<figure><img src="image_1.png" /></figure>

HAL
open science

# Rough differential equations with affine boundary conditions

Antoine Lejay, Renaud Marty

<figure><img src="image_2.png" /></figure>

## To cite this version:

Antoine Lejay, Renaud Marty. Rough differential equations with affine boundary conditions. 2023. [hal-03626402v3](https://hal HAL Id: hal-03626402)

https://inria.hal.science/hal-03626402v3

Preprint submitted on 26 Jun 2023

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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

Distributed under a Creative Commons Attribution 4.0 International License
# Rough differential equations with affine boundary conditions

Antoine Lejay* Renaud Marty†

June 26, 2023

## Abstract

In this paper we consider rough differential equations with affine boundary conditions. Using rough paths and fixed-point techniques, we study properties of the solutions. We also prove a convergence theorem and apply our results to differential equations driven by Brownian motions and fractional Brownian motions.

**Keywords:** Rough path theory, differential equations, affine boundary conditions.

**MSC (2020):** 60H10, 34K10, 60L90

## 1 Introduction

In this paper we study rough differential equations with affine boundary conditions. If $U$ and $V$ are two Banach spaces, we consider the problem

$$ \begin{cases} dY(t) = f(Y(t)) dW(t) & \text{for all } t \in [0, 1] \\ H_0Y(0) + H_1Y(1) = v \end{cases} \qquad (1) $$

where $W$ is a continuous function or a stochastic process with values in $U$, $f: V \to L(U, V)$ is a smooth function, $v$ is a vector in $V$, $H_0$ and $H_1$ are two operators on $V$, and $Y$ is unknown. In the case where $W$ is a finite-dimensional Brownian motion and $dW$ stands for the Stratonovich integration, the existence, the uniqueness and other properties of the solutions of such problems (with possibly other forms of two-point boundary conditions) have been analyzed under various assumptions and with different methods. Let us mention a few examples. Linear equations are studied in [23], where anticipating stochastic calculus (see [21]) is used because the solutions are not

<footnote>*Université de Lorraine, CNRS, IECL, Inria, F-54000 Nancy, France; ORCID: 0000-0003-0406-9550; Antoine.Lejay@univ-lorraine.fr</footnote>

<footnote>†Université de Lorraine, CNRS, IECL, F-54000 Nancy, France; Renaud.Marty@univ-lorraine.fr</footnote>
adapted to W. Nonlinear equations with additive noises and more general two-point boundary conditions are studied in [22] with a pathwise approach. In [11] invariant imbedding (see [3]) is used for the analysis of nonlinear equations. In addition to the works mentioned above, we may also cite for instance [2, 7–9, 24] where various stochastic differential equations with two-point boundary conditions are studied in other frameworks or with other methods.

In our paper we address the problem (1) in a setting where W is a more general continuous process (such as a fractional Brownian motion for instance). We use the theory of rough paths (see for instance [10, 18]). This enable us to avoid any difficulties related to the anticipating stochastic calculus. We use an enhanced Brownian motion or any other suitable rough path W in place of W in order to define rough differential equations $Z(t) = Z(0) + \int_0^t f(Z(s)) dW(s)$. For every $a \in V$, if $f$ is regular enough then there exists a unique solution Z such that $Z(0) = a$. Then the Itô map I which assigns Z to $(a, f, W)$ can be defined and is regular with respect to all its variables.

In our work, setting $\tau \in [0, T]$ we relate (1) to

$$ egin{cases} Y(t) = Y(0) + \int_0^t f(Y(s)) d\mathbf{W}(s) & \text{for } t \in [0, T], \\ H_0Y(0) + H_1Y(\tau) = v. & \end{cases} \qquad (2) $$

and study various properties. Equation (2) links the values of $Y(0)$ and $Y(\tau)$ for a given value of $\tau \in [0, T]$, while the path $\{Y(t)\}_{t \in [0,T]}$ is defined on the whole interval $[0, T]$.

Using the Itô map and a fixed-point method, we prove that there exists a unique solution to (2) if $\tau < C$ where C is a constant that depends on T, $f$, $W$, $H_0$, and $H_1$ (we note that possibly $C \ge T$). In this setting we also establish regularity results, which enable us to prove limit theorems.

For $\tau \in [C, T]$ and when V is finite-dimensional, using the Brouwer degree we show that there exists a solution. While the uniqueness is not ensured, the number of solutions is finite for almost every $v$. The situation is summarized in Figure 1.

<figure><img src="image_4.png" /><figcaption>Figure 1: Summary of the results on existence and uniqueness depending on $\tau$ for fixed time $T$.</figcaption></figure>
**Outline.** In Section 2 we recall results on rough paths theory. In Section 3 we prove the existence and uniqueness of a solution for a small time $\tau$. In Section 4 we study the regularity of the solution with respect to the driving rough noise. In Sections 5 and 6 we establish more results on regularity and convergence. In Section 7 we study the existence of solutions for any time $\tau$. Finally, in Section 8 we consider the case of a driving rough noise with Hölder regularity greater than 1/2. In this context, the rough integrals coincide with the Young integrals.

## 2 Reminders about rough paths

We consider in this section the case of rough differential equations driven by paths of Hölder regularity $\alpha \in (\frac{1}{3}, \frac{1}{2}]$.

For two Banach space U and V whose norms are written $|\cdot|$, we denote by L(U, V) the space of bounded, linear operators from U to V. Here, U and V are finite or infinite dimensional.

We denote by $C^\alpha(W, V)$ the space of $\alpha$-Hölder continuous maps from W to V for $W \subset U$. The space is equipped with the $\alpha$-Hölder semi-norm $\|f\|_\alpha := \sup_{x \neq y} |f(x) - f(y)|/|x - y|^\alpha$. When $W = [0, T] \subset \mathbb{R}$, we write $\|\cdot\|_{\alpha, T}$ instead of $\|\cdot\|_\alpha$.

We denote by $C^k(U, V)$ for some integer $k$ the space of continuous functions which are differentiable up to order $k$ with continuous $k$-th derivatives. We denote by $C_b^k(U, V)$ the space of functions in $C^k(U, V)$ with bounded $j$-th derivatives for all $0 \le j \le k$.

We denote by $C^\alpha([0, T], U)$ the space of $\alpha$-Hölder rough paths $\mathbf{X} = (X, \mathcal{X})$ with values in $U \oplus (U \otimes U)$. The elements of $C^\alpha([0, T], U)$ satisfy the Chen relation

$$X(r, s) + X(s, t) = X(r, t), \quad \mathcal{X}(r, s) + \mathcal{X}(s, t) + X(r, s) \otimes X(s, t) = \mathcal{X}(r, t)$$

for any $0 \le r \le s \le t \le T$. Besides

$$\|X\|_{\alpha, T} = \sup_{0 \le s \neq t \le T} \frac{|X(s, t)|}{|t - s|^\alpha} < \infty \quad \text{and} \quad \|\mathcal{X}\|_{2\alpha, T} = \sup_{0 \le s \neq t \le T} \frac{|\mathcal{X}(s, t)|}{|t - s|^{2\alpha}} < \infty.$$

The metric on $C^\alpha([0, T], U)$ is denoted by $\varrho_{\alpha, T}$ and defined by

$$\varrho_{\alpha,T}(\mathbf{X}_1, \mathbf{X}_2) = \sup_{0 \le s \neq t \le T} \frac{|X_1(s,t) - X_2(s,t)|}{|t-s|^\alpha} + \sup_{0 \le s \neq t \le T} \frac{|\mathcal{X}_1(s,t) - \mathcal{X}_2(s,t)|}{|t-s|^{2\alpha}}$$

for all $\mathbf{X}_1$ and $\mathbf{X}_2$ in $C^\alpha([0, T], U)$.

