Rough differential equations with
path-dependent coefficients

Anna Ananova
Mathematical Institute
University of Oxford.
[email protected]

Abstract

We establish the existence of solutions to path-dependent rough differential equations with non-anticipative coefficients. Regularity assumptions on the coefficients are formulated in terms of horizontal and vertical derivatives.

1 Introduction

The theory of rough paths [26] provides a framework for defining solutions to differential equations driven by a rough signal:

dY(t)=V(Y(t))dX(t)\qquad X(0)=x\in\mathbb{R}^{d}

(1)

where $\displaystyle V$ is a smooth vector field and $\displaystyle X,Y$ are continuous, but non-smooth functions whose lack of regularity prevents an interpretation of (1) in terms of Riemann-Stieltjes or Young integration. A key insight of T. Lyons [24] was to supplement the signal $\displaystyle X$ with a rough path tensor $\displaystyle\mathbb{X}$ constructed above $\displaystyle X$ such that one can construct an integration theory for (1) with respect to the enriched path $\displaystyle(X,\mathbb{X})$ . One of the main results of the theory is that (1) maybe then interpreted as a ‘rough differential equation’ (RDE) using this notion of rough integration.

Since the pioneering work of Lyons [24, 25], the study of such rough differential equations (RDEs) has developed in various directions. Solutions to rough differential equations have been constructed as limits of discrete approximations [11], fixed points of Picard iterations [24, 18, 25, 14] or limits of solutions of certain ODEs [4, 16]. An essential technique underlying many results is Picard iteration in the Banach space of controlled paths, introduced by Gubinelli [18] in the case of Hőlder regularity $\displaystyle\alpha\in(1/3,1/2]$ and extended to the case of arbitrary regularity in [19, 20].

As shown by Lyons [24], for stochastic differential equations driven by Brownian motion, probabilistic (Stratonovich) solutions coincide with RDE solutions constructed for an appropriate choice of Brownian rough path, showing that the theory of RDEs is also relevant for the study of stochastic differential equations (SDEs). SDEs with path-dependent features arise in many problems in stochastic analysis and stochastic modeling [21, 23, 28], and this natural link with RDEs has inspired several studies on rough differential equations with path-dependent features which echo examples of path-dependent SDEs encountered in stochastic models [1, 3, 12, 29].

A classical technique used in the study of path-dependent stochastic equations is to lift them to an infinite-dimensional SDE in the space of paths [28]. This approach has been adapted to RDEs in Banach spaces by Bailleul [3] but requires Fréchet differentiability of the vector fields (coefficients), an assumption which excludes many examples. Neuenkirch et al. [29] show existence and uniqueness for RDEs with delay; Aida [1] and Deya et al. [12] study a class of RDEs with path-dependent bounded variation terms, motivated by reflected SDEs. Although these examples may be represented as Banach space-valued RDEs, the functional coefficients involved fail to have sufficient regularity to apply the Banach space approach [3], and the results in [1, 12, 29] are specific to the class of equations considered.

In this study, we complement these results by revisiting the existence of solutions for a class of path-dependent RDEs using a weaker notion of regularity, based on the non-anticipative functional calculus introduced in [6, 5, 13]. This functional calculus is based on certain directional derivatives and does not require Fréchet differentiability, covering a larger class of examples of ODEs and SDEs with path-dependent coefficients [9].

We consider path-dependent rough differential equations (RDEs) whose coefficients are non-anticipative functionals

\begin{cases}dY(s)=b(s,Y_{s})ds+\sigma(s,Y_{s})dX(s),\\ Y_{t_{0}}=\xi_{t_{0}}.\end{cases}

(2)

where $\displaystyle\bm{X}=(X,\mathbb{X})$ is a $\displaystyle p$ -variation rough path with $\displaystyle p\in[2,3)$ and $\displaystyle b,\sigma$ are non-anticipative functionals allowing for dependence on the (stopped) path $\displaystyle Y_{s}=Y(s\wedge.)$ . We define regularity conditions on the coefficients in terms of the existence and continuity of functional derivatives in the sense of Dupire [5, 13]; these conditions are much weaker than Fréchet differentiability and only involve certain directional derivatives.

