbb{E} \left[ \int_{0}^{T} \tilde{f}^*(t, q_t, \tilde{Y}_t^*, \tilde{Z}_t^*) dt \right] = \mathbb{E} \left[ \int_{0}^{\tau} q_t f^*(t, Y_t^*, Z_t^*) dt \right]. \qquad (95)$$

We also have $\mathbb{E}[q_T^*] < +\infty$ and there exists a constant $C$, which depends on $q$ only via $c_1$, such that

$$\mathbb{E} \left[ \left( \int_{0}^{T} |\tilde{Z}_{s}^{*}|^{2} ds \right)^{1/2} \right] \leq C. \qquad (96)$$

iii. Lastly, if for a certain $\psi \in A_{c_2}$, the pair $(\psi, q) \in A_{c_2} \times \tilde{Q}_{c_1}$ is a solution to $(\tilde{P}')$. Then $\mathbb{P}(\{\inf_{t \in [0, T]} q_t > 0\}) = 1$, and (in particular) $q \in Q_{c_1}$.

**Remark 19.** The following two comments are in order:

1. In dimension $d = 1$, the CIR model, i.e.,

$$d q_t = \sqrt{q_t} d W_t, \quad t \in [0, \tau),$$

provides an interesting example in which $q$ may vanish even if the entropy, which is here equal to $\mathbb{E}[\int_{0}^{\tau \wedge T} q_t^{-1} q_t dt] = \mathbb{E}[\tau \wedge T]$, is finite.

2. When $q$ vanishes, it does not make sense to represent it in the form of a (weighted) Doléans-Dade exponential martingale. This observation causes additional difficulties in the analysis.

**Proof.** Step 1: Representation of $q \in \tilde{Q}_{c_1}$. We recall that $q$ can be represented as

$$q_t = 1 + \int_0^t \tilde{Y}_s^* ds + \int_0^t \tilde{Z}_s^* \cdot dW_s, \quad t \in [0, T],$$
with

$$ \tilde{S}(q) = \mathbb{E} \left[ \int_{0}^{T} \tilde{f}^{*}(t, q_t, \tilde{Y}_{t}^{*}, \tilde{Z}_{t}^{*}) dt \right] \in (-\infty, c_1]. $$

Using (16) together with the bound $\mathbb{E}[q_t^*] \le \exp(\alpha T)$, we deduce that

$$ \mathbb{E} \left[ \int_{0}^{T} \mathbb{1}_{\{q_t > 0\}} q_t \left( \chi_B \left( \frac{\tilde{Y}_t^*}{\alpha q_t} \right) + \frac{1}{2\beta} \left| \frac{\tilde{Z}_t^*}{q_t} \right|^2 \right) dt \right] < +\infty. \quad (97) $$

This proves in particular that

$$ \mathbb{P} \otimes \mathrm{Leb}_{[0,T]} \left( \left\{ (\omega, t) \in \Omega \times [0, T], q_t > 0, \left| \tilde{Y}_t^* \right| > \alpha q_t \right\} \right) = 0. \quad (98) $$

Moreover, recalling the definition (86) of $\tilde{f}^*$, we also have

$$ \mathbb{P} \otimes \mathrm{Leb}_{[0,T]} \left( \left\{ (\omega, t) \in \Omega \times [0, T], q_t = 0, \left| \tilde{Y}_t^* \right| + \left| \tilde{Z}_t^* \right| > 0 \right\} \right) = 0. \quad (99) $$

With the notation (92), (93) and (94) easily follow.

**Step 2: Proving that $q$ stays in $0$ once it has touched it.** Recall that $\tau := \inf\{t \in [0, T], q_t = 0\}$ (with $\inf\emptyset = +\infty$). We want to prove that $\mathbb{P}(\{\sup_{t \in [\tau, T]} q_t > 0\} \cap \{\tau < T\}) = 0$. The proof is as follows. For any $\epsilon > 0$, let $\rho^\epsilon := \inf\{t \in [\tau, T], q_t = \epsilon\}$, with the convention that $\rho^\epsilon = +\infty$ if $\tau = +\infty$ or if $\tau \le T$ and $q$ does not touch $\epsilon$ between $\tau$ and $T$. Using (93) and (94), we then notice that

$$ d(\exp(\alpha t)q_t) \ge \exp(\alpha t)q_t Z_t^* \cdot dW_t, \quad t \in [0, T]. $$

By localization (use (88) together with the fact that $\tilde{Z}_t^* = q_t Z_t^*$), we can find a non-decreasing sequence of stopping times $(\sigma_k)_{k \ge 1}$, converging to $T$ (almost surely), such that

$$ \forall k \ge 1, \quad \mathbb{E} \left[ \int_{0}^{\sigma_k} q_t^2 |Z_t^*|^2 dt \right] < +\infty. $$

And then,

$$ \exp(\alpha \tau \wedge \sigma_k) q_{\tau \wedge \sigma_k} \ge \mathbb{E} [\exp(\alpha \rho^\epsilon \wedge \sigma_k) q_{\rho^\epsilon \wedge \sigma_k} | \mathcal{F}_\tau]. $$

Letting $k$ tend to $+\infty$ and using a conditional version of Fatou's lemma, we obtain, $\mathbb{P}$-almost surely,

$$ \exp(\alpha \tau \wedge T) q_{\tau \wedge T} \ge \mathbb{E} [\exp(\alpha \rho^\epsilon \wedge T) q_{\rho^\epsilon \wedge T} | \mathcal{F}_\tau]. $$

We deduce that there exists a constant $c > 0$, only depending on $T$ and $\alpha$, such that

$$ q_{\tau \wedge T} \ge c \mathbb{E} [q_{\rho^\epsilon \wedge T} | \mathcal{F}_\tau]. $$

Multiply both sides by $\mathbb{1}_{\{\tau < T\}}$ and take expectation under $\mathbb{P}$. Since $q_\tau = 0$ when $\tau < T$, we get

$$ \mathbb{E} [\mathbb{1}_{\{\tau < T\}} q_{\rho^\epsilon \wedge T}] = 0. $$

This shows $\mathbb{P}(\{\tau < T\} \cap \{\rho^\epsilon \le T\}) = 0$. Letting $\epsilon$ tend to 0, we derive the expected claim, that is $\mathbb{P}(\{\tau < T\} \cap \{\sup_{t \in [\tau, T]} q_t > 0\}) = 0$. This makes it possible to prove (95). Indeed, together with (99), we obtain

$$ \mathbb{P} \otimes \mathrm{Leb}_{[0,T]} \left( \left\{ (\omega, t) \in \Omega \times [0, T], t > \tau, \left| \tilde{Y}_t^* \right| + \left| \tilde{Z}_t^* \right| > 0 \right\} \right) = 0, $$
from which we deduce that (recalling (85))

$$ \mathbb{E} \left[ \int_{\tau}^{T} \tilde{f}^{*}(t, q_t, \tilde{Y}_{t}^{*}, \tilde{Z}_{t}^{*}) dt \right] = 0. $$

Identity (95) easily follows.