Let U and V be two Banach spaces, $\mathbf{X} \in C^\alpha([0, T], U)$, $f \in C_b^2(V, L(U, V))$ and $a \in V$. We consider the following differential equation

$$Y(t) = a + \int_{0}^{t} f(Y(s)) \mathrm{d}\mathbf{X}(s) \quad \text{for all } t \in [0, T] \quad (3)$$
with unknown $Y \in C^\alpha([0, T], V)$. Following [6, 10], a solution $Y$ of (9) can be defined as a path such that $Y(0) = a$ and

$$Y(t) - Y(s) = f(Y(s))X(s,t) + Df(Y(s))f(Y(s))\mathbb{X}(s,t) + O(|t-s|^\theta) \quad (4)$$

for some $\theta > 1$.

Following [10, Theorem 8.3], if $f \in C_b^3(V, L(U, V))$, then there exists a unique solution $Y \in C^\alpha([0, T], V)$ to Equation (4) (or (3)). In order to take account of the dependence on $(a, \mathbf{X})$, we set $I(a, \mathbf{X}) := Y$. This defines the Itô map $I$ from $V \times C^\alpha([0, T], U)$ to $C^\alpha([0, T], V)$. The next lemma establishes the boundedness of this solution (see for instance [10, Proposition 8.2]).

**Lemma 1.** There exists a positive constant $K = K(\alpha)$ depending only on $\alpha$ such that for all $a \in V$,

$$\|I(a, \mathbf{X})\|_{\alpha, T} \le K(\alpha) \max \left( \|f\|_{C^2} \|\mathbf{X}\|_{\alpha, T}, (\|f\|_{C^2} \|\mathbf{X}\|_{\alpha, T})^{1/\alpha} \right), \quad (5)$$

where

$$\|\mathbf{X}\|_{\alpha, T} := \|X\|_{\alpha, T} + \sqrt{\|\mathbf{X}\|_{2\alpha, T}}.$$

By T. Lyons' theorem [17] the Itô map $I$ is continuous. Moreover it is locally Lipschitz continuous, as recalled in the following lemma (see for instance [5, 10, 15]). Notice that some extended statements consider also variations of the vector field $f$.

**Lemma 2.** We fix $M > 0$ and consider

$$B_{M,T} = \{\mathbf{X} \in C^\alpha([0, T], U) : \|\mathbf{X}\|_{\alpha, T} \le M\}, \quad (6)$$

where $\|\mathbf{X}\|_{\alpha, T}$ is defined in Lemma 1. There exists a positive constant $C_{\alpha, f, M}$ depending only on $\alpha$, $f$ and $M$ such that for all $a$ and $b \in V$, and for all $\mathbf{X}_1$ and $\mathbf{X}_2 \in B_{M,T}$,

$$\|I(a, \mathbf{X}_1) - I(b, \mathbf{X}_2)\|_{\alpha, T} \le C_{\alpha, f, M} (|a - b| + \varrho_{\alpha, T}(\mathbf{X}_1, \mathbf{X}_2)). \quad (7)$$

In particular, for every $\mathbf{X} \in C^\alpha([0, T], U)$ there exists a positive constant $C_{\alpha, f, \mathbf{X}}$ depending only on $\alpha$, $f$ and $\|\mathbf{X}\|_{\alpha, T}$ such that for all $a$ and $b \in V$,

$$\|I(a, \mathbf{X}) - I(b, \mathbf{X})\|_{\alpha, T} \le C_{\alpha, f, \mathbf{X}} |a - b|. \quad (8)$$

For the sake of simplicity, we do not consider optimal hypotheses on the vector field. The assumptions on the regularity of $f$ can be weakened (for example when $f \in C_b^{2+\gamma}$ with $\alpha(2 + \gamma) > 1$) but to get a global Lipschitz condition with respect to the starting point, we highlight that $f$ shall be bounded. This latter hypothesis is used in several proofs.
### 3 Existence and uniqueness

Let $v \in V$ and $H_0$ and $H_1$ be two bounded operators on $V$ (finite or infinite dimensional). We set $\tau$ and $T$ such that $0 < \tau \le T$.

Throughout the paper and unless otherwise mentioned, we assume that $f \in C_b^3(V, L(U, V))$.

In this section we fix $\mathbf{X} \in C^\alpha([0, T], U)$ and we wish to find a continuous function $Y$ such that

$$ \left\{ \begin{array}{l} Y(t) = Y(0) + \int_0^t f(Y(s)) \, d\mathbf{X}(s) \quad \text{for all } t \in [0, T], \\ H_0Y(0) + H_1Y(\tau) = v. \end{array} \right. \qquad (9) $$

Then we consider the following problem. We look for an element $a \in V$ such that

$$ H_0a + H_1I(a)(\tau) = v, \qquad (10) $$

where $I$ is the Itô map and $I(a, \mathbf{X})$ is denoted by $I(a)$ since $\mathbf{X}$ is fixed. The following lemma establishes the relation between (10) and (9).

**Lemma 3.** If there exists $a \in V$ such that (10) holds, then $Y = I(a)$ satisfies (9). Moreover, if there exists a unique $a \in V$ such that (10) holds, then $Y = I(a)$ is the unique function in $C^\alpha([0, T], V)$ satisfying (9).

*Proof.* The first part of the lemma is obvious. Assume there exists a unique $a \in V$ such that (10) and set $Y = I(a)$. Consider another $Z \in C^\alpha([0, T], V)$ such that

$$ \left\{ \begin{array}{l} Z(t) = Z(0) + \int_0^t f(Z(s)) \, d\mathbf{X}(s) \quad \text{for all } t \in [0, T], \\ H_0Z(0) + H_1Z(\tau) = v. \end{array} \right. $$

By the definition of the Itô map, $Z = I(Z(0))$, so that $H_0Z(0) + H_1I(Z(0))(\tau) = v$. This implies that $Z(0) = a$ (by uniqueness of $a$), and then $Z = I(Z(0)) = I(a) = Y$. $\square$

The following theorem establishes the existence of a unique solution of the problem (9) for small values of $\tau$. It is proved with the use of (10), which is solved with the Banach fixed-point theorem.

For $a \in V$ and $\tau \in [0, T]$, we define

$$ L(a)_\tau := H^{-1}v - H^{-1}H_1(I(a)(\tau) - a) \qquad (11) $$

when $H := H_0 + H_1$ is invertible.

*Remark 1.* The condition that $H_0 + H_1$ is invertible means that our results cannot be applied to deal with a periodic solution.
**Theorem 1.** Let $C_{\alpha,f,\mathbf{X}}$ be the constant of (8). If $H := H_0 + H_1$ is invertible and $\tau$ satisfies

$$ \tau^{\alpha} |H^{-1}H_1| C_{\alpha,f,\mathbf{X}} < 1, \qquad (12) $$

then there exists a unique solution to (9) in $C^{\alpha}([0, T], V)$.

