tochastic Maximum Principle with Defaul

In this paper, we derive sufficient and necessary maximum principles for a stochastic optimal control problem where the system state is given by a controlled stochastic differential equation with default. We prove existence of a unique solution to the controlled default stochastic differential equation. Furthermore, we prove existence and uniqueness of solution to the adjoint backward stochastic differential equation which appears in connection to the maximum principles. Finally, we apply the maximum principles to solve a utility maximisation problem with logarithmic utility functions.

#### Keywords:

Backward Stochastic Differential Equations with Default, Single Jump, Progressive Enlargement of Filtration.

Introduction
============

In this paper we study optimal control of stochastic systems with default. By saying that the system is with default we mean that the system may be stopped at a random time $\tau$, which is neither an $\mathbb{F}$-stopping time nor $\mathcal{F}_{T}$-measurable. The reason for the default comes from outside the Brownian motion driving the system. We denote by $H_{t}=\mathbf{1}_{\tau\leq t},\ t\in\lbrack0,T],$ and consider the filtration $\mathbb{G}$ obtained by enlarging progressively the filtration $\mathbb{F}$ by the process $H$, i.e., $\mathbb{G}$ is the smallest filtration satisfying the usual assumptions of completeness and right-continuity, which contains the filtration $\mathbb{F}$ and has $H$ as an adapted process and that makes $\tau$ a $\mathbb{G}$-stopping time.

In contrast, in this paper, we derive maximum principles under the enlarged filtration rather than the Brownian one. By considering systems with single jumps and using the martingale property we transform the SDE with default to an SDE driven by martingales under the enlarged filtration $\mathbb{G}$. The associated adjoint process is a solution of a BSDE driven by both the Brownian motion and the pure jump martingale $M$ that comes from the martingale property. For information about stochastic control with jumps coming from the compensated Poisson random measure (independent of the Brownian motion), we refer to Øksendal *et al* @FOS [@BO] and for default jumps, we refer to Pham @P.

For more details about enlargement progressive of filtration, we refer to Jacod @Jac, Jeanblanc @aj, Jeulin @Jeulin1980, Jeulin and Yor @JY and Song @songspliting.

In this work, we mostly focus on the book of Aksamit and Jeanblanc @aj. It is organised as follows:

-   In Section 2 we give some preliminaries about enlargement progressive of filtration.

-   Then in Section 3 we prove existence and uniqueness of solutions of systems with default.

-   In Section 4 we study the stochastic maximum principle and we derive sufficient and necessary conditions for optimality. Moreover, we prove existence and uniqueness of BSDEs with a single jump.

-   Finally, we apply our results to solve logarithmic utility maximisation problem for a defaultable wealth.

Framework
=========

Let $(\Omega,\mathcal{G},P)$ be a complete probability space. We assume that this space is equipped with a one-dimensional standard Brownian motion $W$ and we denote by $\mathbb{F}:=(\mathcal{F}_{t})_{t\geq0}$ the right continuous complete filtration generated by $W$. On this space, let $\tau$ be a random time. This random time may represent a default time in credit- or counterparty risk, or a death time in actuarial issues. The random time $\tau$ is not assumed to be an $\mathbb{F}$-stopping time. Therefore, we will use filtration enlargement. Let $\mathbb{G}$ be the smallest right continuous extension of $\mathbb{F}$ that turns $\tau$ into a $\mathbb{G}$-stopping time. More precisely $\mathbb{G}:=(\mathcal{G}_{t})_{t\geq0}$, where ${\mathcal}{G}_t$ is defined by $$\mathcal{G}_{t}:=\bigcap_{\varepsilon>0}\tilde{\mathcal{G}}_{t+\varepsilon
}\;,$$ for all $t\geq0$, where $\tilde{\mathcal{G}}_{s}:=\mathcal{F}_{s}\vee
\sigma(\mathbf{1}_{\tau\leq u}\;,u\in\lbrack0,s])$, for all $s\geq0$.

We let $\mathcal{P}(\mathbb{G})$ denote the $\sigma$-algebra of $\mathbb{G}$-predictable subsets of $\Omega\times\lbrack0,T]$, i.e. the $\sigma$-algebra generated by the left-continuous $\mathbb{G}$-adapted processes.

We shall recall some classical tools in the theory of progressive enlargement. We will state the so-called immersion or hypothesis $(H)$.

$(H)$

:   The process $W$ is a $\mathbb{G}$-Brownian motion. We observe that, since the filtration $\mathbb{F}$ is generated by the Brownian motion $W$, this is equivalent with the fact that all $\mathbb{F}$-martingales are also $\mathbb{G}$-martingales. Moreover, it also follows that the stochastic integral ${\textstyle\int_{0}^{t}}
    X_{s}dW_{s}$ is well defined for all $\mathcal{P}(\mathbb{G})$-measurable processes $X$ such that ${\textstyle\int_{0}^{t}}
    \left\vert X_{s}\right\vert ^{2}ds<\infty,$ for all $t\geq0$.

\[Aksamit and Jeanblanc @aj\]

Suppose there exists a $\mathbb{G}$- predictable (resp. $\mathbb{F}$-predictable) process $\lambda^{\mathbb{G}}$ (resp. $\lambda:=\lambda
^{\mathbb{F}}$) with $\lambda_{s}^{\mathbb{G}}=\mathbf{1}_{s\leq\tau}\lambda_{s}^{\mathbb{F}}$, such that $$M_{t}:=H_{t}-{\textstyle\int_{0}^{t}}
\lambda_{s}^{\mathbb{G}}ds=H_{t}-{\textstyle\int_{0}^{t\wedge\tau}}
\lambda_{s}^{\mathbb{F}}ds,\qquad t\geq0,$$ is a $\mathbb{G}$-martingale with single jump time $\tau$ and the process $\lambda^{\mathbb{G}}$ (resp. $\lambda^{\mathbb{F}}$) is called the ${\mathbb{G}}$-intensity (resp. ${\mathbb{F}}$-intensity) of $\tau$.

In the above, we assume that the process $\lambda^{\mathbb{G}}$ is upper bounded by a constant $c$.

We shall give the predictable representation theorem (PRT).

\[Aksamit and Jeanblanc @aj\]Under hypothesis $(H)$, any ${\mathbb{G}}$-martingale $Y$ admits a representation of the form $$Y_{t}=Y_{0}+{\textstyle\int_{0}^{t}}
\varphi_{s}dW_{s}+{\textstyle\int_{0}^{t}}
\gamma_{s}dM_{s},$$ where $M$ is the compensated martingale of $H$, and $\varphi,\gamma$ are ${\mathbb{G}}$-predictable processes. Moreover, $\varphi_{s}=Y_{s}-Y_{s}^{\mathbb{F}}$ is the predefault value of $Y$.

Throughout this section, we introduce some basic notations and spaces.

-   $S_{\mathbb{G}}^{2}$ is the subset of $\mathbb{R}
    $-valued $\mathbb{G}$-adapted cadlag processes $\left(  Y_{t}\right)
    _{t\in\left[  0,T\right]  }$, such that $$\left\Vert Y\right\Vert _{S_{\mathbb{G}}^{2}}^{2}:=\mathbb{E[}\underset
    {t\in\left[  0,T\right]  }{\sup}\left\vert Y_{t}\right\vert ^{2}]<\infty.$$

-   $H_{\mathbb{G}}^{2}$ is the subset of $\mathbb{R}
    $-valued $\mathcal{P}(\mathbb{G})$-measurable processes $\left(  Z_{t}\right)
    _{t\in\left[  0,T\right]  },$ such that $$\left\Vert Z\right\Vert _{H_{\mathbb{G}}^{2}}^{2}:=\mathbb{E[}{\textstyle\int_{0}^{T}}
    \left\vert Z_{t}\right\vert ^{2}dt]<\infty.$$

-   $L^{2}(\lambda)$ is the subset of $\mathbb{R}
    $-valued $\mathcal{P}(\mathbb{G})$-measurable processes $\left(  U_{t}\right)
    _{t\in\left[  0,T\right]  },$ such that $$\left\Vert U\right\Vert _{L^{2}(\lambda)}^{2}:=\mathbb{E[}{\textstyle\int_{0}^{T\wedge\tau}}
    \lambda_{t}\left\vert U_{t}\right\vert ^{2}dt]<\infty.$$

With these assumptions at hans, we are ready to study the controlled stochastic differential equation (SDE) with default.