We now prove that $\mathbb{E}[q_T^*] < +\infty$. Letting $(\tilde{q}_t := q_t \exp(-\int_0^t Y_s^* ds))_{t \in [0,T]}$, we deduce from (94) that

$$ d\tilde{q}_t = \tilde{q}_t Z_t^* \cdot dW_t, \quad t \in [0, T]. $$

By Itô's formula,

$$ d[\tilde{q}_t \ln(\tilde{q}_t)] = \frac{1}{2} \tilde{q}_t |Z_t^*|^2 dt + [\ln(\tilde{q}_t) + 1] \tilde{q}_t Z_t^* \cdot dW_t, \quad t \in [0, \tau). $$

By a localization argument (together with (97)), we deduce that $\mathbb{E}[\tilde{q}_\tau \ln(\tilde{q}_\tau)] < +\infty$. And then, by $L \log(L)$-Doob's maximal inequality, we obtain $\mathbb{E}[\tilde{q}_T^*] = \mathbb{E}[\sup_{t \in [0, \tau]} \tilde{q}_t] \le C$, for a constant $C$ that depends on $q$ only via $c_1$. We deduce that $\mathbb{E}[q_T^*] < C \exp(\alpha T)$. By Burkholder-Davis-Gundy inequalities, (96) easily follows.

**Step 3: Contradicting the fact that $\tau \le T$, when $(\psi, q)$ is a saddle-point.** We now prove the final result, that is $\mathbb{P}(\{\tau \le T\}) = 0$ when $q$ satisfies the optimality property of a saddle-point. For $\theta \in (0, 1)$, we let $q^\theta := \theta q + (1-\theta)q^0$, where we recall (84) for the definition of $q^0$. We have

$$ \begin{aligned} dq_t^\theta &= [\theta q_t Y_t^* + (1-\theta)q_t^0 \partial_y f(t, 0, 0)] dt + [\theta q_t Z_t^* + (1-\theta)q_t^0 \partial_z f(t, 0, 0)] \cdot dW_t \\ &=: \tilde{Y}_t^{*,\theta} dt + \tilde{Z}_t^{*,\theta} \cdot dW_t, \end{aligned} $$

for any $t \in [0, T]$. Since $q^\theta$ is positive valued, for $\theta \in [0, 1)$, we can let $Y_t^{*,\theta} = \tilde{Y}_t^{*,\theta}/q_t^0$ and $Z_t^{*,\theta} = \tilde{Z}_t^{*,\theta}/q_t^0$, for $t \in [0, T]$. By (83), we know that $\tilde{S}(q^0) = S(q^0) < c_1$. By convexity of $\tilde{S}$ (see Step 2 in the proof of Proposition 20), we deduce that $\tilde{S}(q^\theta) \le c_1$, and then $q^\theta \in \tilde{Q}_{c_1}$ for all $\theta \in [0, 1]$.

By definition of $\tilde{\mathcal{J}}$ (see (89) and (90)),

$$ \tilde{\mathcal{J}}(q^\theta, \psi) = \mathcal{G}(q_T^\theta, X_T^\psi) + \mathbb{E} \left[ \int_0^T q_s^\theta \ell(s, \psi_s) ds \right] - \mathbb{E} \left[ \int_0^T \tilde{f}^* \left( s, q_s^\theta, \tilde{Y}_s^{*,\theta}, \tilde{Z}_s^{*,\theta} \right) ds \right]. $$

We then subtract $\tilde{\mathcal{J}}(q, \psi)$ on both sides. Recalling the definition (86) of $\tilde{f}^*$ we obtain

$$ \begin{aligned} \tilde{\mathcal{J}}(q^\theta, \psi) - \tilde{\mathcal{J}}(q, \psi) &= (\mathcal{G}(q_T^\theta, X_T^\psi) - \mathcal{G}(q_T, X_T^\psi)) + \mathbb{E} \left[ \int_0^T (q_s^\theta - q_s) \ell(s, \psi_s) ds \right] \\ &\quad - \mathbb{E} \left[ \int_0^T \left( \tilde{f}^* (s, q_s^\theta, \tilde{Y}_s^{*,\theta}, \tilde{Z}_s^{*,\theta}) - \tilde{f}^* (s, q_s, \tilde{Y}_s^{*,\theta}, \tilde{Z}_s^{*,\theta}) \right) ds \right] \\ &=: S_1^\theta + S_2^\theta - S_3^\theta. \end{aligned} $$

Using the fact that $q_t^\theta - q_t = (1-\theta)(q_t^0 - q_t)$ together with the regularity of $\mathcal{G}$ in the variable $q$ and the integrability properties of $\psi$, and then applying Lemma 40, we deduce that there exists a positive constant $C$ such that, for any $\theta \in (0, 1)$,

$$ |S_1^\theta| + |S_2^\theta| \le C(1-\theta). $$
Similarly, by strong convexity of $f^*$ (in the last variable), see Remark 2, there exists $c > 0$ (independent of $\theta$) such that

$$
\begin{align*}
& \mathbb{E} \left[ \int_0^T \tilde{f}^* \left( s, q_s^\theta, \tilde{Y}_s^{\star, \theta}, \tilde{Z}_s^{\star, \theta} \right) ds \right] \\
&= \mathbb{E} \left[ \int_0^T q_s^\theta f^* \left( s, \frac{\theta q_s}{q_s^\theta} Y_s^{\star, \theta} + \frac{(1-\theta)q_s^0}{q_s^\theta} \partial_y f_s^0, \frac{\theta q_s}{q_s^\theta} Z_s^{\star, \theta} + \frac{(1-\theta)q_s^0}{q_s^\theta} \partial_z f_s^0 \right) ds \right] \\
&\le \mathbb{E} \left[ \int_0^T \left( \theta \tilde{f}^* \left( s, q_s, \tilde{Y}_s^\star, \tilde{Z}_s^\star \right) + (1-\theta) q_s^0 f_s^0 - c(1-\theta) \frac{\theta q_s q_s^0}{q_s^\theta} |Z_s^\star - \partial_z f_s^0|^2 \right) ds \right],
\end{align*}
$$

where we have used the shorthand notations $\partial_y f_s^0 := \partial_y f(s, 0, 0)$ and $\partial_z f_s^0 := \partial_z f(s, 0, 0)$, and the duality identity $f^*(s, \partial_y f_s^0, \partial_z f_s^0) = f(s, 0, 0) = f_s^0$. The last line, together with the fact that $\|f^0\|_{L^\infty(\mathbb{F})} < +\infty$ and $\|\partial_z f^0\|_{L^\infty(\mathbb{F})} < +\infty$ and the standard inequality $|Z_s^* - \partial_z f_s^0|^2 \ge \frac{1}{2}|Z_s^*|^2 - |\partial_z f_s^0|^2$, yields the following lower bound

$$
-S_3^\theta \geq (1-\theta) \left( -C - \mathbb{E} \left[ \int_0^T \tilde{f}^* (s, q_s, \tilde{Y}_s^*, \tilde{Z}_s^*) ds \right] + \frac{c}{2} \mathbb{E} \left[ \int_0^T \frac{\theta q_s q_s^0}{q_s^\theta} |Z_s^*|^2 ds \right] \right).
$$

Using the fact that $\tilde{S}(q) \le c_1$, we deduce in the end that (for a possibly new value of $C$)

$$
\tilde{\mathcal{J}}(q^\theta, \psi) - \tilde{\mathcal{J}}(q, \psi) \geq (1-\theta) \left( -C + \frac{c}{2} \mathbb{E} \left[ \int_0^T \frac{\theta q_s q_s^0}{q_s^\theta} |Z_s^\star|^2 ds \right] \right). \quad (100)
$$

It then remains to observe that (whether the right-hand side is finite or not)

$$
\lim_{\theta \to 1} \mathbb{E} \left[ \int_0^T \frac{\theta q_s q_s^0}{q_s^\theta} |Z_s^\star|^2 ds \right] = \mathbb{E} \left[ \int_0^\tau q_s^0 |Z_s^\star|^2 ds \right]. \quad (101)
$$

Since $q$ is an optimizer of $\tilde{\mathcal{J}}(\cdot, \psi)$ over $\tilde{Q}_{c_1}$, it holds $\tilde{\mathcal{J}}(q^\theta, \psi) \le \tilde{\mathcal{J}}(q, \psi)$ implying that the left-hand side in (100) is non-positive. Combining the last two lines, this shows that the right-hand side on the above identity is necessarily finite. We claim that this implies that $\mathbb{P}(\{\tau \le T\}) = 0$.