*Proof.* By Lemma 3, it suffices to prove that there exists a unique solution $a \in V$ to (10). We rewrite (10) as

$$ (H_0 + H_1)a + H_1(I(a)(\tau) - a) = v. $$

Then, since $H = H_0 + H_1$ is invertible,

$$ H_0 a + H_1 I(a)(\tau) = v \text{ if and only if } a = H^{-1}v - H^{-1}H_1(I(a)(\tau) - a). $$

With (11), (10) can be reformulated as the fixed-point problem

$$ a = L(a)_{\tau}. \qquad (13) $$

By the Banach fixed-point theorem, it suffices to prove that $L(\cdot)_{\tau}$ is a contraction. For all $a, b \in V$,

$$ \begin{aligned} |L(a)_{\tau} - L(b)_{\tau}| &\le |H^{-1}H_1| \cdot |I(a)(\tau) - a - I(b)(\tau) + b| \\ &\le |H^{-1}H_1| \cdot (|I(a)_0 - a - I(b)_0 + b| + \tau^{\alpha} \|I(a) - a - I(b) + b\|_{\alpha,T}) \\ &= \tau^{\alpha} |H^{-1}H_1| \cdot \|I(a) - a - I(b) + b\|_{\alpha,T} \\ &= \tau^{\alpha} |H^{-1}H_1| \cdot \|I(a) - I(b)\|_{\alpha,T}. \end{aligned} $$

Following (8),

$$ \|I(a) - I(b)\|_{\alpha,T} \le C_{\alpha,f,\mathbf{X}} |a - b|. $$

As a consequence of above,

$$ |L(a)_{\tau} - L(b)_{\tau}| \le \tau^{\alpha} |H^{-1}H_1| C_{\alpha,f,\mathbf{X}} |a - b|. $$

Then for every $\tau$ satisfying (12), the function $L(\cdot)_{\tau}$ is a contraction. Then by the Banach fixed-point theorem, there exists a unique fixed point $a \in V$ such that (13), and then (10) are satisfied. $\square$

As an application of Theorem 1, fix $\tau \in [0, T]$ and consider that $W = (W_1, \dots, W_d)$ is a Brownian motion with values in $U = \mathbb{R}^d$ which is enhanced with $\mathbf{W} = (W, \mathbb{W}) \in C^{\alpha}([0, T], \mathbb{R}^d)$ with for all $1 \le j, k \le d$,

$$ \mathbb{W}_{j,k}(s,t) = \int_s^t (W_j(r) - W_j(s)) \circ dW_k(r) \text{ for any } 0 \le s \le t \le T. \quad (14) $$
Then, Theorem 1 establishes that for every $\omega \in \Omega$ such that $\tau^\alpha|H^{-1}H_1|C_{\alpha,f,\mathbf{W}(\omega)} < 1$, there exists a unique solution $Y(\omega) \in C^\alpha([0, T], V)$ to

$$ egin{cases} Y(\omega)(t) = Y(\omega)(0) + \int_0^t \sum_{j=1}^d f_j(Y(\omega)(s)) \circ dW_j(\omega)(s), \quad t \in [0, T], \\ H_0Y(\omega)(0) + H_1Y(\omega)(\tau) = v. \end{cases} \tag{15} $$

Notice that $\int_0^t \sum_{j=1}^d f_j(Y(\omega)(s)) \circ dW_j(\omega)(s)$ stands for $\int_0^t f(Y(\omega)(s)) d\mathbf{W}(\omega)(s)$ because of the Stratonovich integration in (14) and both rough and Stratonovich integrals coincide. This result follows from Theorem 4 below which is the rough counterpart of the stochastic Wong-Zakai theorem, and the construction of [11] based on characteristics as solutions to some Stratonovich SDE.

This problem can also be solved in an infinite-dimensional setting. Suppose that $U$ is a separable Banach space and consider a $U$-valued Brownian motion $W$ on the Wiener space $(U, H, \mu)$ where $\mu$ is a zero-mean Gaussian measure on $U$ and $H$ is the Cameron-Martin space of $\mu$. Following [14], under suitable conditions, there exists an enhanced Brownian motion $\mathbf{W} = (W, \mathbb{W}) \in C^\alpha([0, T], U)$ obtained as follows. We consider the sequence $(\mathbf{W}(m))_{m \ge 1} = (W(m), \mathbb{W}(m))_{m \ge 1}$ where $(W(m))_{m \ge 1}$ is the dyadic piecewise linear approximation of $W$, and $(\mathbb{W}(m))_{m \ge 1}$ is defined by

$$ \mathbb{W}(m)(s, t) = \int_s^t (W(m)(r) - W(m)(s)) \otimes dW(m)(r) $$

for all $s, t$, and $m$. Assume that the tensor norm $\| \cdot \|_{U \otimes U}$ on $U \otimes U$ is exact for $\mu$, which means that there exist two constants $\gamma \in [1/2, 1)$ and $C > 0$ such that for all independent sequences $(G_n)_{n \ge 1}$ and $(\tilde{G}_n)_{n \ge 1}$ of independent $U$-valued random variables with distribution $\mu$, and for every $N \ge 0$,

$$ \mathbb{E} \left( \left\| \sum_{n=1}^{N} G_n \otimes \tilde{G}_n \right\|_{U \otimes U} \right) \le CN^\gamma. $$

By [14], under this assumption, as $m \to \infty$ the sequence $\mathbf{W}(m)$ converges in $C^\alpha([0, T], U)$ almost surely. Its limit defines an enhanced Brownian motion. Then by Theorem 1, for every $\omega \in \Omega$ such that $\tau^\alpha|H^{-1}H_1|C_{\alpha,f,\mathbf{W}(\omega)} < 1$, there exists a unique solution $Y(\omega) \in C^\alpha([0, T], V)$ to

$$ egin{cases} Y(\omega)(t) = Y(\omega)(0) + \int_0^t f(Y(\omega)(s)) d\mathbf{W}(\omega)(s) \text{ for all } t \in [0, T], \\ H_0Y(\omega)(0) + H_1Y(\omega)(\tau) = v. \end{cases} $$

## 4 Continuity of the solution with respect to the path

In the previous section the driving path $\mathbf{X}$ is fixed. In this section we consider $\mathbf{X}$ as a variable and we establish the continuity of the solution of (9) with respect to $\mathbf{X}$.
We set $M > 0$ and $\tau$ satisfying

$$ \tau^{\alpha} |H^{-1}H_1| C_{\alpha,f,M} < 1. \qquad (16) $$

When $\mathbf{X} \in \mathcal{B}_{M,T}$ (defined by (6)), (16) implies (12).

We define the function

$$ \bar{I}: \mathcal{B}_{M,T} \to C^{\alpha}([0, T], V) $$

such that for every $\mathbf{X} \in \mathcal{B}_{M,T}$, $\bar{I}(\mathbf{X})$ is the unique solution $Y$ to

$$ \begin{cases} Y(t) = Y(0) + \int_{0}^{t} f(Y(s)) \mathrm{d}\mathbf{X}(s) & \text{for all } t \in [0, T], \\ H_{0}Y(0) + H_{1}Y(\tau) = v. \end{cases} $$

The main theorem of this subsection establishes the continuity of $\bar{I}$.