As in [14], a solution of (2) is defined as a controlled path $\displaystyle(Y,Y^{\prime})$ such that $\displaystyle Y^{\prime}(s)=\sigma(s,Y_{s})$ and

\displaystyle Y(t)=\xi_{t_{0}}\,+\,\int_{t_{0}}^{t}b(s,Y)\,{\rm d}s\,+\,\int_{t_{0}}^{t}\sigma(s,Y)d\bm{X}.

where the second integral is a rough integral. Our main result is an existence theorem (Theorem 4.4) for solutions to (2). Detailed definitions, assumptions on coefficients, and precise statements of results are presented below. The proof is based on an adaptation of the proof of Peano’s existence theorem [30] to this setting and a fixed point argument for the map

\displaystyle(Y,Y^{\prime})\mapsto(\xi_{0}+\int_{0}^{\cdot}b(t,Y)dt+\int_{0}^{\cdot}\sigma(\cdot,Y)d\bm{X},\,\sigma(\cdot,Y)).

The main difficulty is to obtain estimates on this map, given the path-dependence in the coefficients.

Outline

Section 2 provides an overview of rough path theory and controlled paths, and recalls the definition of the rough integral and its basic properties. In Section 3, we prove several results on the action of regular functionals on rough paths and controlled paths: Lemmas 3.7, 3.11 and Theorem 3.10. Finally, section 4 presents the setting of the problem and our main result on the existence of solutions to path-dependent RDEs (Theorem 4.4).

Acknowledgements. We thank Rama Cont for fruitful discussions and valuable suggestions that helped to improve the article.

2 Rough paths and rough integration

We begin by recalling some concepts from the theory of rough paths [14, 24, 25]. We will focus on the simplest of continuous paths $\displaystyle X$ with finite $\displaystyle p$ -variation, for $\displaystyle p\in[2,3)$ .

Definition 2.1 (p-variation paths).

We denote by $\displaystyle C^{p-var}([0,T],{\mathbb{R}}^{d})$ the set of continuous paths $\displaystyle X\in C([0,T];{\mathbb{R}}^{d})$ , such that

\displaystyle\|X\|_{p,[0,T]}:=\left(\sup_{\pi\in\mathcal{P}([0,T])}\sum_{t_{k}\in\pi}|X(t_{k+1})-X(t_{k})|^{p}\right)^{\frac{1}{p}}<+\infty.

where the supremum is taken over the set $\displaystyle\mathcal{P}([0,T])$ of all partitions of the interval $\displaystyle[0,T].$ Similarly for functions of two variables $\displaystyle R_{\cdot,\cdot}\colon[0,T]^{2}\to{\mathbb{R}}^{d}$ , we define

\displaystyle\|R\|_{p,[0,T]}=\left(\sup_{\pi\in\mathcal{P}([0,T])}\sum_{t_{k}\in\pi}|R_{t_{k},t_{k+1}}|^{p}\right)^{\frac{1}{p}}.

We denote by $\displaystyle V_{p}(X;t,s)$ the $\displaystyle p$ -variation of the path $\displaystyle{X\in}C^{p-var}([0,T],{\mathbb{R}}^{d})$ on the interval $\displaystyle[t,s]$ :

\displaystyle V_{p}(X;t,s):=\|X\|^{p}_{p,[t,s]}.

One obviously has

\displaystyle\displaystyle|X(s)-X(t)|^{p}\leq V_{p}(X;t,s).

(3)

and $\displaystyle V_{p}$ is superadditive:

\displaystyle\displaystyle V_{p}(X;t,u)+V_{p}(X;u,s)\leq V_{p}(X;t,s),\,\forall t\leq u\leq s.

(4)

As a consequence the function $\displaystyle V_{p}(X;0,\cdot)$ is increasing and continuous.

The above motivates the notion of a superadditive function on the set of the intervals:

Definition 2.2 (Superadditive interval functions).

A map

\displaystyle\omega\colon\{[t,s]\colon 0\leq t\leq s\leq T\}\to{\mathbb{R}}_{+},

with $\displaystyle\omega[t,t]=0,\forall t\in[0,T]$ is called superadditive if for all $\displaystyle t\leq u\leq s$ in $\displaystyle[0,T]$

\displaystyle\omega([t,u])+\omega([u,s])\leq\omega([t,s]).