The Controlled Default Stochastic Differential Equation {#sec: SDE}
=======================================================

In this section, we introduce the controlled SDE with default, and prove existence and uniqueness of a solution to this equation.

Let $(u_t)_{t \geq 0}$ be a control process. We let $\mathcal{A}$ be the set of admissible controls. Assume that $\mathcal{A}$ contains a given set of càdlàg processes in $L^{2}(\Omega\times
\lbrack0,T])$, with values in $\mathcal{V} \subseteq \mathbb{R}$.

Consider the controlled SDE with default under the enlarged filtration $\mathbb{G}$ $$\left\{
\begin{array}
[c]{ll}dX_{t} & =b(t,X_{t},u_{t})dt+\sigma(t,X_{t},u_{t})dW_{t}+\gamma(t,X_{t},u_{t})dH_{t},\\
X_{0} & =x_{0},
\end{array}
\right.  \label{eq: default}$$

where the coefficient functions are as follows: $$\begin{aligned}
b &  :\Omega\times\lbrack0,T]\times\mathbb{R}\times\mathcal{V}\rightarrow
\mathbb{R},\\
\sigma &  :\Omega\times\lbrack0,T]\times\mathbb{R}\times\mathcal{V}\rightarrow\mathbb{R},\\
\gamma &  :\Omega\times\lbrack0,T]\times\mathbb{R}\times\mathcal{V}\rightarrow\mathbb{R},\end{aligned}$$ and the initial value $x_{0}\in\mathbb{R}
.$

Note that we often suppress the $\omega$ for ease of notation. So, for instance, we write $b(t,X_{t},u_{t})$ instead of $b(\omega,t,X_{t}(\omega),u_{t}(\omega))$.

We make the following set of assumptions on these coefficient functions.

\[assumption: E\]

1.  The functions $b(\omega,t,\cdot)$, $\sigma(\omega,t\cdot)$ and $\gamma(\omega,t,\cdot)$ are assumed to be $C^{1}$ for each fixed $\omega,t$.

2.  The functions $b(\cdot,x,u)$ and $\sigma(\cdot,x,u)$ are $\mathbb{G}$-progressively measurable, and $\gamma(\cdot,x,u)$ is $\mathbb{G}-$predictable, for each $x,u$.

3.  *Integrability condition:* The functions $b(\cdot,0,u)$, $\sigma(\cdot,0,u)$ and $\gamma(\cdot,0,u)$ are integrable for each $t,u,\omega$.

4.  *Lipschitz condition:* The functions $b,\sigma$ are Lipschitz continuous in the variable $x$, with the Lipschitz constant independent of the variables $t,u,\omega$.

5.  *Linear growth:* The functions $b,\sigma,\gamma$ satisfy the linear growth condition in the variable $x$, with the linear growth constant independent of the variables $t,u,\omega$.

Note also that under the previous assumption (H), the process $W$ is a $(P,\mathbb{G})$-Brownian motion.

By basic algebra, we can rewrite the SDE as an SDE involving the martingale $M_{t}$ instead of $H_{t}$. $$\left\{
\begin{array}
[c]{ll}dX_{t} & =B(t,X_{t},u_{t})dt+\sigma(t,X_{t},u_{t})dW_{t}+\gamma(t,X_{t},u_{t})dM_{t}\\
X_{0} & =x_{0},
\end{array}
\right.  \label{eq: SDE_default}$$ with $dH_{t}=dM_{t}+\lambda_{t}\mathbf{1}_{t<\tau}dt$ and $B(t,X_{t},u_{t}):=b(t,X_{t},u_{t})+\lambda_{t}(\mathbf{1}_{t<\tau}-H_{t})\gamma(t,X_{t},u_{t}).$ Note also that $M_{t}=H_{t}-{\textstyle\int_{0}^{t\wedge\tau}}
\lambda_{s}ds$ is a pure jump $\mathbb{G}$-martingale.

One can extend this framework to include multiple jumps. Let $\tau_{1},\tau_{2},\ldots,\tau_{n}$ be $n$ random jump times (not necessarily stopping times). Then, let $$H_{t} := \sum_{i=1} ^{n} \boldsymbol{1}_{t <\tau_{i}}.$$ As before, there exists a process $\lambda_{t}$ such that $$M_{t} := H_{t} - \int_{0}^{t\wedge\tau}\lambda_{s}ds$$ is a martingale. Though the framework is very similar, note that we no longer have $H_{t} \in\{0,1\}$ which significantly alters the calculations. The extension to multiple jumps is a work in progress. Note also that this extension is closely related to Hawkes processes and more general self-exciting processes, see e.g. Hawkes @Hawkes1, Hawkes and Oakes @Hawkes2 and Eyjolfsson and Tj<span>ø</span>stheim @EyjolfssonTjostheim. Self-exciting processes, and in particular Hawkes processes, are frequently used to model volatility clustering in stochastic processes.

We introduce some notation, which will be used in the remaining part of the paper:

\[not: delta\]

In the following, $$\begin{array}{llll}
\delta M_t := \lim_{\Delta t \rightarrow 0} \Delta M_t := \lim_{\Delta t \rightarrow 0} \{ M_{t + \Delta t} - M_{t} \}, \\[\smallskipamount]
\delta H_t := \lim_{\Delta t \rightarrow 0} \Delta H_t := \lim_{\Delta t \rightarrow 0} \{ H_{t + \Delta t} - H_{t} \}, 
\end{array}$$

To prove existence and uniqueness of SDE , we check Assumption 3.1 for the new drift term $B(t,x,u):=b(t,x,u)-\lambda_{t}(\mathbf{1}_{t<\tau}-H_{t})\gamma(t,x,u)$:

-   *Integrability condition:*$$\begin{array}
    [c]{ll}E[\int_{0}^{T}|B(t,0,0)|^{2}dt] & =E[\int_{0}^{T}|b(t,0,0)-\lambda
    _{t}(\mathbf{1}_{t<\tau}-H_{t})\gamma(t,0,0)|^{2}dt]\\
    & \leq2E[\int_{0}^{T}|b(t,0,0)|^{2}dt+\int_{0}^{T}\lambda_{t}^{2}|\gamma(t,0,0)|^{2}dt].
    \end{array}$$ Since $b,\gamma$ are integrable and $\lambda$ is bounded, we get the result$$E[\int_{0}^{T}|B(t,0,0)|^{2}dt]<\infty,$$ for each $t\in\lbrack0,T].$

-   *Linear growth:* There exists $L>0$ such that for for each $t\in
    \lbrack0,T]$ and $x\in\mathbb{R}
    $, we have$$\begin{array}
    [c]{ll}|B(t,x,u)| & =|b(t,x,u)-\lambda_{t}(\mathbf{1}_{t<\tau}-H_{t})\gamma(t,x,u)|\\
    & \leq|b(t,x,u)|+\lambda_{t}(\mathbf{1}_{t<\tau}-H_{t})|\gamma(t,x,u)|\\
    & \leq L(1+|x|),
    \end{array}$$ where the last inequality follows from the linear growth of $b,\gamma$ and the boundedness of $\lambda.$

-   *Lipschitz condition:* There exists $\psi^{\prime}>0$ such that for for each $t\in\lbrack0,T]$ and $x,x^{\prime}\in\mathbb{R}
    $, we have $$\begin{array}
    [c]{l}|B(t,x,u)-B(t,x^{\prime},u)|\\
    =|b(t,x,u)-b(t,x^{\prime})-\lambda_{t}(\mathbf{1}_{t<\tau}-H_{t})(\gamma(t,x,u)-\gamma
    (t,x^{\prime},u)|\\
    \leq|b(t,x,u)-b(t,x^{\prime},u)|+\lambda_{t}(\mathbf{1}_{t<\tau}-H_{t})|\gamma
    (t,x,u)-\gamma(t,x^{\prime},u)|\\
    \leq\psi(\left\vert x-x^{\prime}\right\vert +\lambda_{t}(\mathbf{1}_{t<\tau}-H_{t})|x-x^{\prime}|)\\
    \leq\psi(1+\lambda_{t})\left\vert x-x^{\prime}\right\vert \\
    \leq\psi^{\prime}\left\vert x-x^{\prime}\right\vert ,
    \end{array}
    \nonumber$$ where we have used the fact that $b,\gamma$ satisfy the Lipschitz condition above and $\lambda$ is bounded.