Assume by a way of contradiction that $\mathbb{P}(\{\tau \le T\}) > 0$. For $t \in [0, \tau)$, we can expand $\ln(q_t)$ by means of Itô's formula. We get

$$
\begin{align*}
\mathrm{d} \ln(q_t) &= \left(Y_t^* - \frac{1}{2} |Z_t^*|^2\right) \mathrm{d}t + Z_t^* \cdot \mathrm{d}W_t \\
&= \left(Y_t^* - \frac{1}{2} |Z_t^*|^2 + Z_t^* \cdot \partial_z f_t^0\right) \mathrm{d}t + Z_t^* \cdot \mathrm{d}\left(W_t - \int_0^t \partial_z f_s^0 \mathrm{d}s\right), \quad t \in [0, \tau).
\end{align*}
$$

Let $\mathbb{Q}^0 := \mathcal{E}_T(\int_0^T \partial_z f_s^0 \cdot \mathrm{d}W_s)$. Setting $\sigma^\epsilon := \inf\{t \ge 0, q_t \le \epsilon\}$ for any $\epsilon \in (0, 1)$ (with the convention that $\inf\emptyset = +\infty$), we have

$$
\begin{align*}
-\mathbb{E}^{\mathbb{Q}^0}[\ln(q_{\sigma^\epsilon \wedge T})] &\le \frac{1}{2}\mathbb{E}^{\mathbb{Q}^0}\left[\int_0^T |Z_t^*|^2 \mathrm{d}t\right] + \mathbb{E}^{\mathbb{Q}^0}\left[\int_0^T |Z_t^*| \|\partial_z f_t^0\| \mathrm{d}t\right] + \alpha T, \\
&\le \mathbb{E}^{\mathbb{Q}^0}\left[\int_0^T |Z_t^*|^2 \mathrm{d}t\right] + C,
\end{align*}
$$

for a constant $C$ depending on $\alpha$, $T$ and $\|\partial_z f^0\|_{L^\infty(\mathbb{F})}$. Now, if $\mathbb{P}(\{\tau \le T\}) > 0$, then $\mathbb{Q}^0(\{\tau \le T\}) > 0$, and $\sup_{\epsilon>0}[-\ln(\epsilon)\mathbb{Q}^0(\{\sigma^\epsilon \le T\})] = +\infty$, and thus the right-hand side is also infinite, which contradicts (100) and (101). $\square$
### 5.1.2 Solvability of the perspective min-max problem

**Proposition 20.** Let $\psi \in L^\infty(\mathbb{F}, \mathbb{R}^n)$ and $c_1 > 0$. Viewing $\tilde{Q}_{c_1}$ as a subset of $L^1(\Omega \times [0, T], \mathbb{P} \otimes \text{Leb}_{[0,T]})$ equipped with the weak topology $\sigma(L^1, L^\infty)$, $\tilde{Q}_{c_1}$ is (weakly) compact and convex, and satisfies

$$ \sup_{q \in \tilde{Q}_{c_1}} \sup_{t \in [0,T]} \mathbb{E}[h(q_t)] < +\infty. \quad (102) $$

In addition, the mapping $\tilde{Q}_{c_1} \ni q \mapsto \tilde{\mathcal{J}}(\psi, q)$ is strictly concave and upper semi-continuous (w.r.t. the weak topology).

**Remark 21.** As a corollary of the proof, we obtain that the functional $\tilde{\mathcal{S}}$ is convex, which has further applications. Indeed, for any $\theta \in [0, 1]$, let (as in the proof of Lemma 18) $q^\theta := (1-\theta)q^0 + \theta q$, where $q^0$ is defined as in (84). We observe that $q^\theta$ is positive valued for each $\theta \in [0, 1]$. By Lemma 18, it is easy to see that, for every $\theta \in [0, 1)$, $q^\theta \in \mathcal{Q}$. Moreover, by convexity of $\tilde{Q}_{c_1}$, we have $q^\theta \in \mathcal{Q}_{c_1}$, provided that $c_1$ is large enough, which is not a restriction here. This shows that $q^\theta \in \mathcal{Q}_{c_1}$ and, more generally, that $\mathcal{S}(q^\theta) \le (1-\theta)\mathcal{S}(q^0) + \theta \tilde{\mathcal{S}}(q)$. Since $\mathcal{S}(q^0) \in \mathbb{R}$, we deduce that

$$ \limsup_{\theta \to 1} \mathcal{S}(q^\theta) \le \tilde{\mathcal{S}}(q). $$

This observation allows us to extend results holding on $\mathcal{Q}_{c_1}$ to $\tilde{\mathcal{Q}}_{c_1}$, such as the duality inequality (14), and Lemmas 40 and 41. Similarly, Lemmas 43 and 44, which are invoked in Step 5 of the proof below, can also be extended to sequences with values in $\tilde{\mathcal{Q}}$, using the additional fact that the function $h$ is convex (recall (10) for the definition of $h$).

**Proof.** The proof is divided into six steps. In Step 1, we show the relative weak compactness of $\tilde{Q}_{c_1}$. In Step 2, we establish the (strict) concavity of the mapping $\tilde{\mathcal{J}}$ and we prove the convexity of $\tilde{Q}_{c_1}$. In Steps 3 and 4, we show the weak lower semi-continuity of $\tilde{\mathcal{S}}$ and the weak compactness of $\tilde{Q}_{c_1}$. In Step 5, we establish the weak continuity of $\mathcal{R}$. In Step 6, we conclude the proof. Throughout, the value of $\psi$ is fixed. For this reason, we omit it in many notations. For instance, we just write $\tilde{\mathcal{J}}(q)$ for $\tilde{\mathcal{J}}(q, \psi)$.