A basic example of a superadditive function is $\displaystyle\omega([t,s])=\|X\|^{p}_{p,[t,s]}$ for any $\displaystyle X\in C^{p-var}([0,T],{\mathbb{R}}^{d}).$ A very useful fact about superadditive functions, which will be used in the paper, is that for $\displaystyle\omega_{1},\omega_{2}$ superadditive, so are $\displaystyle\omega_{1}^{r}$ and $\displaystyle\omega_{1}^{\theta}\omega_{2}^{1-\theta},$ for all $\displaystyle r\geq 1,\,\theta\in(0,1).$

The notion of superadditive functions allows us to formulate an alternative definition of the space of $\displaystyle p$ -variation paths:

Proposition.

$\displaystyle X\in C^{p-var}([0,T],{\mathbb{R}}^{d})$ if and only if there exists a superadditive function $\displaystyle\omega$ such that

\displaystyle|X(s)-X(t)|\leq\omega([t,s])^{\frac{1}{p}},\,\forall t\leq s\in[0,T].

The above definition is closer to the definition of Hőlder continuous paths, and corresponds to the latter in the case $\displaystyle\omega([t,s])=|s-t|.$

We now define the space of rough paths (see e.g. [15][Sec. 1.2.4]):

Definition 2.3 (Space of $\displaystyle p$ -rough paths).

For $\displaystyle p\in[2,3)$ we define the space $\displaystyle\mathcal{C}^{p-var}([0,T],{\mathbb{R}}^{d})$ of continuous $\displaystyle p$ -rough paths as the set of pairs $\displaystyle\mathbf{X}:=(X,\mathbb{X})$ of $\displaystyle{\mathbb{R}}^{d}\times{\mathbb{R}}^{d\times d}$ -valued paths such that

(i)

$\displaystyle\mathbb{X}_{t,u}-\mathbb{X}_{t,s}-\mathbb{X}_{s,u}=X_{t,s}\otimes X_{s,u},\quad\forall t,s,u\in[0,T].$
(ii)

$\displaystyle\|X\|_{p,[0,T]}+\|\mathbb{X}\|_{{\frac{p}{2}},[0,T]}<+\infty.$

As shown by Lyons and Victoir [27], any Hőlder continuous path $\displaystyle X\in C^{\alpha}([0,T],{\mathbb{R}}^{d})$ can be associated with a rough path, but this association is far from canonical and in fact for $\displaystyle\alpha<1/2$ there are infinitely many such rough paths.

Now, we define the analog of weakly controlled paths [18, Def.1]:

Definition 2.4 (Controlled paths).

Let $\displaystyle p>q>1$ and $\displaystyle X\in C^{p-var}([0,T],{\mathbb{R}}^{d})$ . A pair $\displaystyle(Y,Y^{\prime})\in C^{p-var}([0,T],{\mathbb{R}}^{k})\times C^{p-var}([0,T],{\mathbb{R}}^{d\times k})$ of finite $\displaystyle p$ -variation a $\displaystyle(p,q)$ -controlled path with respect to $\displaystyle X$ if

\displaystyle R^{Y}_{t,s}{:=}Y_{t,s}-Y^{\prime}_{t}X_{t,s}

has a finite $\displaystyle q$ -variation. We denote by $\displaystyle\mathcal{D}^{p,q}_{{}_{X}}([0,T],{\mathbb{R}}^{k})$ the set of all $\displaystyle(p,q)$ -controlled paths with respect to $\displaystyle X$ .

The path $\displaystyle X$ is called the control or reference path. Typical examples of controlled paths arise from smooth functions $\displaystyle f:{\mathbb{R}}^{d}\to{\mathbb{R}}$ of $\displaystyle X$ :

\displaystyle Y(t)=f(X(t)),\qquad Y^{\prime}(t)=\nabla f(X(t)).