Now that we know the standard assumptions are satisfied for the new drift term $B$ as well, we are ready to prove existence of a unique solution to the SDE with default . The idea of the proof is classical: We will show that $X$ is a contraction in a suitable Banach space and use Banach's fixed point theorem to get existence of a unique solution of the SDE .

\[Existence of unique solution to the SDE with default\] \[prop\] Under the above assumptions, there exists a unique solution $X\in
S_{\mathbb{G}}^{2}$ of SDE .

<span>Proof.</span> Let us first introduce a norm in the Banach space $V:=S_{\mathbb{G}}^{2}$ for $\beta >0$, for $X\in  V$: $\left\Vert X\right\Vert _{V}:=\mathbb{E[}{\textstyle\int_{0}^{T}}
e^{-\beta s}\left\vert X_{s}\right\vert ^{2}ds]\text{. }$ Setting $\Phi\left(  x\right)  :=X$ we define a mapping $\Phi:V\rightarrow V,$ for fixed $u\in\mathcal{A}$, as follows $$X_{t}:=x_{0}+{\textstyle\int_{0}^{t}}
B\left(  s,x_{s}\right)  ds+{\textstyle\int_{0}^{t}}
\sigma\left(  s,x_{s}\right)  dW_{s}+{\textstyle\int_{0}^{t}}
\gamma\left(  s,x_{s}\right)  dM_{s}.$$ We are going to prove that $\Phi:(V,\left\Vert
\cdot\right\Vert _{V})\rightarrow(V,\left\Vert \cdot\right\Vert _{V
})$ is contracting. Indeed, we consider arbitrary $x^{i}\in V$, $i=1,2,$ and we put $X^{i}:=\Phi\left(  x^{i}\right)  $, $i=1,2$. Let $\bar{x}:=x^{1}-x^{2}$ and $\bar{X}:=X^{1}-X^{2}.$ Then, by applying Itô's formula to $(e^{-\beta t}\left\vert \bar{X}_{t}\right\vert ^{2})_{t\geq0}$, and taking the expectation, leads to $$\begin{array}
[c]{l}\mathbb{E[}e^{-\beta t}\left\vert \bar{X}_{t}\right\vert ^{2}]+\mathbb{E[}{\textstyle\int_{0}^{t}}
\beta e^{-\beta s}\left\vert \bar{X}_{s}\right\vert ^{2}ds]=2\mathbb{E[}{\textstyle\int_{0}^{t}}
e_{s}^{-\beta s}\bar{X}_{s} \{b(s,x_{s}^{1})-b(s,x_{s}^{2})\}ds]\\[\smallskipamount]
+\mathbb{E[}{\textstyle\int_{0}^{t}}
e^{-\beta s}|\sigma(s,x_{s}^{1})-\sigma(s,x_{s}^{2})|^{2}ds]+\mathbb{E[}{\textstyle\int_{0}^{t}}
e^{-\beta s}\lambda_{s}|\gamma(s,x_{s}^{1})-\gamma(s,x_{s}^{2})|^{2}ds],
\end{array}$$ where we have used the fact that $M_{t}=H_{t}-{\textstyle\int_{0}^{t\wedge\tau}}
\lambda_{s}ds$ is a pure jump martingale. Note that $\left(  \delta M_{s}\right)  ^{2}=\delta M_{s}=\delta H_{s}$ (see Notation \[not: delta\]). Thus, the covariation process, we refer to Definition 2.1, of $M$ is $H:$ $$\left[
M\right]  _{t}=\underset{0\leq s\leq t}{\sum}\left(  \delta
M_{s}\right)  ^{2}=\underset{0\leq s\leq t}{\sum}\left(  \delta
H_{s}\right)  ^{2}=H_{t},$$ The quadratic variation is: $$\left\langle M\right\rangle _{t}=\int_{0}^{t\wedge\tau}\lambda_{s}ds=\int
_{0}^{t}\lambda_{s}(\mathbf{1}_{s<\tau}-H_{s})ds,t\geq0.$$ Consequently,$$\begin{aligned}
{\displaystyle\int_{0}^{t}}
|\gamma(s,x_{s}^{1})-\gamma(s,x_{s}^{2})|^{2}d\left\langle M\right\rangle _{s}
&  ={\displaystyle\int_{0}^{t}}
\lambda_{s}(\mathbf{1}_{s<\tau}-H_{s})|\gamma(s,x_{s}^{1})-\gamma(s,x_{s}^{2})|^{2}ds\\
&  \leq{\displaystyle\int_{0}^{t}}
\lambda_{s}|\gamma(s,x_{s}^{1})-\gamma(s,x_{s}^{2})|^{2}ds.\end{aligned}$$ Using the assumptions$$\begin{array}
[c]{l}\mathbb{E[}e^{-\beta t}\left\vert \bar{X}_{t}\right\vert ^{2}]+\mathbb{E[}{\textstyle\int_{0}^{t}}
\beta e^{-\beta s}\left\vert \bar{X}_{s}\right\vert ^{2}ds]\leq C\mathbb{E[}{\textstyle\int_{0}^{t}}
e^{-\beta s}\left\vert \bar{X}_{s}\right\vert \left\vert \bar{x}_{s}\right\vert ds]\\[\smallskipamount]
+C^{2}\mathbb{E[}{\textstyle\int_{0}^{t}}
e^{-\beta s}\left\vert \bar{x}_{s}\right\vert )^{2}ds]\leq\frac{C^{2}}{\varepsilon}\mathbb{E[}{\textstyle\int_{0}^{t}}
e^{-\beta s}\left\vert \bar{X}_{s}\right\vert ^{2}ds]+(\varepsilon
+C^{2})\mathbb{E[}{\textstyle\int_{0}^{t}}
e^{-\beta s}\left\vert \bar{x}_{s}\right\vert ^{2}ds],
\end{array}$$ where for the last inequality, we have used the inequality $2ab\leq\varepsilon
a^{2}+\frac{1}{\varepsilon}b^{2}$ for all $\varepsilon >0$ (which follows by basic algebra from $(\varepsilon a - b)^2 \geq 0$).

Choosing $\beta=\frac{C^{2}}{\varepsilon
}+1$, we have$$\mathbb{E[}e^{-\beta t}\left\vert \bar{X}_{t}\right\vert ^{2}]+\mathbb{E[}{\textstyle\int_{0}^{t}}
e^{-\beta s}\left\vert \bar{X}_{s}\right\vert ^{2}ds]\leq(\varepsilon
+C^{2})\mathbb{E[}{\textstyle\int_{0}^{t}}
e^{-\beta s}\left\vert \bar{x}_{s}\right\vert ^{2}ds].$$ Then, $\left\Vert \bar{X}\right\Vert _{V}^{2}\leq(\varepsilon
+C^{2})\left\Vert \bar{x}\right\Vert _{V},$ i.e.,$$\left\Vert \Phi\left(  x^{1}\right)  -\Phi\left(  x^{2}\right)  \right\Vert
_{V}\leq(\varepsilon+C^{2})\left\Vert x^{1}-x^{2}\right\Vert _{V
},\text{ for all }x^{1},x^{2}\in V.$$ Choosing $\varepsilon >0$ such that $\varepsilon+C^{2}<1$, we obtain that $\Phi:(V,\left\Vert \cdot\right\Vert _{V})\rightarrow(V,\left\Vert
\cdot\right\Vert _{V})$ is a contraction on the Banach space $\left(
V,\left\Vert \cdot\right\Vert _{V}\right)  $. Hence, from Banach's fixed point theorem, there exists a unique fixed point $X\in H,$ such that $X=\Phi\left(  X\right)  ,$ i.e., $$X_{t}:=x_{0}+{\textstyle\int_{0}^{t}}
B\left(  s,x_{s}\right)  ds+{\textstyle\int_{0}^{t}}
\sigma\left(  s,x_{s}\right)  dW_{s}+{\textstyle\int_{0}^{t}}
\gamma\left(  s,x_{s}\right)  dH_{s}.$$

As a consequence, there also exists a unique solution to the SDE with default .