**Step 1: relative weak compactness of $\tilde{Q}_{c_1}$**. We first establish the relative weak compactness, in $L^1(\Omega \times [0, T], \mathbb{P} \otimes \text{Leb}_{[0,T]})$ equipped with the weak topology $\sigma(L^1, L^\infty)$, of any subset $\mathcal{D}$ of $L \log L(\mathbb{F})$ that is bounded in the sense that

$$ \sup_{q \in \mathcal{D}} \sup_{t \in [0,T]} \mathbb{E}[h(q_t)] < +\infty. $$

The argument is classic and goes as follows. By de la Vallée Poussin Theorem [47, Theorem VI] (see also [51, Theorem 22, page 24-II]), the set $\mathcal{D}$ is uniformly integrable on $\Omega \times [0, T]$ equipped with $\mathbb{P} \otimes \text{Leb}_{[0,T]}$. Then, by the Dunford-Pettis Theorem [21, Theorem 4.30], the set is weakly relatively compact. The set is weakly sequentially relatively compact by the Eberlein-Smulian Theorem [54, Section V.6.1, p.430]. The last two statements can also be found combined into a single statement, see [51, Theorem 25, page 27-II].

We now prove that $\tilde{Q}_{c_1}$ is a bounded subset of $L \log L(\mathbb{F})$ (which corresponds to (102) in the statement). Indeed, for any $q \in \tilde{Q}_{c_1}$, letting $\tau := \inf\{t \in [0, T], q_t = 0\}$,
we know from Lemma 18 that $q$ stays equal to 0 after $\tau$. Moreover, it satisfies (95) (under the notation (93)), from which we deduce that

$$c_1 \ge \tilde{\mathcal{S}}(q) = \mathbb{E} \left[ \int_0^\tau q_t f^*(t, Y_t^*, Z_t^*) dt \right] \ge -C + \frac{1}{2\beta} \mathbb{E} \left[ \int_0^\tau q_t |Z_t^*|^2 dt \right], \quad (103)$$

where the process $(Y_t^*, Z_t^*)_{t \in [0,T]}$ denotes $(1_{\{q_t>0\}} \tilde{Y}_t^*/q_t, 1_{\{q_t>0\}} \tilde{Z}_t^*/q_t)_{t \in [0,T]}$, and where we used the coercivity inequality (16) of $f^*$. The constant $C$ only depends on the $L^\infty$ bound for $f^0$.

We now expand $(h(q_t))_{t \in [0, \tau)}$ by means of Itô's formula. We get

$$dh(q_t) = \left( (h(q_t) + q_t)Y_t^* + \frac{1}{2}q_t|Z_t^*|^2 \right) dt + (h(q_t) + q_t)Z_t^* \cdot dW_t, \quad t \in [0, \tau). \quad (104)$$

Recalling that $Y^*$ is bounded by $\alpha$ and using a standard localization argument, we deduce that

$$\sup_{t \in [0, T]} \mathbb{E}[h(q_{t \wedge \tau})] \le C + C \mathbb{E} \left[ \int_0^T q_t |Z_t^*|^2 dt \right],$$

for a constant $C$ independent of $q$. Observing that $h(0) = 0$, the left-hand side is also equal to $\sup_{t \in [0, T]} \mathbb{E}[h(q_t)]$. Back to (103), we deduce that $\tilde{Q}_{c_1}$ is a bounded subset of $L \log L(\mathbb{F})$.

**Step 2: (strict) concavity of $\tilde{Q} \ni q \mapsto \tilde{J}(q)$**. Recalling the definition of $\tilde{Q}$ on the line below (90), we first notice that $\tilde{Q}$ is convex. Indeed, the decomposition (87) is linear and thus stable by convex combinations. Moreover, the non-negativity constraint and the bound $\mathbb{E}[q_T^*] \le \exp(\alpha T)$, which are both required in the definition of $\tilde{Q}$, are also stable by convex combinations. It remains to see that, for any $\theta \in (0, 1)$, the convex combination $q^\theta := \theta q^1 + (1-\theta)q^2$ of any two $q^1, q^2 \in \tilde{Q}$ satisfies $\tilde{\mathcal{S}}(q^\theta) < +\infty$. Denoting by $(\tilde{Y}^{*,\theta}, \tilde{Z}^{*,\theta})$, $(\tilde{Y}^{*,1}, \tilde{Z}^{*,1})$ and $(\tilde{Y}^{*,2}, \tilde{Z}^{*,2})$ the respective representation processes of $q^\theta$, $q^1$ and $q^2$ in (87), we have

$$\tilde{\mathcal{S}}(q^\theta) = \mathbb{E} \left[ \int_0^T \tilde{f}^*(s, q_s^\theta, Y_s^{*,\theta}, Z_s^{*,\theta}) ds \right],$$

with $(\tilde{Y}^{*,\theta}, \tilde{Z}^{*,\theta}) = \theta(\tilde{Y}^{*,1}, \tilde{Z}^{*,1}) + (1-\theta)(\tilde{Y}^{*,2}, \tilde{Z}^{*,2})$. Since $f^*$ is strictly convex in its last two variables, see Remark 2, $\tilde{f}^*$ is (strictly) jointly convex in its last three arguments, see [15, Lemma 1.157]. It easily follows that $\tilde{\mathcal{S}}(q^\theta) < +\infty$, i.e., $\tilde{Q}$ is convex. Moreover, $\tilde{\mathcal{S}}$ is strictly convex on $\tilde{Q}$.

It remains to see that, by the concavity Assumption A8, $\tilde{Q} \ni q \mapsto \mathcal{R}(q)$ defined in (1) is concave (with the shorthand notation $\mathcal{R}(q)$ for $\mathcal{R}(q, \psi)$). We deduce that $\tilde{Q} \ni q \mapsto \tilde{J}(q)$ is strictly convex. By convexity of $\tilde{\mathcal{S}}$, $\tilde{Q}_{c_1}$ is also convex.

**Step 3: weak lower semi-continuity of $\tilde{\mathcal{S}}$**. We begin with further compactness properties of $\tilde{Q}_{c_1}$. We thus consider a sequence $(q^k)_{k \in \mathbb{N}}$ lying in $\tilde{Q}_{c_1}$. We denote by $(\tilde{Y}^{*,k}, \tilde{Z}^{*,k})_{k \in \mathbb{N}}$ the sequence of processes associated to $(q^k)_{k \in \mathbb{N}}$, as given by (87). We first prove that the two sequences $(\tilde{Y}^{*,k})_{k \in \mathbb{N}}$ and $(\tilde{Z}^{*,k})_{k \in \mathbb{N}}$ are relatively compact with respect to the weak topologies on $L^1(\Omega \times [0, T], \mathbb{R}, \mathbb{P} \otimes \text{Leb}_{[0,T]})$ and $L^1(\Omega \times [0, T], \mathbb{R}^d, \mathbb{P} \otimes \text{Leb}_{[0,T]})$ respectively. The relative compactness of $(\tilde{Y}^{*,k})_{k \in \mathbb{N}}$ is established as in Step 1, by proving that the sequence is uniformly integrable. The latter is quite obvious: by (93), we know that the sequence $(|\tilde{Y}^{*,k}|)_{k \in \mathbb{N}}$ is dominated by $(\alpha q^k)_{k \in \mathbb{N}}$; by Step 1, the sequence $(q^k)_{k \in \mathbb{N}}$ is uniformly integrable and, therefore,
the sequence $(\tilde{Y}^{*,k})_{k \in \mathbb{N}}$ is also uniform integrable. To prove the relative compactness of the sequence $(Z^{*,k})_{k \in \mathbb{N}}$ in $L^1(\Omega \times [0, T], \mathbb{P} \otimes \text{Leb}_{[0,T]}, \mathbb{R}^d)$, we proceed as follows. By (103), we observe that, for any $k \in \mathbb{N}$, any subset $A \subset \Omega \times [0, T]$ in the progressive $\sigma$-field, and any real $\varepsilon > 0$,

$$ 
\begin{aligned}
\mathbb{E} \left[ \int_0^T 1_A(t) |\tilde{Z}_t^{k, \star}| dt \right] &= \mathbb{E} \left[ \int_0^T 1_A(t) 1_{\{q_t^k > 0\}} |\tilde{Z}_t^{k, \star}| dt \right] \\
&\leq \frac{1}{\varepsilon} \mathbb{E} \left[ \int_0^T 1_A(t) q_t^k dt \right] + \varepsilon \mathbb{E} \left[ \int_0^T 1_{\{q_t^k > 0\}} \frac{1}{q_t^k} |\tilde{Z}_t^{k, \star}|^2 dt \right].
\end{aligned}
$$