$\displaystyle R^{Y}(t,s)$ is then given by the remainder in a first order Taylor expansion. By analogy any $\displaystyle Y^{\prime}$ satisfying Def. 2.3 is called a ‘Gubinelli derivative’ for $\displaystyle Y$ . $\displaystyle R^{Y}(s,t)$ plays the role of a remainder in a first order expansion of $\displaystyle Y$ , and $\displaystyle Y^{\prime}(s)$ plays the role of a ‘derivative’ of $\displaystyle Y$ with respect to $\displaystyle X$ . The requirement is that the remainder $\displaystyle R^{Y}$ is smoother than $\displaystyle Y$ itself: we go from $\displaystyle p$ to $\displaystyle q<p$ in the finite variation regularity scale. The above definition corresponds to weakly-controlled paths in [18], for our convenience, throughout the paper we will use the name “ $\displaystyle(p,q)-$ controlled paths” or “controlled paths” if the exponents $\displaystyle p,q$ are apparent from the context.

One can check that $\displaystyle\mathscr{D}^{p,q}_{X}([0,T],{\mathbb{R}}^{d})$ is a Banach space under the norm

\displaystyle\|(Y,Y^{\prime})\|_{\mathscr{D}^{p,q}_{X}}=|Y_{0}|+|Y^{\prime}_{0}|+\underbrace{\|Y^{\prime}\|_{p,[0,T]}\,+\,\|R^{Y}\|_{q,[0,T]}}_{:=\|Y,Y^{\prime}\|_{p,q,X}}.

The next theorem establishes that controlled paths are proper integrands for rough integration:

Theorem 2.5 (c.f. [14][Theorem 4.10], [18][Theorem 1]).

Let $\displaystyle p\in[2,3),\,q\geq p/2$ , $\displaystyle\mathbf{X}=(X,\mathbb{X})\in\mathcal{C}^{p-var}([0,T],{\mathbb{R}}^{d})$ . Let also $\displaystyle p^{-1}+q^{-1}>1$ and $\displaystyle(Y,Y^{\prime})\in\mathcal{D}^{p,q}_{{}_{X}}([0,T],{\mathbb{R}}^{k})$ be a controlled path. Define the compensated Riemann sums

\displaystyle S(\pi):=\sum_{[t,s]\in\pi}Y_{t}X_{t,s}+Y^{\prime}_{t}\mathbb{X}_{t,s}.

Then the limit

\displaystyle\int_{0}^{T}Yd\mathbf{X}:=\lim_{|\pi|\to 0}S(\pi)

exist and satisfies the estimate

\displaystyle\displaystyle\left|\int_{t}^{s}YdX-Y_{t}X_{t,s}-Y^{\prime}_{t}{\mathbb{X}}_{t,s}\right|\leq C\left(\|{R}^{Y}\|_{q,[t,s]}\|X\|_{p,[t,s]}+\|Y^{\prime}\|_{p,[t,s]}\|{\mathbb{X}}\|_{p/2,[t,s]}\right).

Moreover, the map

\displaystyle(Y,Y^{\prime})\mapsto(Z,Z^{\prime}):=\left(\int_{0}^{\cdot}Yd\mathbf{X},\,Y\right),

\displaystyle\mathscr{D}^{p,q}_{X}([0,T],\mathcal{L}({\mathbb{R}}^{d},{\mathbb{R}}^{k}{))\rightarrow}\mathscr{D}^{p,q}_{X}([0,T],\mathcal{L}({\mathbb{R}}^{d},{\mathbb{R}}^{k}{))}

is continuous and

\displaystyle\displaystyle\|Z,Z^{\prime}\|_{p,q,X}\leq\|Y\|_{p}+\|Y^{\prime}\|_{\infty}\|\mathbb{X}\|_{p/2}+C\left(\|X\|_{p}\|R^{Y}\|_{q}+\|Y^{\prime}\|_{p}\|\mathbb{X}\|_{p/2}\right).