<span>$\square$ </span>

Now that we know that there exists a unique solution to the controlled SDE with default , we can move on to study a stochastic optimal control problem with default.

Stochastic Maximum Principles
=============================

In this section, we present two stochastic maximum principles which can be used to solve stochastic optimal control problems where the system state is determined by the controlled SDE with default . The performance functional, which we would like to maximise over all strategies $u\in\mathcal{A}$, is defined as $$J(u)=\mathbb{E}\left[  \int_{0}^{T}h(t,X_{t},u_{t})dt+g(X_{T})\right]  .$$ We would like to derive stochastic maximum principles for this problem.

In order to maximize the performance functional we define the Hamiltonian function$$\label{eq: Hamiltonian}
\mathcal{H}(t,x,u,p,q):=h(t,x,u)+\big(b(t,x,u)+\lambda^{\mathbb{G}}\gamma
((t,x,u)\big)p+\sigma(t,x,u)q+\lambda^{\mathbb{G}}\gamma(t,x,u)w,$$ where $(p,q,w)$ are called the adjoint processes.

\[not: shorthand\] For ease of notation, we define the following shorthand for some given control $u_t$ with corresponding $X_t, p_t, q_t, w_t$ $$b_t := b(t, X_t, u_t), \\[\smallskipamount]
\sigma_t := \sigma(t, X_t, u_t), \\[\smallskipamount]
\gamma_t := \gamma(t, X_t, u_t),$$ and similarly define $\hat{X}_t, \hat{p}_t, \hat{q}_t, \hat{w}_t, \hat{q}_t$ for the control $\hat{u}_t$. Also, define $\mathcal{H}_t:= \mathcal{H}(t, X_t, u_t)$ and similarly $\hat{\mathcal{H}}_t :=\mathcal{H}(t, \hat{X}_t, \hat{u}_t)$.

The adjoint processes are given as the solution of the adjoint BSDE$$\left\{
\begin{array}
[c]{ll}dp_{t} & =-\frac{\partial\mathcal{H}_{t}}{\partial x}dt +q_{t}dW_{t}+w_{t}dM_{t},\\
p_{T} & =g^{\prime}(X_{T}).
\end{array}
\right.$$

Existence of a unique solution to BSDEs of this type is proved in Section \[sec: BSDE\]. The adjoint BSDE can be rewritten as: $$\left\{
\begin{array}
[c]{llll}dp_{t} & =&-[\frac{\partial h_{t}}{\partial x}+(\frac{\partial b_{t}}{\partial
x}+\lambda^{\mathbb{G}}_{t}\frac{\partial\gamma_{t}}{\partial x})p_{t}+\frac{\partial
\sigma_{t}}{\partial x}q_{t}+\frac{\partial\gamma_{t}}{\partial x}w_{t}]dt\\[\smallskipamount]
&& +q_{t}dW_{t}+w_{t}dM_{t},\\[\medskipamount]
p_{T} & =&g^{\prime}(X_{T}).
\end{array}
\right.  \label{eq: adjoint}$$ Note that this adjoint BSDE is linear. In the next section, we derive an explicit solution to linear BSDEs with default.

Explicit solution of linear BSDEs with default
----------------------------------------------

In this section, we derive an explicit solution formula for linear BSDEs with default of a particular form. To do so, consider a linear BSDE with default of the following form: $$\label{eq: lin_BSDE_default}
\begin{array}{llll}
dp_t &=& (\phi_t + (\alpha_t - \pi_t + \lambda^{\mathbb{G}}_t \mu_t)p_t  + \beta_t q_t + \lambda^{\mathbb{G}}_t \mu_t w_t)dt + q_t dW_t + w_t dM_t \\[\smallskipamount]
p_T &=& F
\end{array}$$ where $\alpha, \pi, \mu, \beta$ and $\phi$ : $\mathbb{R} \rightarrow \mathbb{R}$ are $\mathbb{G}$-adapted stochastic processes and $F$ is a ${\mathcal}{G}_T$-measurable square integrable random variable. Note that this is of the same form as the adjoint BSDE arising in connection to the stochastic maximum principles, see .

The following theorem gives an explicit formula for the solution of the linear BSDE with default :

\[lemma: BSDE\_linear\] Consider a linear BSDE of the form . This BSDE has a unique solution $(p,q,w)$, where $p(t)$ is explicitly given by $$p_t= E[F\Gamma_{T,t} + \int_t^{T} \Gamma_{s,t} \phi_sds | {\mathcal}{G}_t].$$ Here, $$\label{eq: gamma}
\begin{array}{llll}
 d\Gamma_{t} & =&\Gamma_{t}[(\alpha_t + \lambda^{\mathbb{G}}_t \mu_t) dt +\beta_t dW_t + \mu_tdM_t],\\
\Gamma_{0} & =&1.
\end{array}$$ and we denote $\Gamma_{s,t}:=\frac{\Gamma_{s}}{\Gamma_{t}} \mbox{ for all } t, s \in [0,T].$

<span><span>Proof.</span> </span>(Sketch)

By the It<span>ô</span> product formula,

$$\begin{array}{llll}
d(\Gamma_tp_t) &=& p_td\Gamma_t + \Gamma_tdp_t + dp_td\Gamma_t \\[\smallskipamount]
&=& \big(p_t\Gamma_t (\alpha_t + \lambda^{\mathbb{G}}_t) - \Gamma_t\phi_t - \Gamma_tp_t(\alpha_t + \lambda^{\mathbb{G}} \mu_t) \\[\smallskipamount]
&&-\Gamma_t\beta(t)q_t - \Gamma_t\lambda^{\mathbb{G}} \mu(t)w_t + \Gamma \beta_t q_t+ \Gamma_t \mu_tw_t\lambda^{\mathbb{G}}  \big)\\[\smallskipamount] 
&&+ \mathbb{G}\mbox{-martingale} \\[\smallskipamount]
&=& -\Gamma_t\phi_tdt +\mathbb{G}\mbox{-martingale}
\end{array}$$ where the first equality follows by inserting the form of $dp$ from and $d\Gamma$ from .

Hence, $\Gamma_tp_t + \int_0^t \Gamma_s\phi_sds$ is a $\mathbb{G}$-martingale, i.e.,

$$\Gamma_tp_t + \int_0^t \Gamma_s\phi_sds = E[\Gamma_Tp_T + \int_0^T \Gamma_s\phi_sds \mid \mathcal{G}_t].$$

That is,

$$\Gamma_tp_t= E[\Gamma_T F + \int_t^T \Gamma_s\phi_sds \mid \mathcal{G}_t].$$

<span>$\square$ </span>

Sufficient Stochastic Maximum Principle
---------------------------------------

Now, we are ready to prove the following sufficient maximum principle for optimal control of an SDE with default of the form .

\[thm: suff\_max\_princ\] Let $\hat{u}$ be an admissible performance strategy with corresponding solution $\hat{X}$ of the SDE and adjoint processes $\hat{p}, \hat{q}, \hat{w}$ to equation . Assume

1.  <span>The functions </span> $x \rightarrow g(x)$ and $(x, u) \rightarrow
    \mathcal{H}(t, x, u, \hat{p}, \hat{q}, \hat{w})$ are concave a.s. for every $t
    \in[0,T]$.

2.  <span>For every $v \in\mathcal{V}$</span>, $$E\big[  \frac{\partial}{\partial u} \mathcal{H}(t, \hat{X}_{t}, \hat{u}_{t},
    \hat{p}_{t}, \hat{q}_{t}, \hat{w}_{t}) \big] (v - \hat{u}_{t}) \leq0 \mbox{ }
    \mbox{ } dt \times P \mbox{ } a.s.$$

Then, $\hat{u}$ is an optimal control for the stochastic optimal control problem with default.