**Theorem 2.** The map $\bar{I}$ is Lipschitz continuous with respect to $\varrho_{\alpha,T}$ on $\mathcal{B}_{M,T}$, with constant

$$ C_{\alpha,f,M} + \frac{|H^{-1}H_1| C_{\alpha,f,M} \tau^{\alpha}}{1 - |H^{-1}H_1| C_{\alpha,f,M} \tau^{\alpha}}. $$

*Proof.* Previously we proved that for every $\mathbf{X}$, $\bar{I}(\mathbf{X})$ can be written as $\bar{I}(\mathbf{X}) = I(a_{\mathbf{X}}, \mathbf{X})$ where $I$ is the Itô map and $a_{\mathbf{X}}$ is the fixed point of the function $L(\cdot, \mathbf{X})_{\tau}: a \mapsto L(a, \mathbf{X})_{\tau} = H^{-1}v - H^{-1}H_1(I(a, \mathbf{X})(\tau) - a)$.

By (7), for all $\mathbf{X}$ and $\mathbf{Y} \in \mathcal{B}_{M,T}$,

$$ \begin{aligned} \|\bar{I}(\mathbf{X}) - \bar{I}(\mathbf{Y})\|_{\alpha,T} &= \|I(a_{\mathbf{X}}, \mathbf{X}) - I(a_{\mathbf{Y}}, \mathbf{Y})\|_{\alpha,T} \\ &\le C_{\alpha,f,M} (|a_{\mathbf{X}} - a_{\mathbf{Y}}| + \varrho_{\alpha,T}(\mathbf{X}, \mathbf{Y})). \end{aligned} \qquad (17) $$

Moreover,

$$ \begin{aligned} |a_{\mathbf{X}} - a_{\mathbf{Y}}| &= |L(a_{\mathbf{X}}, \mathbf{X})_{\tau} - L(a_{\mathbf{Y}}, \mathbf{Y})_{\tau}| \\ &\le |L(a_{\mathbf{X}}, \mathbf{X})_{\tau} - L(a_{\mathbf{X}}, \mathbf{Y})_{\tau}| + |L(a_{\mathbf{X}}, \mathbf{Y})_{\tau} - L(a_{\mathbf{Y}}, \mathbf{Y})_{\tau}| \\ &= |H^{-1}H_1| \cdot |I(a_{\mathbf{X}}, \mathbf{X})_{\tau} - I(a_{\mathbf{X}}, \mathbf{Y})_{\tau}| \\ &\quad + |H^{-1}H_1| \cdot |I(a_{\mathbf{X}}, \mathbf{Y})_{\tau} - I(a_{\mathbf{Y}}, \mathbf{Y})_{\tau} - a_{\mathbf{X}} + a_{\mathbf{Y}}| \\ &\le |H^{-1}H_1| \cdot \|I(a_{\mathbf{X}}, \mathbf{X}) - I(a_{\mathbf{X}}, \mathbf{Y})\|_{\alpha,T} \tau^{\alpha} \\ &\quad + |H^{-1}H_1| \cdot \|I(a_{\mathbf{X}}, \mathbf{Y}) - I(a_{\mathbf{Y}}, \mathbf{Y})\|_{\alpha,T} \tau^{\alpha} \\ &\le |H^{-1}H_1| C_{\alpha,f,M} \varrho_{\alpha,T}(\mathbf{X}, \mathbf{Y}) \tau^{\alpha} \\ &\quad + |H^{-1}H_1| C_{\alpha,f,M} |a_{\mathbf{X}} - a_{\mathbf{Y}}| \tau^{\alpha}. \end{aligned} $$

As a consequence, for all $\mathbf{X}$ and $\mathbf{Y} \in \mathcal{B}_{M,T}$,

$$ |a_{\mathbf{X}} - a_{\mathbf{Y}}| \le \frac{|H^{-1}H_1| C_{\alpha,f,M} \tau^{\alpha}}{1 - |H^{-1}H_1| C_{\alpha,f,M} \tau^{\alpha}} \varrho_{\alpha,T}(\mathbf{X}, \mathbf{Y}) $$
and, by (17),

$$ \|\bar{I}(\mathbf{X}) - \bar{I}(\mathbf{Y})\|_{\alpha, T} \le \left( C_{\alpha, f, M} + \frac{|H^{-1}H_1| C_{\alpha, f, M} \tau^{\alpha}}{1 - |H^{-1}H_1| C_{\alpha, f, M} \tau^{\alpha}} \right) \varrho_{\alpha, T}(\mathbf{X}, \mathbf{Y}). $$

This completes the proof. □

# 5 Regularity and numerical analysis

In Section 4 we proved the continuity of the solution with respect to the path $\mathbf{X}$. In this section, with the implicit function theorem, we consider the regularity with respect to the other variables. Then we briefly present an application to numerical analysis.

For $\gamma \in (0, 1]$ and $k \in \mathbb{N}^*$, we denote by $C^{k+\gamma}$ the class of functions $f$ of class $C^k$ such that $D^k f$ is $\gamma$-Hölder continuous. We denote by $C_b^{k+\gamma}$ the space of functions in $C^{k+\gamma}$ with bounded $j$-th derivatives for all $0 \le j \le k$. This space is equipped with

$$ \|f\|_{C^{k+\gamma}} := \max\{\|f\|_{\infty}, \dots, \|D^k f\|_{\infty}, \|D^k f\|_{\gamma}\}. $$

Let $\mathbf{X} \in C^{\alpha}([0, T], U)$. We define

$$ \mathcal{P} := V \times C_b^{k+1+\gamma}(V, L(U, V)), $$

$$ \text{and } Q := C_b^{k+1+\gamma}(V, L(U, V)) \times V \times L(V, V) \times L(V, V). $$

**Proposition 1** ( [5, Theorem 3]). For any $\kappa \in (0, 1)$ such that $\alpha(2 + \kappa\gamma) > 1$, the Itô map $I: \mathcal{P} \to C^{\alpha}([0, T], V)$ is locally of class $C^{k+(1-\kappa)\gamma}$.

Let us write

$$ P(a, q)_{\tau} = H_{0}a + H_{1}I(a, f, \mathbf{X})(\tau) - v \text{ with } q = (f, v, H_{0}, H_{1}) \in \mathcal{Q}. $$

If $H_0 + H_1$ is invertible, the problem $L(a)_{\tau} = a$ is equivalent to $P(a, q)_{\tau} = 0$. Besides, by Proposition 1, $P$ is locally of class $C^{k+(1-\kappa)\gamma}$ on $\mathcal{Q}$ when $\alpha(2 + \kappa\gamma) > 1$. Moreover,

$$ D_a P(a, q)_{\tau} = H_0 + H_1 D_a I(a, f, \mathbf{X})(\tau), \quad (18) $$

where

$$ D_a I(a, f, \mathbf{X})(t) = \mathrm{Id} + \int_0^t Df(I(a, f, \mathbf{X})(s)) \cdot D_a I(a, f, \mathbf{X})(s) \mathrm{d}s. $$

The regularity is established by the following result, which is a generalized implicit function theorem (see e.g., [1]) applied to $P$.
**Theorem 3.** Let $\tau \in [0, T]$, $a^\dagger \in V$ and $q^\dagger \in Q$ such that $P(a^\dagger, q^\dagger)_\tau = 0$ and $D_a P(a^\dagger, q^\dagger)_\tau \in L(V, V)$ is an isomorphism. Then there exist neighborhoods $U^\dagger$ of $q^\dagger$ and $V^\dagger$ of $a^\dagger$ as well as a function $A \in C_b^{k+(1-\kappa)\gamma}(U^\dagger, V^\dagger)$ with $A(q^\dagger) = a^\dagger$ such that $P(A(q), q) = 0$ for any $q \in U^\dagger$.