Letting $Z_t^{k, \star} = 1_{\{q_t^k > 0\}} \tilde{Z}^{k, \star}/q_t^k$, we deduce from (103) that there exists a constant $C$, independent of $k$, $A$ and $\varepsilon$, such that

$$ \mathbb{E} \left[ \int_0^T 1_A(t) |\tilde{Z}_t^{k, \star}| dt \right] \leq \frac{1}{\varepsilon} \mathbb{E} \left[ \int_0^T 1_A(t) q_t^k dt \right] + C\varepsilon. $$

By the uniform integrability property established in the first step, we know that the first term on the right-hand side can be made as small as desired by choosing $\text{Leb}_{[0,T]} \otimes \mathbb{P}(A)$ small enough. This proves that the collection $(\tilde{Z}^{*,k})_{k \in \mathbb{N}}$ is uniformly integrable, on $\Omega \times [0, T]$, equipped with $\mathbb{P} \otimes \text{Leb}_{[0,T]}$.

We now come back to the proof of the lower semi-continuity of $\tilde{S}$. Let $(q^k)_{k \in \mathbb{N}}$ be a sequence in $\tilde{Q}_{c_1}$ that converges for the weak topology on $L^1(\Omega \times [0, T], \mathbb{P} \otimes \text{Leb}_{[0,T]})$. The relative compactness of $(\tilde{Y}^{*,k}, \tilde{Z}^{*,k})_{k \in \mathbb{N}}$ allows us to extract a subsequence (still indexed by $k$) that converges in $L^1(\Omega \times [0, T], \mathbb{R}, \mathbb{P} \otimes \text{Leb}_{[0,T]}) \times L^1(\Omega \times [0, T], \mathbb{R}^d, \mathbb{P} \otimes \text{Leb}_{[0,T]})$ equipped with the product of the weak topologies (on each factor). Since the objective is to prove the lower semi-continuity of the convex functional $\tilde{S}$, we can replace $(q^k, \tilde{Y}^{*,k}, \tilde{Z}^{*,k})_{k \in \mathbb{N}}$ by a convex combination that converges for the strong topologies. We denote by $(q, \tilde{Y}^{*}, \tilde{Z}^{*})$ its (strong) limit. The objective is to pass to the limit in (87) and to prove that $q$ is a non-negative valued process that can be expanded as

$$ \forall t \in [0, T], \quad q_t = 1 + \int_0^t \tilde{Y}_s^* ds + \int_0^t \tilde{Z}_s^* \cdot dW_s. \qquad (105) $$

In fact, the difficulty is to pass to the limit in the stochastic integrals appearing in the expansion of each $q^k$. By (96),

$$ \sup_{k \in \mathbb{N}} \mathbb{E} \left[ \left( \int_0^T |\tilde{Z}_t^{*,k}|^2 dt \right)^{1/2} \right] < +\infty. $$

By Fatou's lemma, we deduce that $\tilde{Z}^*$ satisfies the same bound. And then, by a new uniform integrability argument (combining with the convergence in $L^1$),

$$ \forall \eta \in (0, 1), \quad \lim_{k \to \infty} \mathbb{E} \left[ \left( \int_0^T |\tilde{Z}_t^{*,k} - \tilde{Z}_t^*|^2 dt \right)^{\eta/2} \right] = 0, $$

which shows that

$$ \forall \eta \in (0, 1), \quad \lim_{k \to \infty} \mathbb{E} \left[ \sup_{t \in [0, T]} \left| \int_0^t (\tilde{Z}_s^{*,k} - \tilde{Z}_s^*) \cdot dW_s \right|^\eta \right] = 0. $$
This makes it possible to derive (105). Moreover, using Fatou's lemma, we deduce from the following three inequalities (with the last one following from (103))

$$ \mathbb{E}[q_T^{k,*}] \le \exp(\alpha T), \quad \mathbb{E}\left[\int_0^T \frac{|\tilde{Y}^{*,k}|}{q_t^k} dt\right] \le T\alpha, \quad \mathbb{E}\left[\int_0^T \frac{|\tilde{Z}^{*,k}|^2}{q_t^k} dt\right] \le C + 2\beta c_1, $$

that

$$ \mathbb{E}[q_T^*] \le \exp(\alpha T), \quad \mathbb{E}\left[\int_0^T 1_{\{q_t=0, \tilde{Y}_t^* \ne 0\}} dt\right] = 0, \quad \mathbb{E}\left[\int_0^T 1_{\{q_t=0, \tilde{Z}_t^* \ne 0\}} dt\right] = 0. $$

Letting $Y_t^* := 1_{\{q_t>0\}}\tilde{Y}_t^*/q_t$ and $Z_t^* := 1_{\{q_t>0\}}\tilde{Z}_t^*/q_t$, we deduce that (98) and (99) hold true (even though we do not have yet that $q \in \tilde{Q}_{c_1}$ which prevents us from applying Lemma 18 at this stage of the proof). We also have

$$ \mathbb{E}\left[\int_0^T q_t |Z_t^*|^2 dt\right] < +\infty. $$

In particular, $|Z_t^*|$ is (a.e.) finite when $q_t > 0$, and we can write, $\text{Leb}_{[0,T]} \otimes \mathbb{P}$ almost everywhere, $\tilde{Y}_t^* = q_t Y_t^*$ and $\tilde{Z}_t^* = q_t Z_t^*$. By repeating the proof of the first claim in item ii of Lemma 18, we also have that $\mathbb{P}(\{\sup_{t \in [0,T]} q_t > 0\} \cap \{\tau < T\}) = 0$, where $\tau := \inf\{t \in [0,T]; q_t = 0\}$.