Before we move on to the proof, note that the theorem says that if $g$ and the Hamiltonian are concave, then we may maximize the Hamiltonian instead of the performance functional in order to find the optimal control of our problem. This essentially reduces the stochastic optimal control problem to the problem of solving the SDE and the adjoint BSDE . The idea of the proof is to show that $$J(u) - J(\hat{u}) = E\big[  \frac{\partial}{\partial u} \mathcal{H}(t, \hat
{X}_{t}, \hat{u}_{t}, \hat{p}_{t}, \hat{q}_{t}, \hat{w}_{t}) \big] (v -
\hat{u}_{t}).$$ From this, the maximum principle follows.

<span><span>Proof.</span> </span>Fix $\hat{u} \in \mathcal{A}$ with corresponding $\hat{X}_t, \hat{b}_t, \hat{\sigma}_t, \hat{\gamma}_t, \hat{p}_t, \hat{q}_t, \hat{w}_t$. Write $$J(u) - J(\hat{u}) = A_1 + A_2,$$

where

$$\begin{array}{llll}
A_1 &:=& E[ \int_0^T \{ h(t, X_t, u_t) - h(t, \hat{X}_t, \hat{u}_t) \}dt] \\[\smallskipamount]
A_2 &:=& E[ g(X_T) - g(\hat{X_T})].
\end{array}$$ Then, from the definition of the Hamiltonian $$\label{eq: A_1}
\begin{array}{llll}
A_1 &=& E[ \int_0^T \{ \mathcal{H}_t -  \hat{\mathcal{H}}_t   - \hat{p}_t \big( (b_t- \hat{b}_t) + \lambda^{\mathbb{G}}_t (\gamma_t - \hat{\gamma}_t) \big) - \hat{q}_t (\sigma_t - \hat{\sigma}_t) \\[\medskipamount]
&&- \lambda^{\mathbb{G}}_t \hat{w}_t (\gamma_t - \hat{\gamma}_t)  \} dt] \\[\medskipamount]
&\leq& E \big [\int_0^T \{ \frac{\partial \hat{\mathcal{H}}_t}{\partial x} (X_t- \hat{X}_t +  \frac{\partial \hat{\mathcal{H}}_t}{\partial u} (u_t- \hat{u}_t) - \hat{p}_t \big( (b_t- \hat{b}_t) + \lambda^{\mathbb{G}}_t (\gamma_t - \hat{\gamma}_t) \big)  \\[\smallskipamount]
&&- \hat{q}_t (\sigma_t - \hat{\sigma}_t)- \lambda^{\mathbb{G}}_t \hat{w}_t (\gamma_t - \hat{\gamma}_t) dt\} \big] ,
\end{array}$$ where the second equality follows because the Hamiltionan $\mathcal{H}$ is concave. Similarly from the concavity of $g$, we have$$\begin{array}{lll}
A_{2} =E[  g(X_{T})-g(\hat{X}_{T})] \leq E \big[  g^{\prime}(\hat{X}_{T})\left(  X_{T}-\hat
{X}_{T}\right)  \big]
= E \big[  \hat{p}_{T}\left(  X_{T}-\hat{X}_{T}\right)
\big],  \nonumber\\
\end{array}$$ where we have used the terminal condition of the adjoint BSDE in the final equality. From Itô's product rule, $$\label{eq: A_2}
\begin{array}{lll}
E\big[  \hat{p}_{T}\left(  X_{T}-\hat{X}_{T}\right)\big] &=& E\big[\int_0^T (X_t - \hat{X}_t) d\hat{p}_t + \int_0^T \hat{p}_t d(X_t - \hat{X}_t)  \\[\smallskipamount]
&&+ \int_0^T d\hat{p}(t) d (X_t - \hat{X}_t)\big]  \\
&=& E[\int_0^T (X_t - \hat{X}_t) (-\frac{\partial\mathcal{\hat{H}}}{\partial x}(t)+\hat{q}_{t}dW_{t}+\hat{w}_{t}dM_{t}) \\[\medskipamount]
&&+ \int_0^T \hat{p}_t \{ \big( (b_t- \hat{b}_t) + \lambda^{\mathbb{G}}_t (\gamma_t - \hat{\gamma}_t) \big) dt + (\sigma_t - \hat{\sigma}_t)dW_t\\[\medskipamount]
&&+ (\gamma_t - \hat{\gamma}_t) dM_t \}
+ \int_0^T \hat{q}_t(\sigma_t - \hat{\sigma}_t)dt \\[\medskipamount]
&&+ \int_0^T \hat{w}_t (\gamma_t - \hat{\gamma}_t) (dM_t)^2].
\end{array}$$ Note that $$\begin{array}{lcll}
\Delta M_t &=& H_{t + \Delta t} - \int_0^{t+\Delta t} \lambda^{\mathbb{G}}_s ds - H_t + \int_0^t \lambda^{\mathbb{G}}_s ds \\[\smallskipamount]
&=& \Delta H_t - \int_t^{t + \Delta t} \lambda^{\mathbb{G}}_s ds.
\end{array}$$ This implies that $$\label{eq: M_and_H}
 \delta M_t = \lim_{\Delta t \rightarrow 0} \Delta M_t = \lim_{\Delta t \rightarrow 0} \{ \Delta H_t - \int_t^{t + \Delta t} \lambda^{\mathbb{G}}_s ds\} = \delta H_t ,$$ where the final equality follows because $\lambda^{\mathbb{G}}$ is bounded. Therefore, since $H_t$ per definition is always either $0$ or $1$, it follows that $$\label{eq: dM}
(dM_t)^2 =(\delta M_t)^2 = (\delta H_t)^2 = \delta H_t= dH_t.$$ By combining the expressions for $A_1$ and $A_2$ found in equations and respectively, we find that $$\label{eq: A1_plus_A2}
A_1 + A_2 \leq E\big[\int_0^T \frac{\partial \hat{\mathcal{H}}_t} {\partial u} (u_t- \hat{u}_t) dt\big] + E\big[\int_0^T \hat{w}_t (\gamma_t - \hat{\gamma}_t) dH_t\big] - E\big[\int_0^T \lambda^{\mathbb{G}}_t \hat{w}_t (\gamma_t - \hat{\gamma}_t)dt \big].$$ Note that $$\label{eq: calculation_1}
\begin{array}{lll}
E\big[\int_0^T \hat{w}_t (\gamma_t - \hat{\gamma}_t) dH_t\big] &=& E\big[\int_0^T \hat{w}_t (\gamma_t - \hat{\gamma}_t) dM_t\big] + E\big[\int_0^T \hat{w}_t (\gamma_t - \hat{\gamma}_t) \lambda^{\mathbb{G}}_t dt\big] \\[\smallskipamount]
&=& E\big[\int_0^T \lambda^{\mathbb{G}}_t \hat{w}_t (\gamma_t - \hat{\gamma}_t)dt\big].
\end{array}$$ So, by combining and , we see that $$J(u) - J(\hat{u}) = A_1 +A_2 = E \big [\int_0^T \frac{\partial \hat{\mathcal{H}}_t}{\partial u} (u_t- \hat{u}_t) dt \big ] \leq0,$$ where the final inequality follows from the assumptions. Hence, $J(u) \leq J(\hat{u})$, so since $\hat{u} \in \mathcal{A}$ it is an optimal control.

<span>$\square$ </span>

Equivalence Maximum Principle
-----------------------------

A problem with the sufficient maximum principle from the previous section is that the concavity condition is quite strict, and may not hold in applications. In this section, we derive an alternative maximum principle, called a necessary maximum principle or equivalence principle for the optimal control of the SDE with default.