**Remark 2.** At $\tau = 0$, $D_a I(a, f, \mathbf{X})(0) = \text{Id}$ and then $D_a P(a, q)_0 = H_0 + H_1$. To apply the implicit function theorem, $D_a P(a, q)_0$ has to be invertible. Hence, $H_0 + H_1$ shall be invertible which is an hypothesis we used to get the existence and uniqueness of a solution.

**Remark 3.** If $H_0 + H_1$ is invertible, then we can deduce from (18) that $D_a P(a, q)_\tau$ is invertible when $\tau$ is small enough. This not ensured for large $\tau$.

Theorem 3 implies that the solution $a$ is locally Lipschitz continuous with respect to $q = (f, v, H_0, H_1)$. Notice that we can also establish that $a$ is globally Lipschitz continuous with respect to $v$. More precisely, using (8) we get

$$|a(v_1) - a(v_2)| \le (1 - \tau^\alpha |H^{-1}H_1| C_{\alpha, f, \mathbf{X}}) |v_1 - v_2| \quad (19)$$

for all $v_1$ and $v_2$.

We now present an application to numerical analysis. Let us consider that for any starting point $a$ we have a numerical scheme $J(a)$ such that $J(a) = I(a, f, \mathbf{X})(\tau) + R(a)$ for a Lipschitz continuous error $a \mapsto R(a)$ with a Lipschitz norm $\ell$ smaller than 1. Several numerical schemes exist for rough paths and for RDE driven by Brownian motion or fractional Brownian motion (see for instance [13, 16, 20]).

Besides, let us assume that we are able to solve the fixed-point problem $H_0 a + H_1 J(a) - v = 0$ at precision $\eta$, that is we find $a'$ such that

$$H_0 a' + H_1 J(a') = v + \eta.$$

This is the case for example when we use a finite number of iterations in the fixed-point iteration, or the Newton scheme. Then

$$H_0 a' + H_1 I(a', f, \mathbf{X})(\tau) = v + \eta - H_1 R(a').$$

It follows from Theorem 3 or (19) that there exists a constant $K$, depending on $\tau$, $\ell$, $f$, $\mathbf{X}$, $H_0$ and $H_1$ such that

$$|a' - a| \le K|\eta| + K|R(a)|.$$

This implies that the numerical scheme converges.

# 6 Limit theorems

In this section we deal with limit theorems. We fix $\tau$ and $M$ such that (16), namely

$$\tau^\alpha |H^{-1}H_1| C_{\alpha, f, M} < 1.$$
For every $T > 0$ we denote by $C^{0,\alpha}([0, T], U)$ the closure of smooth rough paths in $C^\alpha([0, T], U)$. We assume that the Banach space $U$ is separable and recall that, as a consequence, $C^{0,\alpha}([0, T], U)$ is a Polish space (see Exercise 2.8 of [10] for instance). The main result is the following.

**Theorem 4.** Suppose (16) and let $\mathbf{X}$ and $(\mathbf{X}^n)_n$ be a random variable and a sequence of random variables with values in $C^{0,\alpha}([0, T], U)$ such that $\mathbf{X}^n$ converges in distribution to $\mathbf{X}$ in $C^{0,\alpha}([0, T], U)$ and $\mathbb{P}(\|\mathbf{X}\|_{\alpha,T} = M) = 0$. For every continuous bounded function $h: C^\alpha([0, T], V) \to \mathbb{R}$,

$$ \lim_{n \to \infty} \mathbb{E}(h(\bar{I}(\mathbf{X}^n))) \mathbf{1}_{\|\mathbf{X}^n\|_{\alpha,T} \le M} = \mathbb{E}(h(\bar{I}(\mathbf{X}))) \mathbf{1}_{\|\mathbf{X}\|_{\alpha,T} \le M}. $$

*Proof.* Since $C^{0,\alpha}([0, T], U)$ is a Polish space, by the Skorokhod representation theorem there exists a probability space $(\Omega^*, \mathcal{T}^*, \mathbb{P}^*)$ and random variables $\mathbf{X}^{*,n}$ and $\mathbf{X}^*$ from $(\Omega^*, \mathcal{T}^*, \mathbb{P}^*)$ to $C^{0,\alpha}([0, T], U)$ such that $\mathbf{X}^{*,n} \stackrel{\text{law}}{=} \mathbf{X}^n$, $\mathbf{X}^* \stackrel{\text{law}}{=} \mathbf{X}$ and $\mathbf{X}^{*,n}$ converges $\mathbb{P}^*$-almost surely to $\mathbf{X}^*$. Then $\mathbb{P}^*$-almost surely, $h(\bar{I}(\mathbf{X}^{*,n})) \mathbf{1}_{\|\mathbf{X}^{*,n}\|_{\alpha,T} \le M}$ converges to $h(\bar{I}(\mathbf{X}^*)) \mathbf{1}_{\|\mathbf{X}^*\|_{\alpha,T} \le M}$ as $n \to \infty$. Moreover, by the bounded convergence theorem, since for all $n$,

$$ h(\bar{I}(\mathbf{X}^{*,n})) \mathbf{1}_{\|\mathbf{X}^{*,n}\|_{\alpha,T} \le M} \le \|h\|_{\infty}, $$

we deduce that

$$ \lim_{n \to \infty} \mathbb{E}(h(\bar{I}(\mathbf{X}^{*,n}))) \mathbf{1}_{\|\mathbf{X}^{*,n}\|_{\alpha,T} \le M} = \mathbb{E}(h(\bar{I}(\mathbf{X}^*)) \mathbf{1}_{\|\mathbf{X}^*\|_{\alpha,T} \le M}). $$

This completes the proof. $\square$

<figure><img src="image_5.png" /></figure>

As an application, we consider for every $n \in \mathbb{N}^*$ the problem

$$ \left\{ \begin{array}{l} \frac{dY^n(t)}{dt} = \sqrt{n} \sum_{j=1}^{d} \xi_{j,[nt]} f_j(Y^n(t)) \quad \text{for all } t \in [0, T], \\ H_0 Y^n(0) + H_1 Y^n(\tau) = v, \end{array} \right. $$

where $\xi := (\xi_{j,k})_{(j,k) \in \{1, \dots, d\} \times \mathbb{N}}$ is a discrete family of independent random variables with mean zero, variance 1, same distribution and finite moments. We define the sequence of $\mathbb{R}^d$-valued processes $W^n = (W_1^n, \dots, W_d^n)$ such that for all $j = 1, \dots, d$,

$$ W_j^n(t) = \frac{1}{\sqrt{n}} \left( \sum_{k=0}^{[nt]-1} \xi_{j,k} + (nt - [nt]) \xi_{j,[nt]} \right). $$

Notice that $W^n$ is Lipschitz-continuous. Therefore, $W^n$ can be enhanced with $\mathbf{W}^n = (W^n, \mathbb{W}^n) \in C^\alpha([0, T], \mathbb{R}^d)$ where

$$ \mathbb{W}^n(s, t) = \int_s^t (W^n(r) - W^n(s)) \otimes dW^n(r). $$

The following theorem establishes the convergence in distribution of $Y^n$ as $n$ goes to infinity.
**Theorem 5.** For every continuous bounded function $h: C^\alpha([0, T], V) \to \mathbb{R}$,

$$ \lim_{n \to \infty} \mathbb{E}(h(Y^n)\mathbf{1}_{\mathbf{W}^n \in \mathcal{B}_{M,T}}) = \mathbb{E}(h(Y)\mathbf{1}_{\mathbf{W} \in \mathcal{B}_{M,T}}), $$

where $\mathbf{W}$ is the enhanced Brownian motion defined by (14) and $Y$ is the solution of

$$ \left\{ \begin{array}{l} dY(t) = \sum_{j=1}^{d} f_j(Y(t)) \circ dW_j(t) \quad \text{for all } t \in [0, T], \\ H_0Y(0) + H_1Y(\tau) = v. \end{array} \right. $$

This theorem is similar to [11, Theorem 7] but its proof is different.