We now come back to the definition of $\tilde{Q}_{c_1}$. Using the fact that $q^k \in \tilde{Q}_{c_1}$ and letting $\tau^k := \inf\{t \in [0,T]; q_t^k = 0\}$, for each $k \in \mathbb{N}$, we deduce from (95) (together with the first claim in item ii of Lemma 18, which allows us to derive the third line below) that

$$ \begin{aligned} \tilde{S}(q^k) &= \mathbb{E}\left[\int_0^{\tau^k} q_s^k f^*(s, Y_s^{*,k}, Z_s^{*,k}) ds\right] \\ &\ge \mathbb{E}\left[\int_0^{\tau^k} q_s^k \left(Y_s Y_s^{*,k} + Z_s \cdot Z_s^{*,k} - f(s, Y_s, Z_s)\right) ds\right] \quad (106) \\ &= \mathbb{E}\left[\int_0^T \left(Y_s \tilde{Y}_s^{*,k} + Z_s \cdot \tilde{Z}_s^{*,k} - q_s^k f(s, Y_s, Z_s)\right) ds\right], \end{aligned} $$

where $(Y, Z)$ is taken, for a certain $R > 0$, in the ball of center 0 and radius $R$ of the space $L^\infty(\mathbb{F}) \times L^\infty(\mathbb{F}, \mathbb{R}^d)$, i.e. for almost every $(\omega, s) \in \Omega \times [0, T]$,

$$ |Y_s(\omega)| + |Z_s(\omega)| \le R. $$

Letting $k$ tend to $+\infty$ in (106), we deduce that, for any $R > 0$,

$$ c_1 \ge \sup_{(Y,Z) \in \mathcal{B}_R} \mathbb{E} \left[ \int_0^T q_s (Y_s Y_s^* + Z_s \cdot Z_s^* - f(s, Y_s, Z_s)) \, ds \right]. \quad (107) $$

This prompts us to define, for any $R > 0$, the mapping $f_R^*: \Omega \times [0, T] \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}$, by letting

$$ f_R^*(t, y^*, z^*) := \sup_{|y|+|z| \le R} \{y^*y + z^* \cdot z - f(t, y, z)\}. \quad (108) $$
Let us remark that $(f_R^*)_{R \in \mathbb{N}}$ is a non-decreasing sequence, converging pointwise to $f^*$. By [15, Proposition 3.78], we have that

$$ 
\begin{aligned}
& \sup_{(Y,Z) \in \mathcal{B}_R} \mathbb{E} \left[ \int_0^T q_s (Y_s Y_s^* + Z_s \cdot Z_s^* - f(s, Y_s, Z_s)) \, ds \right] \\
& = \mathbb{E} \left[ \int_0^T q_s f_R^*(s, Y_s^*, Z_s^*) \, ds \right] = \mathbb{E} \left[ \int_0^T q_s f_R^*(s, Y_s^*, Z_s^*) \, ds \right],
\end{aligned}
\quad (109) $$

where we recall that $\tau = \inf\{t \in [0, T]; q_t = 0\}$. Combining (107) and (109) together with Fatou's lemma, we obtain

$$ 
\begin{aligned}
c_1 &\ge \liminf_{R \to +\infty} \mathbb{E} \left[ \int_0^T q_s f_R^*(s, q_s, \tilde{Y}_s^*, \tilde{Z}_s^*) \, ds \right] \\
&\ge \mathbb{E} \left[ \int_0^T q_s f^*(s, q_s, \tilde{Y}_s^*, \tilde{Z}_s^*) \, ds \right] = \tilde{S}(q),
\end{aligned}
\quad (110) $$

with the last identity following from (95) (which holds true here, thanks to the analysis achieved in the first part of this step). This shows that $\tilde{S}$ is weakly lower semi-continuous.

**Step 4: weak compactness of $\tilde{\mathcal{Q}}_{c_1}$**. Most of the work has been done in the previous steps. We know from the first step that any sequence $(q^k)_{k \in \mathbb{N}}$ is relatively compact for the weak topology on $L^1(\Omega \times [0, T], \mathbb{P} \otimes \text{Leb}_{[0,T]})$. By (105) in Step 3, we know that any weak limit $q$ can be expanded as in (87), is non-negative valued and satisfies $\mathbb{E}[q_T^*] \le \exp(\alpha T)$. And by (110), $\tilde{S}(q) \le c_1$.

**Step 5: weak continuity of $\mathcal{R}$**. We are thus given a sequence $(q^k)_{k \in \mathbb{N}} \in \tilde{\mathcal{Q}}_{c_1}$. By the previous step, there exists $q \in \tilde{\mathcal{Q}}_{c_1}$ such that, up to a subsequence, $(q^k)_{k \in \mathbb{N}}$ weakly converges (i.e., with respect to $\sigma(L^1, L^\infty)$) to $q$ as $k \to +\infty$. By concavity of $\mathcal{G}$, we have (with the shorthand notation $\mathcal{G}(q)$ for $\mathcal{G}(q, X_T^\psi)$, and similarly for the derivative $\delta_q \mathcal{G}(q_T)$)

$$ 
\begin{aligned}
\mathcal{R}(q^k) &= \mathcal{G}(q_T^k) + \mathbb{E} \left[ \int_0^T q_s^k \ell_s \, ds \right] \\
&\le \mathcal{G}(q_T) + \mathbb{E} \left[ (q_T^k - q_T) \delta_q \mathcal{G}(q_T) + \int_0^T q_s^k \ell_s \, ds \right] \quad (111) \\
&= \mathcal{R}(q) + \mathbb{E} \left[ (q_T^k - q_T) \delta_q \mathcal{G}(q_T) + \int_0^T (q_s^k - q_s) \ell_s \, ds \right].
\end{aligned}
$$

Recalling the growth Assumption A6 on $\mathcal{G}$, we have

$$ |\delta_q \mathcal{G}(q_T)| = |\delta_q \mathcal{G}(q_T, X_T^\psi)| \le L \left( 1 + |X_T^\psi|^{2-r} + \mathbb{E} \left[ q_T |X_T^\psi|^{2-r} \right] \right), $$

and then, $\delta_q \mathcal{G}$ admits exponential moments of all orders since $\psi$ is assumed to be bounded. Thanks to Step 1, Lemma 43 (applicable by Remark 21) yields that the sequences

$$ (q_T^k \delta_q \mathcal{G}(q_T))_{k \in \mathbb{N}}, \quad (q_s^k \ell_s)_{k \in \mathbb{N}} $$

are uniformly integrable. Since $(q^k)_{k \in \mathbb{N}}$ is assumed to converge weakly with respect to $\sigma(L^1, L^\infty)$, we deduce from Lemma 44 (which is applicable even though the sequence $(q^k)_{k \in \mathbb{N}}$ takes values in $\tilde{\mathcal{Q}}_{c_1}$ and possibly not in $\mathcal{Q}_{c_1}$) that

$$ \lim_{k \to +\infty} \mathbb{E} \left[ (q_T^k - q_T) \delta_q \mathcal{G}(q_T) + \int_0^T (q_s^k - q_s) \ell_s \, ds \right] = 0. $$
Then $\limsup_{k \to \infty} \mathcal{R}(q^k) \le \mathcal{R}(q)$ concluding the step.

**Step 6: Conclusion.** The statement follows from the combination of Steps 1 to 6. $\square$

**Lemma 22.** Let $c_1, c_2 > 0$ and $q \in \tilde{\mathcal{Q}}_{c_1}$. The mapping $\mathcal{A}_{c_2} \ni \psi \mapsto \tilde{\mathcal{J}}(q, \psi)$ is convex, lower semi-continuous. In addition, $\mathcal{A}_{c_2}$ is convex and weakly compact in $L^2(\mathbb{F}, \mathbb{Q}^0, \mathbb{R}^n)$, where $\mathbb{Q}^0 := \mathcal{E}_T(\int_0^T \partial_z f(t, 0, 0) \cdot dW_t)\mathbb{P}$. In particular, it is bounded in $M^\eta(\mathbb{F}, \mathbb{R}^n, \mathbb{P})$, for any $\eta \in (0, 1)$.