In order to do this we need some additional notation and assumptions:\
For $u \in {\mathcal}{A}$ let ${\mathcal}{V}(u)$ denote the set of $\mathbb{G}$-adapted processes $\beta$ of finite variation such that there exists $\delta=\delta(u) > 0$ satisfying $$u + y \beta \in {\mathcal}{A}, \text{ for all  } y \in [0,\delta].$$

Assume that for all $u \in {\mathcal}{A}$ and for all $\beta \in {\mathcal}{V}(u)$ the following derivative process exists and belongs to $L^2([0,T] \times \Omega)$: $$\label{eq33}
\begin{array}{lll}
x_t:=& \frac{d}{dy} X_t^{u+y\beta}|_{y=0} = \lim_{y \rightarrow 0^+} \frac{X_t^{u + y \beta} - X_t^{u}}{y}. \\
\end{array}$$

The existence and $L^2$-features of these derivative process is a non-trivial issue, and we do not discuss conditions for this in our paper. Here we will just assume that these properties hold. We refer to Pr<span>é</span>v<span>ô</span>t and R<span>ö</span>ckner @PR for a study of this issue in a related setting.

The following equivalence principle says that for a control to be a critical point for the performance functional $J$ is equivalent to that the expected value of $\beta$ times the derivative of the Hamiltonian is zero for all admissible $\beta$'s.

\[thm: equivalence\_principle\] The following two statements are equivalent:

1.  <span> $$\label{eq: i}\frac{d J(u+y\beta)}{dy}\big|_{y=0} = 0, \mbox{  for all } \beta
    \in\mathcal{A}.$$ </span>

2.  <span> $$E\big[\frac{\partial\mathcal{H}_t}{\partial u}\beta_t\big] = 0, \mbox{  for all }
    \beta\in\mathcal{A}.$$ </span>

<span><span>Proof.</span> </span>Note that $$\frac{d J(u+y\beta)}{dy}|_{y=0} = \frac{d}{dy} E\big[\int_0^T h(t, X_t^{u + y\beta}, u_t+y_t \beta_t)dt + g(X_T^{u+y\beta})\big].$$ Define $I_1 := \frac{d}{dy}E\big[\int_0^T h(t, X_t^{u + y\beta}, u_t+y_t \beta_t)dt\big]$ and $I_2 :=\frac{d}{dy} E\big[g(X_T^{u+y\beta})\big]$. Then, $$I_1 = E\big[\int_0^T \{ \frac{\partial h_t}{\partial x}x_t + \frac{\partial h_t}{\partial u}\beta_t \} dt\big].$$ Here, the equality follows by changing the order of differentiation and integration (by the dominated convergence theorem). Also, $$\begin{array}{lll}
I_2 &=& E[g'(X_T^{u+y\beta}) x_T] \\[\smallskipamount]
&=& E[p_Tx_T] \\[\smallskipamount]
&=& E[\int_0^T p_tdx_t + \int_0^T x_tdp_t+ \int_0^T dp_tdx_t],
\end{array}$$ where the first equality follows by changing the order of differentiation and integration (again, using the dominated convergence theorem), the second equality follows from the adjoint equation \[eq: adjoint\] and the third equation follows from the Ito product rule. Now, note that from the FSDE , the adjoint BSDE and equation respectively, $$\begin{array}{lll}
dx_t &=& [\frac{\partial b_t}{\partial x}x_t + \frac{\partial b_t}{\partial u}\beta_t + \lambda^{\mathbb{G}}_t (\frac{\partial \gamma_t}{\partial x} x_t + \frac{\partial \gamma_t}{\partial u} \beta_t)] dt \\[\smallskipamount]
&& + (\frac{\partial \sigma_t}{\partial x} x_t + \frac{\partial \sigma_t}{\partial u} \beta_t) dW_t + (\frac{\partial \gamma_t}{\partial x} x_t + \frac{\partial \gamma_t}{\partial u} \beta_t) dM_t,
\end{array}$$ $$dp_t dx_t= q_t (\frac{\partial \sigma_t}{\partial x} x_t + \frac{\partial \sigma_t}{\partial u} \beta_t) dt + w_t (\frac{\partial \gamma_t}{\partial x} x_t + \frac{\partial \gamma_t}{\partial u} \beta_t)dH_t,$$ where we have used that $(\delta M_t)^2=\delta H_t$, so $(d M_t)^2=d H_t$ . So, by the previous expression for $dx_t$ and $dp_tdx_t$, as well as the expression for $dp_t$ from the adjoint BSDE and transforming $dH$ into $dM + \lambda^{\mathbb{G}} dt$, $$\begin{array}{llll}
I_2 &=& E[\int_0^T p_t \{  \frac{\partial b_t}{\partial x} x_t + \frac{\partial b_t}{\partial u} \beta_t + \lambda^{\mathbb{G}}_t (\frac{\partial \gamma_t}{\partial x} x_t + \frac{\partial \gamma_t}{\partial u} \beta_t) \}]dt \\[\smallskipamount]
&&- \int_0^T x_t \frac{\partial \mathcal{H}_t}{\partial x}dt + \int_0^T q_t (\frac{\partial \sigma_t}{\partial x} x_t + \frac{\partial \sigma_t}{\partial u}\beta_t)dt\\[\smallskipamount]
&& + \int_0^T \lambda^{\mathbb{G}}_t w_t (\frac{\partial \gamma_t}{\partial x} x_t + \frac{\partial \gamma_t}{\partial u} \beta_t)dt].
\end{array}$$ By collecting the $\beta_t$- and $x_t$-terms and using the definition of the Hamiltonian to cancel all $x_t$-terms against $x_t \frac{\partial \mathcal{H}_t}{\partial x}$, we find that $$\begin{array}{llll}
I_1 + I_2 &=& E[\int_0^T \beta_t (\frac{\partial h_t}{\partial u} + p_t(\frac{\partial b_t}{\partial u}  + \lambda^{\mathbb{G}}_t \frac{\partial \gamma_t}{\partial u}) + q_t \frac{\partial \sigma_t}{\partial u} + w_t \lambda^{\mathbb{G}}_t \frac{\partial \gamma_t}{\partial u}) dt] \\[\smallskipamount]
&=& E[\int_0^T \beta_t \frac{\partial \mathcal{H}_t}{\partial u}dt].
\end{array}$$ So, $$\frac{d J(u+y\beta)}{dy}|_{y=0} = E\big[\beta_t\frac{\partial \mathcal{H}_t}{\partial u}\big], \mbox{  for all } \beta \in \mathcal{A},$$ and hence the theorem follows. <span>$\square$ </span>

Existence and Uniqueness of Solution to the BSDE with Default
=============================================================

We study the BSDE with default $$Y_{t}=\xi+{\textstyle\int_{t}^{T}}
f(s,Y_{s},Z_{s},K_{s})ds-{\textstyle\int_{t}^{T}}
Z_{s}dW_{s}-{\textstyle\int_{t}^{T}}
K_{s}dH_{s},\qquad t\in\left[  0,T\right]  , \label{4}$$ where $f$ is $\mathcal{G}_{t}\otimes\mathcal{B}\left(  \left[  0,T\right]
\right)  \otimes\mathcal{B}\left(
\mathbb{R}
^{3}\right)  $-measurable, and the terminal condition $\xi$ is $\mathcal{G}_{T}$-measurable.

We call the triplet $(Y,Z,K)$ solution of the BSDE (\[4\]).

Let us impose the following set of assumptions.

(*i) Assumption on the terminal condition:*

-   $\xi\in L^{2}\left(  \Omega,\mathcal{G}_{T}\right)  $.

\(ii) *Assumptions on the generator function* $f:\Omega\times\left[
0,T\right]  \times\mathcal{\mathbb{R}}^{3}\rightarrow\mathbb{R}$ is such that

-   $\mathbb{G}$-predictable and satisfies the integrability condition, such that $$\mathbb{E[}{\textstyle\int_{0}^{T}}
    |f(t,0,0,0)|^{2}dt]<0\text{, } \label{int}$$

for all $t\in\left[  0,T\right]  .$

-   Lipschitz in the sense that, there exists $C>0,$ such that$$|f(t,y,z,k)-f(t,y^{\prime},z^{\prime},k^{\prime})|\leq C(\left\vert
    y-y^{\prime}\right\vert +|z-z^{\prime}|+\lambda_{t}^{\mathbb{G}}|k-k^{\prime}|),\nonumber$$

for all $t\in\left[  0,T\right]  $ and all $y,y^{\prime},z,z^{\prime
},k,k^{\prime}\in\mathbb{R}
.$

We give the existence of the solution to a BSDE in the enlarged filtration $\mathbb{G}.$ The existence follows from the property of martingale representation, and a standard approach like any classical BSDE. However, for existence and uniqueness of BSDE with default by using the decomposition approach, we refer to Kharoubi and Lim @kha and to Jeanblanc *et al* @jla.