*Proof.* It is a direct consequence of [4, Theorem 1] and of Theorem 4. $\square$

# 7 Global existence result

In Section 3 we prove that (9) (and (10)) has a unique solution for every $\tau$ small enough. In this section we use the Brouwer degree in order to prove that (9) has a solution (maybe more than one) for every $\tau \in [0, T]$, yet only when $V$ is finite-dimensional.

Throughout the section, we assume that $V = \mathbb{R}^N$ for some $N \in \mathbb{N}^*$. We equip $V$ with the Lebesgue measure.

Before we establish the existence of solution for (9) in Theorem 6, we remind the definition and some properties of the Brouwer degree. For a more complete exposition on the degree theory we refer the reader for instance to [12].

Consider a bounded open subset $O \subset \mathbb{R}^N = V$, a continuous function $\Phi: \bar{O} \to \mathbb{R}^N$ and $y \in V$ such that $y \notin \Phi(\partial O)$. The Brouwer degree $\deg(\Phi, O, y)$ with $\Phi \in C^1(O, V)$ can be defined as follows (see for instance [12, Definition 13.2.9]).

**Definition 1.** If $y$ is a regular value of $\Phi$ (namely such that $\det J_\Phi(x) \neq 0$ for all $x \in \Phi^{-1}(\{y\})$), then $\deg(\Phi, O, y)$ is defined by

$$ \deg(\Phi, O, y) = \sum_{x \in \Phi^{-1}(\{y\})} \operatorname{sign}(\det J_{\Phi}(x)). $$

If $y$ is a critical value of $\Phi$ ($y$ is not a regular value of $\Phi$), then $\deg(\Phi, O, y)$ is defined by

$$ \deg(\Phi, O, y) = \deg(\Phi, O, y'), $$

where $y'$ is a regular value of $\Phi$ such that $|y - y'| < \operatorname{dist}(y, \Phi(\partial O))$.
The Brouwer degree has the following property (see for instance [12, Theorem 13.2.11(iv)]).

**Lemma 4.** Let $y \in V$ such that $y \notin \Phi(\partial O)$. If $\deg(\Phi, O, y) \neq 0$ then there exists a solution $x \in O$ to the equation $\Phi(x) = y$.

The invariance by homotopy is another important property of the Brouwer degree and can be stated as follows (see for instance [12, Theorem 13.2.11(ii)]).

**Lemma 5.** We fix $\tau > 0$ and consider a continuous function $H : [0, \tau] \times \overline{O} \to V$. If we assume that $y \notin H(t, \partial O)$ for all $t \in [0, \tau]$, then

$$ \deg(H(0, \cdot), O, y) = \deg(H(\tau, \cdot), O, y). $$

The existence of solutions of (9) can now be established in the following theorem.

**Theorem 6.** If $H := H_0 + H_1$ is invertible, then for every $\tau \in [0, T]$, there exists $Y \in C^\alpha([0, T], V)$ such that (9) is satisfied. Moreover, for almost every $v \in V$ the number of solutions of (9) is finite.

*Proof.* As a consequence of Lemma 3 and the beginning of the proof of Theorem 1, it suffices to prove that there exists $a \in V$ such that $a = L(a)_\tau$. With $\mathcal{L}$ defined by

$$ \mathcal{L}(\tau, a) = a + H^{-1}H_1(I(a)(\tau) - a) \text{ for } (\tau, a) \in [0, T] \times V, $$

the problem $a = L(a)_\tau$ is equivalent to

$$ \mathcal{L}(\tau, a) = \tilde{v} \text{ where } \tilde{v} = H^{-1}v. \qquad (20) $$

Hence, we look for $a \in V$ solving (20). In order to prove the existence of solutions it suffices to prove that for $r > 0$ large enough and for all $\tau \in [0, T]$ the Brouwer degree of $\mathcal{L}(\tau, \cdot)$ exists and satisfies

$$ \deg(\mathcal{L}(\tau, \cdot), B_r(\tilde{v}), \tilde{v}) \neq 0, \qquad (21) $$

where $B_r(\tilde{v})$ stands for the ball in $V$ of center $\tilde{v}$ and radius $r$. For all $r > 0$,

$$ \deg(\mathcal{L}(0, \cdot), B_r(\tilde{v}), \tilde{v}) = \deg(\mathrm{Id}, B_r(\tilde{v}), \tilde{v}) = 1. $$

By Lemma 1, for all $(\tau, x) \in [0, T] \times V$ such that $\mathcal{L}(\tau, x) = \tilde{v}$, we have

$$ \begin{aligned} |x - \tilde{v}| &= |H^{-1}H_1(I(x)(\tau) - x)| \\ &\le |H^{-1}H_1| \cdot |I(x)(\tau) - I(x)(0)| \\ &\le |H^{-1}H_1| \cdot \|I(x)\|_{\alpha} \tau^{\alpha} \\ &\le |H^{-1}H_1| c_{\alpha, f, \mathbf{X}} T^{\alpha}. \end{aligned} $$
This implies that for all $r > |H^{-1}H_1|c_{\alpha,f,\mathbf{X}}T^\alpha$ and all $\tau \in [0, T]$,

$$ \tilde{v} \notin \mathcal{L}(\tau, \partial B_r(\tilde{v})) $$

and then the degree $\deg(\mathcal{L}(\tau, \cdot), B_r(\tilde{v}), (\tilde{v}))$ is well-defined. Moreover, by the continuity of $I$, the function $\mathcal{L}$ is a homotopy. This implies that for all $\tau \in [0, T]$,

$$ \deg(\mathcal{L}(\tau, \cdot), B_r(\tilde{v}), \tilde{v}) = \deg(\mathcal{L}(0, \cdot), B_r(\tilde{v}), \tilde{v}) = 1 $$

and then (21) and the existence of a solution $a \in B_r(\tilde{v})$ to $a = L(a)_\tau$.

It remains to prove that for every $\tau \in [0, T]$ the number of solutions of $a = L(a)_\tau$ is finite for almost every $v$, or equivalently for almost every $\tilde{v}$. We fix $r > |H^{-1}H_1|c_{\alpha,f,\mathbf{X}}T^\alpha$. Since $f \in C_b^3$ the function $a \mapsto I(a)$ is $C^1$. Following the Sard theorem [12, Lemma 13.2.1, p. 331], the Lebesgue measure of the set of critical values of $\mathcal{L}(\tau, \cdot)$ is equal to 0. Consider a regular value $\tilde{v}$ of $\mathcal{L}(\tau, \cdot)$. The set $\overline{B_r(\tilde{v})} \cap \mathcal{L}(\tau, \cdot)^{-1}(\tilde{v})$ is closed and bounded, and consequently compact because $V$ is finite-dimensional. Since $\tilde{v}$ is a regular value, $\mathcal{L}(\tau, \cdot)$ is locally invertible on a neighborhood of each point of $\overline{B_r(\tilde{v})} \cap \mathcal{L}(\tau, \cdot)^{-1}(\tilde{v})$. Then each point of $\overline{B_r(\tilde{v})} \cap \mathcal{L}(\tau, \cdot)^{-1}(\tilde{v})$ is isolated. Since $|L(\tau, \cdot)|$ is bounded (by $|H^{-1}v| + |H^{-1}H_1|\tau^\alpha c_{\alpha,f,\mathbf{X}}$ for instance), we choose $r$ large enough and deduce that $\mathcal{L}(\tau, \cdot)^{-1}(\tilde{v})$ is finite. This completes the proof. $\square$