*Proof. Step 1: convexity and weak compactness of $\mathcal{A}_{c_2}$.* The convexity of $\mathcal{S}^*$, regarded as a $[0, +\infty]$-valued mapping defined on the space of $\mathbb{F}$-progressively measurable $\mathbb{R}^n$-valued processes, is a direct consequence of its definition (6): for each $q \in \mathcal{Q}$, the mapping $\psi \mapsto \mathbb{E}[\int_0^T q_s |\psi_s|^2 ds]$ is convex since $q$ takes non-negative values. Therefore, $\mathcal{S}^*$ is convex as the supremum of a family of convex mappings. As a result, the set $\mathcal{A}_{c_2}$ is convex.

The relative weak compactness of $\mathcal{A}_{c_2}$ in $L^2(\mathbb{F}, \mathbb{Q}^0, \mathbb{R}^n)$ is a direct consequence of the fact that

$$c_2 \ge \mathcal{S}^*(\psi) = \sup_{q \in \mathcal{Q}} \left\{ \mathbb{E} \left[ \int_0^T q_t |\psi_t|^2 dt \right] - \frac{1}{\gamma} \mathcal{S}(q) \right\} \ge \mathbb{E} \left[ q_T^0 \int_0^T |\psi_t|^2 dt \right] - \frac{1}{\gamma} \mathcal{S}(q^0),$$

with $q^0$ as in (84). Then, we are left to prove the weak closure property of $\mathcal{A}_{c_2}$. By convexity of the latter, it suffices to show that it is closed for the strong topology (on $L^2(\mathbb{F}, \mathbb{R}^n, \mathbb{Q}^0)$). We thus consider an $\mathcal{A}_{c_2}$-valued sequence $(\psi^k)_{k \in \mathbb{N}}$ that (strongly) converges in $L^2(\mathbb{F}, \mathbb{R}^n, \mathbb{Q}^0)$ to some limit $\bar{\psi}$. Then, for all $q \in \mathcal{Q}$,

$$\mathbb{E} \left[ \int_0^T q_t |\psi_t^k|^2 dt \right] - \frac{1}{\gamma} \mathcal{S}(q) \le c_2.$$

By Fatou's lemma, the mapping $\psi \mapsto \mathbb{E}[\int_0^T q_t |\psi_t|^2 dt]$ is lower semi-continuous for the strong topology on $L^2(\mathbb{F}, \mathbb{R}^n, \mathbb{Q}^0)$. We deduce that

$$\mathbb{E} \left[ \int_0^T q_t |\psi_t|^2 dt \right] - \frac{1}{\gamma} \mathcal{S}(q) \le c_2,$$

which implies that $\bar{\psi} \in \mathcal{A}_{c_2}$, as required.

Observing that $d\mathbb{P}/d\mathbb{Q}^0$ has finite exponential moments of any order under $\mathbb{Q}^0$ and using again the fact that $\sup_{\psi \in \mathcal{A}_{c_2}} \mathbb{E}^{\mathbb{Q}^0}[\int_0^T |\psi_t|^2 dt] < +\infty$, we deduce that $\mathcal{A}_{c_2}$ is a bounded subset of $M^\eta(\mathbb{F}, \mathbb{R}^n, \mathbb{P})$ for any $\eta \in (0, 2)$.

**Step 2: convexity of $\mathcal{A}_{c_2} \ni \psi \mapsto \tilde{\mathcal{J}}(q, \psi)$, for a fixed $q \in \tilde{\mathcal{Q}}_{c_1}$.** The convexity of the mapping

$$\mathcal{A}_{c_2} \ni \psi \mapsto \mathbb{E} \left[ \int_0^T q_s \ell(s, \psi_s) ds \right],$$

is a direct consequence of the convexity of $\ell$ and the non-negativity of $q$. We now turn to the convexity of the mapping $\mathcal{A}_{c_2} \ni \psi \mapsto \mathcal{G}(q_T, X_T^\psi)$. For $\psi^0, \psi^1 \in \mathcal{A}_{c_2}$, we denote by $(X^i)_{i=0,1}$ the solutions to the two state equations associated with $\psi^0$ and $\psi^1$ respectively. Moreover, for any $\theta \in (0, 1)$, we call $X^\theta$ the solution to the state equation associated with $\theta\psi^1 + (1 - \theta)\psi^0$. We notice that

$$X_T^\theta = \theta X_T^1 + (1 - \theta) X_T^0.$$
Using the fact that $\mathcal{G}$ is convex with respect to its second variable by Assumption A8 together with the last equality, it is easy to deduce that $[0, 1] \ni \theta \mapsto \mathcal{G}(q_T, X_T^\theta)$ is convex. This concludes the step.

*Step 3: lower semi-continuity of $A_{c_2} \ni \psi \mapsto \tilde{\mathcal{J}}(q, \psi)$, for a fixed $q \in \tilde{\mathcal{Q}}_{c_1}$.* The proof relies on the fact that, as explained in Remark 21, the duality inequality (14) extends to elements $q \in \tilde{\mathcal{Q}}_{c_1}$.

Let $(\psi^k)_{k \in \mathbb{N}}$ be a sequence with values in $A_{c_2}$. By the extended version of (14), we have, for every $k \in \mathbb{N}$,

$$ \gamma \mathbb{E} \left[ \int_0^T q_t |\psi_t^k|^2 dt \right] \le \mathcal{S}^*(\psi^k) + \tilde{\mathcal{S}}(q) \le c_1 + c_2, $$

from which we deduce that the sequence $(\psi^k)_{k \in \mathbb{N}}$ lies in a weakly compact subset of the space of $\mathbb{F}$-progressively measurable $\mathbb{R}^n$-valued processes that are square integrable under the measure $\mathbb{Q}$, defined by $\mathbb{Q}(E) := \mathbb{E} \int_0^T 1_{E} q_t dt$, for any event of $\Omega \times [0, T]$; with a slight abuse of notation, we will denote this space by $L^2(\mathbb{F}, \mathbb{R}^n, \mathbb{Q})$. Up to a subsequence, there exists a weak limit in $L^2(\mathbb{F}, \mathbb{R}^n, \mathbb{Q})$, which we denote $\bar{\psi}$. As the purpose is to prove that $\mathcal{J}(q, \bar{\psi}) \le \liminf_{k \to +\infty} \mathcal{J}(q, \psi^k)$, and the functional $\mathcal{J}$ is convex with respect to the first argument, we can replace the sequence $(\psi^k)_{k \in \mathbb{N}}$ by a sequence of convex combinations (of the $(\psi_k)_{k \in \mathbb{N}}$'s) that converges in $L^2(\mathbb{F}, \mathbb{R}^n, \mathbb{Q})$ (equipped with the strong topology) to $\bar{\psi}$. Following the second step in the proof of Lemma 18, we know that there exists a constant $c > 0$ such that $q_t \ge c \mathbb{E}[q_T | \mathcal{F}_t]$, for all $t \in [0, T]$. In particular, if $E$ is in the progressive $\sigma$-field, then

$$ \mathbb{Q}(E) \ge c \mathbb{E}[q_T 1_E]. \qquad (112) $$

By convexity of the function $\mathcal{G}$ in the second argument, see A8, we also have

$$ \forall k \in \mathbb{N}, \quad \mathcal{J}(q, \psi^k) \ge \mathcal{J}(q, \bar{\psi}) + \mathbb{E} \left[ A^k + B^k \right], \qquad (113) $$

where

$$ A^k := \delta_X \mathcal{G} (q_T, X_T^{\bar{\psi}}) \cdot (X_T^{\psi^k} - X_T^{\bar{\psi}}), \quad B^k := \int_0^T q_s (\ell(s, \psi_s^k) - \ell(s, \bar{\psi}_s)) \, ds. $$