Under our assumptions we know that equation (\[4\]) is equivalent to$$\begin{array}
[c]{l}Y_{t}=\xi+{\textstyle\int_{t}^{T}}
F(s,Y_{s},Z_{s},K_{s})ds-{\textstyle\int_{t}^{T}}
Z_{s}dW_{s}-{\textstyle\int_{t}^{T}}
K_{s}dM_{s},
\end{array}
\label{5}$$ with $dH_{s}=dM_{s}+\lambda_{s}\mathbf{1}_{s<\tau}ds$, and $$F(s,y,z,k):=f(s,y,z,k)-\lambda_{s}^{\mathbb{G}}k.$$ By assumption, the process $\lambda^{\mathbb{G}}$ is bounded.

In order to get existence and uniqueness for the BSDE (\[5\]), let us check that the generator $F$ satisfies the same assumption as $f:$ The function ** $F:\Omega\times\left[  0,T\right]  \times\mathcal{\mathbb{R}}^{5}\rightarrow\mathbb{R}$ is such that

(i)

:   $\mathbb{G}$-predictable and integrable in the sense that, for all $t\in\left[  0,T\right]  $, by inequality (\[int\]), we have $$\mathbb{E[}{\textstyle\int_{0}^{T}}
    |F(t,0,0,0)|^{2}dt]=\mathbb{E[}{\textstyle\int_{0}^{T}}
    |f(t,0,0,0)|^{2}dt]<\infty.$$

(ii)

:   Lipschitz in the sense that there exists a constant $C^{\prime}>0$, such that$$\begin{array}
    [c]{l}|F(t,y,z,k)-F(t,y^{\prime},z^{\prime},k^{\prime})|\\
    =|f(t,y,z,k)-f(t,y^{\prime},z^{\prime},k^{\prime})-\lambda_{t}^{\mathbb{G}}(k-k^{\prime})|\\
    \leq|f(t,y,z,k)-f(t,y^{\prime},z^{\prime},k^{\prime})|+\lambda_{t}^{\mathbb{G}}|k-k^{\prime}|\\
    \leq C(\left\vert y-y^{\prime}\right\vert +|z-z^{\prime}|+\lambda_{t}^{\mathbb{G}}|k-k^{\prime}|)+\lambda_{t}^{\mathbb{G}}|k-k^{\prime}|\\
    \leq C^{\prime}(\left\vert y-y^{\prime}\right\vert +|z-z^{\prime}|+\lambda_{t}^{\mathbb{G}}|k-k^{\prime}|),
    \end{array}
    \nonumber$$

for all $t\in\left[  0,T\right]  $ and all $y,z,k,y^{\prime},z^{\prime
},k^{\prime}\in\mathbb{R}
,$where we have used that $f$ is Lipschitz.

(iii)

:   The terminal value: $\xi\in L^{2}\left(  \Omega,\mathcal{G}_{T}\right)  $.

\[ex-uni\]Under the above assumptions (i)-(iii), the BSDE $\left(
\ref{5}\right)  $ admits a unique solution $\left(  Y,Z,K\right)  \in
S_{\mathbb{G}}^{2}\times H_{\mathbb{G}}^{2}\times L^{2}(\lambda).$

<span>Proof.</span> We define the mapping** **$$\Phi:H_{\mathbb{G}}^{2}\times H_{\mathbb{G}}^{2}\times L^{2}(\lambda
)\rightarrow H_{\mathbb{G}}^{2}\times H_{\mathbb{G}}^{2}\times L^{2}(\lambda),$$ for which we will show that it is contracting under a suitable norm. For this we note that for any $\left(  Y,Z,K\right)  \in H_{\mathbb{G}}^{2}\times
H_{\mathbb{G}}^{2}\times L^{2}(\lambda)$ there exists a unique quadruple $(\hat{Y},\hat{Z},\hat{U},\hat{K})\in S_{\mathbb{G}}^{2}\times H_{\mathbb{G}}^{2}\times L^{2}(\lambda),$ such that$$\hat{Y}_{t}=\xi+{\textstyle\int_{t}^{T}}
F(s,Y_{s},Z_{s},K_{s})ds-{\textstyle\int_{t}^{T}}
\hat{Z}_{s}dW_{s}-{\textstyle\int_{t}^{T}}
K_{s}dM_{s}, \label{n1}$$ Let $\Phi(Y,Z,K):=(\hat{Y},\hat{Z},K)$. For given $\left(  Y^{i},Z^{i},K^{i}\right)  $** **$\in H_{\mathbb{F}}^{2}\times H_{\mathbb{F}}^{2}\times L^{2}(\lambda),$** **for $i=1,2$, we use the simplified notations: $$\begin{array}
[c]{ll}(\hat{Y}^{i},\hat{Z}^{i},K^{i}) & :=\Phi(Y^{i},Z^{i},K^{i}),\\
(\tilde{Y},\tilde{Z},K) & :=(\hat{Y}^{1},\hat{Z}^{1},K^{1})-(\hat{Y}^{2},\hat{Z}^{2},K^{2}),\\
(\bar{Y},\bar{Z},K) & :=(Y^{1},Z^{1},K^{1})-(Y^{2},Z^{2},K^{2}).
\end{array}$$ The triplet of processes $\left(  \tilde{Y},\tilde{Z},\tilde{K}\right)  $ satisfies the equation$$\begin{array}
[c]{c}\tilde{Y}_{t}={\textstyle\int_{t}^{T}}
\mathbb{\{}F(s,Y_{s}^{1},Z_{s}^{1},K_{s}^{1})-F(s,Y_{s}^{2},Z_{s}^{2},K_{s}^{2})\}ds\\
-{\textstyle\int_{t}^{T}}
\tilde{Z}_{s}dW_{s}-{\textstyle\int_{t}^{T}}
\tilde{K}_{s}dM_{s}.
\end{array}$$ We have that $M_{t}=H_{t}-{\textstyle\int_{0}^{t}}
\lambda_{s}^{\mathbb{G}}ds$ which is a pure jump martingale. Then,$$\left[  M\right]  _{t}=\underset{0\leq s\leq t}{{\textstyle\sum}
}\left(  \delta M_{s}\right)  ^{2}=\underset{0\leq s\leq t}{{\textstyle\sum}
}\left(  \delta H_{s}\right)  ^{2}=H_{t},$$ and$$\left\langle M\right\rangle _{t}={\textstyle\int_{0}^{t}}
\lambda_{s}^{\mathbb{G}}ds,$$$${\textstyle\int_{t}^{T}}
|\tilde{U}_{s}|^{2}d\left\langle M\right\rangle _{s}={\textstyle\int_{t}^{T}}
\lambda_{s}^{\mathbb{G}}|\tilde{U}_{s}|^{2}ds.$$ Applying It<span>ô</span>'s formula to** **$e^{\beta t}|\tilde{Y}_{t}|^{2}$**,** taking conditional expectation and using the Lipschitz condition, we get $$\begin{array}
[c]{l}\mathbb{E[}{\textstyle\int_{0}^{T}}
e^{\beta s}(\beta|\tilde{Y}_{s}|^{2}+|\tilde{Z}_{s}|^{2}+\lambda_{s}^{\mathbb{G}}|\tilde{K}_{s}|^{2})ds]\\
\leq10\rho C^{2}\mathbb{E[}{\textstyle\int_{0}^{T}}
e^{\beta s}\left\vert \bar{Y}_{s}\right\vert ^{2}ds]+\tfrac{1}{2\rho
}\mathbb{E[}{\textstyle\int_{0}^{T}}
e^{\beta s}\{\left\vert \bar{Z}_{s}\right\vert ^{2}+(\lambda_{s}^{\mathbb{G}}) ^{2}\left\vert
\tilde{K}_{s}\right\vert ^{2}\}ds],
\end{array}$$ Since $\lambda^{\mathbb{G}}$ is bounded, we get that $(\lambda^{\mathbb{G}})^{2}\leq c\lambda^{\mathbb{G}}$ and by choosing $\beta=1+10\rho C^{2}\ $we obtain$$||(\tilde{Y},\tilde{Z},\tilde{K})||^{2}\leq\tfrac{1}{2\rho}||(\bar{Y},\bar{Z},\tilde{K})||^{2}$$ which means for $\rho\geq1$, there exists a unique fixed point that is a solution for our BSDE $\left(  \ref{5}\right)  . \qquad  \qquad \qquad \qquad \qquad  \qquad  \qquad   \qquad  \qquad  \qquad  \qquad  \qquad  \qquad\square$