**Remark 4.** If $f$ is not bounded, the existence of solutions is not ensured for every $\tau$. Consider for instance the real-valued differential equation

$$ dY(t) = Y(t) dW(t) $$

where $W$ is a Hölder function and $a \in \mathbb{R}$. The solutions are written as

$$ Y(t) = a \exp(W(t) - W(0)). $$

where $a$ is a constant. The condition

$$ H_0Y(0) + H_1Y(T) = v $$

can be rewritten as

$$ H_0a + H_1a \exp(W(T) - W(0)) = v $$

and then

$$ a = \frac{v}{H_0 + H_1 \exp(W(T) - W(0))} $$

which is defined if and only if $H_0 + H_1 \exp(W(T) - W(0)) \neq 0$. The solution is then

$$ Y(t) = \frac{v \exp(W(t) - W(0))}{H_0 + H_1 \exp(W(T) - W(0))} = \frac{v \exp(W(t))}{H_0 \exp(W(0)) + H_1 \exp(W(T))}. $$
If $H_0$ and $H_1$ have different signs and are different from 0, then the condition $H_0 + H_1 \exp(W(T) - W(0)) \neq 0$ is equivalent to

$$W(T) - W(0) \neq \ln \left( -\frac{H_0}{H_1} \right).$$

This is not ensured even when $W$ is bounded.

# 8 Young differential equations

In this section we wish to extend previous results to a framework with long-range dependence. This generally involves more regular processes and the Young integration suffices. We consider in this section $\alpha \in (\frac{1}{2}, 1]$. If $U$ is a Banach space we denote by $C^\alpha([0, T], U)$ the space of $\alpha$-Hölder rough paths $X$ and define

$$\|X\|_{\alpha,T} = \sup_{0 \le s \ne t \le T} \frac{|X(s,t)|}{|t-s|^\alpha} < \infty.$$

The metric on $C^\alpha([0, T], U)$ is defined by

$$\varrho_{\alpha,T}(X_1, X_2) = \sup_{0 \le s \ne t \le T} \frac{|X_1(s,t) - X_2(s,t)|}{|t-s|^\alpha}$$

for all $X_1$ and $X_2$ in $C^\alpha([0, T], U)$. If $V$ is another Banach space, $f \in C_b^2(V, L(U, V))$ and $a \in V$ then the following differential equation

$$Y(t) = a + \int_{0}^{t} f(Y(s)) dX(s) \text{ for all } t \in [0, T]$$

(in the Young sense) has a unique solution $Y \in C^\alpha([0, T], V)$. If we denote this solution by $I(a, X)$ to take account of the dependence on $(a, X)$, then it defines the Itô map $I$ from $V \times C^\alpha([0, T], U)$ to $C^\alpha([0, T], V)$. Moreover it can be proved that, if we set

$$B_{M,T} = \{X \in C^\alpha([0, T], U) \mid \|X\|_{\alpha,T} \le M\},$$

that there exists a positive constant $C_{\alpha,f,M}$ depending only on $\alpha$, $f$ and $M$ such that for all $a, b \in V$, and for all $X_1$ and $X_2 \in B_{M,T}$,

$$\|I(a, X_1) - I(b, X_2)\|_{\alpha,T} \le C_{\alpha,f,M} (|a-b| + \varrho_{\alpha,T}(X_1, X_2)). \quad (22)$$

In particular, for every $X \in C^\alpha([0, T], U)$ there exists a positive constant $C_{\alpha,f,X}$ depending only on $\alpha$, $f$ and $\|X\|_{\alpha,T}$ such that for all $a$ and $b \in V$,

$$\|I(a, X) - I(b, X)\|_{\alpha,T} \le C_{\alpha,f,X} |a - b|. \quad (23)$$

From (22), we can deduce the following result whose proof is similar to that of Theorem 1.
**Theorem 7.** Let $\tau > 0$ such that

$$ \tau < \frac{1}{(|H^{-1}H_1|C_{\alpha,f,X})^{1/\alpha}}. \qquad (24) $$

If $H := H_0 + H_1$ is invertible, then there exists a unique solution $Y \in C^{\alpha}([0, T], V)$ to

$$ \left\{ \begin{array}{l} dY(t) = f(Y(t)) dX(t) \quad \text{for all } t \in [0, T], \\ H_0 Y(0) + H_1 Y(\tau) = v. \end{array} \right. \qquad (25) $$

We fix $M > 0$ and $\tau > 0$ such that

$$ \tau < \frac{1}{(|H^{-1}H_1|C_{\alpha,f,M})^{1/\alpha}}. \qquad (26) $$

Theorem 7 enables us to define the function $\bar{I} : \mathcal{B}_{M,T} \to C^{\alpha}([0, T], V)$ such that for every $X \in C^{\alpha}([0, T], U)$, $\bar{I}(X)$ is the unique solution of (25). The following theorem states that $\bar{I}$ is continuous. Its proof is similar to that of Theorem 2.

**Theorem 8.** The map $\bar{I}$ is Lipschitz continuous with respect to $\varrho_{\alpha,T}$ on $\mathcal{B}_{M,T}$, with constant

$$ C_{\alpha,f,M} + \frac{|H^{-1}H_1|C_{\alpha,f,M}\tau^{\alpha}}{1 - |H^{-1}H_1|C_{\alpha,f,M}\tau^{\alpha}}. $$

We can deduce the following limit theorem.

**Theorem 9.** We assume that $U$ is separable and that (26) is satisfied. Let $X$ and $(X^n)_n$ be a random variable and a sequence of random variables with values in $C^{0,\alpha}([0, T], U)$ such that $X^n$ converges in distribution to $X$ in $C^{0,\alpha}([0, T], U)$ and $\mathbb{P}(\|X\|_{\alpha,T} = M) = 0$. For every continuous bounded function $h : C^{\alpha}([0, T], V) \to \mathbb{R}$,

$$ \lim_{n \to \infty} \mathbb{E}(h(\bar{I}(X^n)) \mathbf{1}_{X^n \in \mathcal{B}_{M,T}}) = \mathbb{E}(h(\bar{I}(X)) \mathbf{1}_{X \in \mathcal{B}_{M,T}}). $$

Now we give an application of the above result. We consider for every $\varepsilon \in (0, 1]$ the problem

$$ \left\{ \begin{array}{l} \displaystyle \frac{dY^{\varepsilon}(t)}{dt} = \sum_{j=1}^{d} \frac{1}{\varepsilon^{\gamma_j}} m_j \left( \frac{t}{\varepsilon^2} \right) f_j(Y^{\varepsilon}(t)) \quad \text{for all } t \in [0, T], \\ H_0 Y^{\varepsilon}(0) + H_1 Y^{\varepsilon}(\tau) = v, \end{array} \right. \qquad (27) $$

where for every $j$, $\gamma_j \in (0, 1)$ and $m_j$ is a continuous stationary Gaussian process with mean zero and such that there exists $c_j$ such that the covariance satisfies as $t \to \infty$,

$$ \mathbb{E}(m_j(0)m_j(t)) \sim \frac{c_j}{t^{\gamma_j}}. $$
Moreover we assume that $m_1, m_2, \dots, m_d$ are independent, and without loss of generality, that $\gamma_1 \ge \dots \ge \gamma_d$.