We study the two sequences of random variables $(A^k)_{k \in \mathbb{N}}$ and $(B^k)_{k \in \mathbb{N}}$ separately. We start with $(A^k)_{k \in \mathbb{N}}$. By the growth Assumption A6 on $\delta_X \mathcal{G}$, we have

$$ |A^k| \le L q_T \left( 1 + |X_T^{\bar{\psi}}|^{1-r} + \mathbb{E} \left[ q_T |X_T^{\bar{\psi}}|^{2-r} \right] \right) |X_T^{\psi^k} - X_T^{\bar{\psi}}| \qquad (114) $$

By Lemma 40 in Appendix B (together with Remark 21), we know that

$$ \mathbb{E} [q_T |X_T^{\bar{\psi}}|^{2-r}] \le C (1 + \tilde{\mathcal{S}}(q) + \mathcal{S}^*(\bar{\psi})) \le C, $$

since $\bar{\psi} \in L^2(\mathbb{F}, \mathbb{R}^n, \mathbb{Q})$ and $q \in \tilde{\mathcal{Q}}_{c_1}$. Taking the expectation both sides of (114) yields

$$ \mathbb{E}[|A^k|] \le C \mathbb{E} \left[ q_T \left( 1 + |X_T^{\bar{\psi}}|^{1-r} \right) |X_T^{\psi^k} - X_T^{\bar{\psi}}| \right], \qquad (115) $$

We distinguish between the cases $r = 0$ and $r = 1$.
Sub-step 3a: analysis of $(A^k)_{k \in \mathbb{N}}$ when $r = 0$. When $r = 0$, we have

$$
\begin{align*}
\mathbb{E}[|A^k|] &\le L \mathbb{E} \left[ q_T \left( 1 + |X_T^{\bar{\psi}}| \right) |X_T^{\psi^k} - X_T^{\bar{\psi}}| \right] \\
&\le C \mathbb{E} \left[ q_T \left( 1 + |X_T^{\bar{\psi}}| \right) \int_0^T |\psi_t^k - \bar{\psi}_t| dt \right] \\
&\le C \mathbb{E} \left[ q_T \left( 1 + |X_T^{\bar{\psi}}|^2 \right) \right]^{1/2} \mathbb{E} \left[ q_T \int_0^T |\psi_t^k - \bar{\psi}_t|^2 dt \right]^{1/2},
\end{align*}
$$

where the last line follows from the Cauchy-Schwarz inequality, for some constant $C > 0$. Then, $\lim_{k \to +\infty} \mathbb{E}[|A^k|] = 0$, by (112) and strong convergence of $(\psi^k)_{k \in \mathbb{N}}$ in $L^2(\mathbb{F}, \mathbb{R}^n, \mathbb{Q})$.

Sub-step 3b: analysis of $(A^k)_{k \in \mathbb{N}}$ when $r = 1$. When $r = 1$, there exists a constant $C > 0$ such that

$$
\begin{align}
\mathbb{E}[|A^k|] &\le C \mathbb{E} \left[ q_T \left| X_T^{\psi^k} - X_T^{\bar{\psi}} \right| \right] \nonumber \\
&\le C \left( \mathbb{E} \left[ q_T \int_0^T \left| \psi_t^k - \bar{\psi}_t \right| dt \right] + \mathbb{E} \left[ q_T \left| \int_0^T \left( \sigma(t, \psi_t^k) - \sigma(t, \bar{\psi}_t) \right) dW_t \right| \right] \right) \nonumber \\
&=: a_1^k + a_2^k. \tag{116}
\end{align}
$$

Clearly, $(a_1^k)_{k \in \mathbb{N}}$ converges to 0 as $k \to +\infty$. To handle $(a_2^k)_{k \in \mathbb{N}}$, we use the same family $(q^\theta)_{\theta \in [0,1]}$ as in Remark 21. We recall that each $q^\theta$ is positive valued and belongs to $Q_{c_1}$ (provided that $c_1$ is large enough, which is not a restriction here). Below, we write the expansion (4) of $q^\theta$, for $\theta \in [0, 1)$, in the form

$$
dq_t^\theta = q_t^\theta Y_t^{\star, \theta} dt + q_t^\theta Z_t^{\star, \theta} \cdot dW_t, \quad t \in [0, T],
$$

with initial condition $q_0^\theta = 1$. We notice that

$$
a_2^k \le \frac{1}{1-\theta} \mathbb{E} \left[ q_T^\theta \left| \int_0^T \left( \sigma(t, \psi_t^k) - \sigma(t, \bar{\psi}_t) \right) dW_t \right| \right] =: \frac{1}{1-\theta} a_2^{k,\theta}.
$$

By (102), we have $\sup_{\theta \in [0,1]} \mathbb{E}[h(q_t^\theta)] < +\infty$. By Lemma 39, we also know that $\mathbb{Q}^\theta := \mathcal{E}_T(\int_0^T Z_t^{\star,\theta} \cdot dW_t)\mathbb{P}$ is a probability measure. It satisfies $\exp(-\alpha T)q_T^\theta \le \mathbb{d}\mathbb{Q}^\theta/\mathbb{d}\mathbb{P} \le \exp(\alpha T)q_T^\theta$. Therefore, letting $(\tilde{W}_t^\theta := W_t - \int_0^t Z_s^{\star,\theta}ds)_{t \in [0,T]}$, we deduce from Girsanov's theorem that

$$
\begin{align}
a_2^{k,\theta} &\le C \mathbb{E}^{\mathbb{Q}^\theta} \left[ \left| \int_0^T \left( \sigma(t, \psi_t^k) - \sigma(t, \bar{\psi}_t) \right) d\tilde{W}_t^\theta \right| \right] \nonumber \\
&\quad + C \mathbb{E}^{\mathbb{Q}^\theta} \left[ \left| \int_0^T \left( \sigma(t, \psi_t^k) - \sigma(t, \bar{\psi}_t) \right) Z_t^{\star,\theta} dt \right| \right] \tag{117} \\
&\le C \mathbb{E}^{\mathbb{Q}^\theta} \left[ \left| \int_0^T |\psi_t^k - \bar{\psi}_t|^2 dt \right| \right]^{1/2} \left( 1 + \mathbb{E}^{\mathbb{Q}^\theta} \left[ \int_0^T |Z_t^{\star,\theta}|^2 dt \right]^{1/2} \right). \nonumber
\end{align}
$$

By Lemma 39 and thanks to the bound $\sup_{\theta \in [0,1]} \mathbb{E}[h(q_T^\theta)] < +\infty$, we have

$$
\sup_{\theta \in [0,1)} \mathbb{E}^{\mathbb{Q}^{\theta}} \left[ \int_{0}^{T} |Z_{t}^{\star, \theta}|^{2} dt \right] < +\infty. \qquad (118)
$$