Application: Log-Utility Maximisation with Default
==================================================

In this section, we apply the stochastic maximum principles Theorem \[thm: suff\_max\_princ\] and Theorem \[thm: equivalence\_principle\] to a logarithmic utility maximisation problem. Consider the wealth process with default $$\label{eq: SDE_ex}
dS_{t}=S_{t-}[(\alpha_{t}-\pi_{t})dt+\beta_{t}dW_{t}+\mu_{t}dH_{t}], S_{0}>0,$$ where the coefficients $\alpha,\beta,\mu$ are $\mathbb{G}$-predictable processes and we assume that $\mu_{t}\geq-1$ for all $t \in [0,T]$ a.s. Note that is a linear SDE.

Also, note that in the SDE , the control $\pi_t$ corresponds to a consumption process because of its negative impact on the wealth process $S_t$. The default term $\mu_{t}dH_{t}$ implies that the wealth process will grow w.r.t. $\mu_t$ until the default time $\tau$. From that point on, $\mu_t$ has no impact on the wealth. This may correspond to investing in a defaultable firm.

By Definition 2.1, the SDE can be rewritten as$$\label{eq: rewritten_sde_ex}
dS_{t}=S_{t-}[(\alpha_{t}-\pi_{t}+\lambda^{\mathbb{G}}_{t}\mu_{t})dt+\beta_{t}dW_{t}+\mu_{t}dM_{t}].$$

If the coefficients are constants (as in the classical Black-Scholes case), i.e. $\alpha_t=\alpha, \beta_t=\beta$ and $\mu_t=\mu$ for all times $t \geq 0$ and all $\omega \in \Omega$, we can solve the SDE explicitly. We distinguish between three cases:

-   If $t<\tau$: $dS_{t}=S_{t}[\alpha dt+\beta dW_{t}]-S_{t}\pi_{t}dt$, then $$S_{t}=S_{0}e^{\alpha t}e^{\int_{0}^{t}\pi_{s}ds}e^{\beta W_{t}-\frac{1}{2}\beta^{2}t}.\label{explic1}$$

-   If $t=\tau$: $\triangle S_{t}=\mu S_{t -}$, thus $$\ S_{\tau}=(1+\mu)S_{\tau -}.$$

-   If $t>\tau$, from (\[explic1\]), we get   $$S_{t}=S_{\tau}e^{\alpha(t-\tau)}e^{\beta(W_{t}-W_{\tau})-\frac{1}{2}\beta
    ^{2}(t-\tau)}.$$

In fact, we can write up explicit solutions to the SDE as long as $\mu$ is constant, but $\alpha_t, \beta_t$ vary with time. In this case, we get:

-   If $t<\tau$, then

    $$S_{t}=S_{0}e^{\int_0^t \alpha_s ds}e^{\int_{0}^{t}\pi_{s}ds}e^{\int_0^t \beta_s dW_{s}-\frac{1}{2}\int_0^t \beta_s^{2} ds}.\label{explic2}$$

-   If $t>\tau$, then we get from (\[explic2\])   $$S_{t}=S_{\tau}e^{\int_{\tau}^t \alpha_s ds}e^{\int_{\tau}^t \beta_s dW_s-\frac{1}{2}\int_{\tau}^t \beta_s
    ^{2}ds}.$$

The solution in the case where $t=\tau$ is as above.

The performance function we want to maximize is $$J(u)=E[{\textstyle\int_{t}^{T}}
U^{1}(S_{t},\pi_{t})dt+\theta U^{2}( S_{T})],$$

where $U^{1},U^{2}$ are some given deterministic utility functions and $\theta
:=\theta(\omega)$ is a $\mathcal{G}_{T}$-measurable, square integrable random variable which expresses the importance of the terminal value. To be able to find explicit solutions for our optimal control, we consider logarithmic utilities. Hence, the performance function is $$J(\pi)=E[{\textstyle\int_{t}^{T}}
\log(S_{t}\pi_{t})dt+\theta\log( S_{T})].$$ The corresponding Hamiltonian functional, see , is $$\mathcal{H}(t,s,\pi,p,q,w)=\log(s\pi)+s(\alpha-\pi+\lambda^{\mathbb{G}} \mu)p+s\beta
q+\lambda^{\mathbb{G}} s\mu w.$$ The adjoint BSDE, see , has the form$$\label{eq: BSDE_example}
\left\{
\begin{array}
[c]{ll}dp_{t} & =-\frac{\partial\mathcal{H}}{\partial s}(t)dt+q_{t}dW_{t}+w_{t}dM_{t},\\
p_{T} & =\frac{\theta}{S_{T}},
\end{array}
\right.$$ such that$$\label{eq: partial_H}
\frac{\partial\mathcal{H}_t} {\partial s}=\frac{1}{S_{t}}+(\alpha_{t}-\pi
_{t}+\lambda^{\mathbb{G}}_{t}\mu_{t})p_{t}+\beta_{t}q_{t}+\lambda^{\mathbb{G}}_{t}\mu_{t}w_{t}.$$ Using the first order necessary condition of Theorem \[thm: equivalence\_principle\], we obtain$$-S_{t}p_{t}+\frac{1}{\pi_{t}}=0.$$ Consequently,$$\hat{ \pi_{t}}=\frac{1}{S_{t}p_{t}}.\label{pi}$$ To derive an explicit expression for the optimal control, we do the following computations:

Note that by the It<span>ô</span> product rule,

$$d(p_t S_t) = p_t d S_t + S_t dp_t + dp_t dS_t.$$

Hence,

$$\label{eq: product}
p_T S_T - p_t S_t = \int_t^T p_u dS_u + \int_t^T S_u dp_u + \int_t^T 1 dp_u dS_u.$$

By taking the conditional expectation with respect to ${\mathcal}{G}_t$ on both sides of equation , we see that

$$\begin{array}{lllll}
p_t S_t &=& E[p_T S_T | {\mathcal}{G}_t]  - E[\int_t^T p_u S_{u-}[(\alpha_{u}-\pi_{u}+\lambda^{\mathbb{G}}_{u}\mu_{u})du+\beta_{u}dW_{u}+\mu_{u}dM_{u}] \\[\smallskipamount]
&& + \int_t^T S_u (-\frac{\partial\mathcal{H}_u}{\partial s}du+q_{u}dW_{u}+w_{u}dM_{u} )+ \int_t^T 1 dp_u dS_u | {\mathcal}{G}_t]. \\[\smallskipamount]
&=& E[\theta | {\mathcal}{G}_t] + E[\int_t^T 1 du | {\mathcal}{G}_t] \\[\smallskipamount]
&=& E[\theta+T-t | \mathcal{G}_{t}]
\end{array}$$ where the second to last equality follows by inserting expression for $p_T$ from the adjoint BSDE , using the expression for the partial derivative of the Hamiltonian in as well as the fact that the expectation of the integrals w.r.t. martingales are $0$. Note also that we use equation when calculating the $(dM_t)^2$-part of the $dp_u dS_u$ term.

Hence, an explicit expression for the stochastic optimal control is $$\hat{ \pi_{t}}=\frac{1}{E[\theta+T-t | \mathcal{G}_{t}]} = \frac{1}{E[\theta | \mathcal{G}_t] + T - t}.\label{pi_explicit}$$

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{'source': 'arXiv_out/2001'}