We define for every $\varepsilon$ the Lipschitz continuous process $w^\varepsilon$ such that for all $(s, t)$,

$$w^\varepsilon(s, t) = (w_1^\varepsilon(s, t), \dots, w_d^\varepsilon(s, t))$$

where for all $j$,

$$w_j^\varepsilon(s, t) = \int_s^t \frac{1}{\varepsilon^{\gamma_j}} m_j \left( \frac{r}{\varepsilon^2} \right) dr.$$

The following result deals with the asymptotic behavior of (27).

**Theorem 10.** Consider $\alpha \in (\frac{1}{2}, \frac{2-\gamma_1}{2})$. For every continuous bounded function $h: C^\alpha([0, T], V) \to \mathbb{R}$,

$$\lim_{n \to \infty} \mathbb{E}(h(Y^\varepsilon) \mathbf{1}_{w^\varepsilon \in \mathcal{B}_{M,T}}) = \mathbb{E}(h(Y) \mathbf{1}_{W \in \mathcal{B}_{M,T}}), \quad (28)$$

where $Y$ is the solution of

$$\begin{cases} dY(t) = \sum_{j=1}^{d} f_j(Y(t)) dW_j(t) & \text{for all } t \in [0, T], \\ H_0Y(0) + H_1Y(\tau) = v & \end{cases}$$

and $W = (W_1, \dots, W_d)$ is a multidimensional fractional Brownian motion such that for all $(j, k)$,

$$\mathbb{E}(W_j(t)W_k(s)) = \delta_{j=k} \frac{\rho_j}{2} (|t|^{2H_j} + |s|^{2H_j} - |t-s|^{2H_j})$$

with $H_j = (2 - \gamma_j)/2$ and $\rho_j = H_j^{-1}(2H_j - 1)^{-1}c_j$.

*Proof.* Since $\alpha \in (\frac{1}{2}, \frac{2-\gamma_1}{2})$ and by [19, Theorem 5], $w^\varepsilon$ converges in distribution to $W$ in $C^{0,\alpha}([0, T], V)$ as $\varepsilon \to 0$. By Theorem 9, the proof is completed. $\square$

# 9 Conclusion

We reduced the affine boundary value problem (9) to a fixed-point problem. This approach may be used whenever the Itô map is well defined. This is especially suitable for a large range of driving rough signals (semi-martingales, fractional Brownian motion and many more) and enable us to avoid any technical difficulties related to anticipating stochastic calculus.

For every time $\tau$ small enough, using properties of the Itô map we established the existence and uniqueness of a solution, regularity results, and limit theorems.
Using the Brouwer degree we proved a global existence result. The number of solutions is finite for almost every $v$. We also gave counterexample showing that, if $f$ is not bounded, then the existence of a solution for every time $\tau$ is not ensured. This is related to the behavior of the Itô map and to the operators $H_0$ and $H_1$.

Further studies could concern sharper hypotheses, counterexamples for the existence or the uniqueness of a solution, as well as asymptotic results and their applications in multiscale modeling.

Finally, let us mention that the use of regularity structures [10] could lead to similar results where the ordinary/rough differential equations are replaced by (rough) partial differential equations.

## References

[1] R. Abraham, J.E. Marsden, and T.S. Raţiu. *Manifolds, tensor analysis, and applications*. Addison-Wesley Publishing Co., Reading, Mass., 1983.

[2] A. Alabert and M. A. Marmolejo. Differential equations with boundary conditions perturbed by a Poisson noise. *Stochastic Process. Appl.*, 91(2):255-276, 2001.

[3] R. Bellman and G. M. Wing. *An introduction to invariant imbedding*. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1975.

[4] E. Breuillard, P. Friz, and M. Huesmann. From random walks to rough paths. *Proc. Amer. Math. Soc.*, 137(10):3487-3496, 2009.

[5] L. Coutin and A. Lejay. Sensitivity of rough differential equations: an approach through the omega lemma. *J. Differential Equations*, 264(6):3899-3917, 2018.

[6] A. M. Davie. Differential equations driven by rough paths: an approach via discrete approximation. *Appl. Math. Res. Express. AMRX*, 2008:abm009, 2008.

[7] C. Donati-Martin. Équations différentielles stochastiques dans $\mathbf{R}$ avec conditions aux bords. *Stochastics Stochastics Rep.*, 35(3):143-173, 1991.

[8] M. Ferrante and D. Nualart. On the Markov property of a stochastic difference equation. *Stochastic Process. Appl.*, 52(2):239-250, 1994.

[9] J.-P. Fouque and E. Merzbach. A limit theorem for linear boundary value problems in random media. *Ann. Appl. Probab.*, 4(2):549-569, 1994.
[10] P. K. Friz and M. Hairer. *A course on rough paths*. Universitext. Springer, Cham, 2nd edition, 2020.

[11] J. Garnier. Stochastic invariant imbedding. Application to stochastic differential equations with boundary conditions. *Probab. Theory Related Fields*, 103(2):249–271, 1995.

[12] V. Hutson and J. S. Pym. *Applications of functional analysis and operator theory*, volume 146 of Mathematics in Science and Engineering. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980.

[13] D. Kelly. Rough path recursions and diffusion approximations. *Ann. Appl. Probab.*, 26(1):425–461, 2016.

[14] M. Ledoux, T. Lyons, and Z. Qian. Lévy area of Wiener processes in Banach spaces. *Ann. Probab.*, 30(2):546–578, 2002.

[15] A. Lejay. Global solutions to rough differential equations with unbounded vector fields. In *Séminaire de Probabilités XLIV*, volume 2046 of Lecture Notes in Math., pages 215–246. Springer, Heidelberg, 2012.

[16] Y. Liu and S. Tindel. First-order Euler scheme for SDEs driven by fractional Brownian motions: the rough case. *Ann. Appl. Probab.*, 29(2):758–826, 2019.

[17] T. Lyons. Differential equations driven by rough signals. *Rev. Mat. Iberoamericana*, 14(2):215–310, 1998.

[18] T. Lyons and Z. Qian. *System control and rough paths*. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2002.

[19] R. Marty. Asymptotic behavior of differential equations driven by periodic and random processes with slowly decaying correlations. *ESAIM Probab. Stat.*, 9:165–184, 2005.

[20] A. Neuenkirch and T. Shalaiko. The relation between mixed and rough SDEs and its application to numerical methods. *Stoch. Anal. Appl.*, 33(5):927–942, 2015.

[21] D. Nualart and É. Pardoux. Stochastic calculus with anticipating integrands. *Probab. Theory Related Fields*, 78(4):535–581, 1988.

[22] D. Nualart and É. Pardoux. Boundary value problems for stochastic differential equations. *Ann. Probab.*, 19(3):1118–1144, 1991.

[23] D. Ocone and É. Pardoux. Linear stochastic differential equations with boundary conditions. *Probab. Theory Related Fields*, 82(4):489–526, 1989.
[24] O. Zeitouni and A. Dembo. A change of variables formula for Stratonovich integrals and existence of solutions for two-point stochastic boundary value problems. *Probab. Theory Related Fields*, 84(3):411–425, 1990.