In the theorem $\displaystyle Y^{\prime}_{t}\mathbb{X}_{t,s}$ is interpreted via the natural inclusion:

	$\displaystyle\displaystyle\mathcal{L}({\mathbb{R}}^{d},\mathcal{L}({\mathbb{R}}^{d},{\mathbb{R}}^{k}))\hookrightarrow\mathcal{L}({\mathbb{R}}^{d}\otimes{\mathbb{R}}^{d},{\mathbb{R}}^{k})$
	$\displaystyle\displaystyle Y^{\prime}(v\otimes w):=\underbrace{Y^{\prime}(v)}_{\in\mathcal{L}({\mathbb{R}}^{d},{\mathbb{R}}^{k})}(w),\,v,w\in{\mathbb{R}}^{d}$

(in coordinates as $\displaystyle(Y^{\prime}_{t}\mathbb{X}_{t,s})_{l}:=\sum_{i,j=1}^{d}(Y^{\prime})^{lj}_{i}\mathbb{X}^{ij},\,l=1,\ldots,k$ ).

Proof.

The proof of the first estimate is similar to [15, Theorem 31] so we omit it here. The second estimate of the theorem follows from the first one and the triangle inequality by noting that $\displaystyle{R^{Z}(t,s)}=\int_{t}^{s}Yd\textbf{X}-Y_{t}X_{t,s}$ , and

	$\displaystyle\displaystyle\|{R^{Z}(t,s)}\|\leq\Big{\|}\int_{t}^{s}Y\,d\textbf{X}-Y_{t}X_{t,s}-Y^{\prime}_{t}\mathbb{X}_{t,s}\Big{\|}\,+\,\|Y^{\prime}_{t}\mathbb{X}_{t,s}\|$
	$\displaystyle\displaystyle\leq C\left(\\|{R}^{Y}\\|_{q,[t,s]}\\|X\\|_{p,[t,s]}+\\|Y^{\prime}\\|_{p,[t,s]}\\|\tilde{\mathbb{X}}\\|_{p/2,[t,s]}\right)\ +\\|Y^{\prime}\\|_{\infty}\\|\mathbb{X}\\|_{p/2,[t,s]}.$		(5)

hence, using that the $\displaystyle q$ -th power of the right-hand side is superadditive, we obtain

\displaystyle\displaystyle\|R^{Z}\|_{q,[0,T]}\leq C\left(\|{R}^{Y}\|_{q,[0,T]}\|X\|_{p,[0,T]}+\|Y^{\prime}\|_{p,[0,T]}\|\tilde{\mathbb{X}}\|_{p/2,[0,T]}\right)+\|Y^{\prime}\|_{\infty}\|\mathbb{X}\|_{p/2,[0,T]}.

Consequently,

	$\displaystyle\displaystyle\\|Z,Z^{\prime}\\|_{p,q,X}=\\|Y\\|_{p}+\\|R^{Z}\\|_{q}\leq\\|Y\\|_{p}+\\|Y^{\prime}\\|_{\infty}\\|\mathbb{X}\\|_{p/2}$
	$\displaystyle\displaystyle+C\left(\\|X\\|_{p}\\|R^{Y}\\|_{q}+\\|Y^{\prime}\\|_{p}\\|\mathbb{X}\\|_{p/2}\right).$

∎

3 Non-anticipative functionals of rough paths

In this section, we study the behaviour of rough paths under the actions of regular non-anticipative functionals. We construct a rough integral for integrands given by sufficiently regular non-anticipative functionals.

3.1 Non-anticipative functionals

Let us recall briefly the definition of non-anticipative functionals and their derivatives [5]. A functional $\displaystyle F:[0,T]\times D([0,T],\mathbb{R}^{d})\rightarrow\mathbb{R}$ on the space $\displaystyle D([0,T],\mathbb{R}^{d})$ of càdlàg paths is called non-anticipative if it satisfies a causality property:

\displaystyle\displaystyle F(t,{x})=F(t,{x}_{t})\qquad\forall{x}\in\Omega,

(6)

where $\displaystyle{x}_{t}$ represents the path $\displaystyle{x}$ stopped at time $\displaystyle t$ . It turns out that it is convenient to define these non-anticipative functionals on the space of stopped paths, where we define a stopped path as an equivalence class in $\displaystyle[0,T]\times D([0,T],\mathbb{R}^{d})$ with respect to the following equivalence relation:

\displaystyle\displaystyle(t,{x})\sim(t^{\prime},{x}^{\prime})\Longleftrightarrow(t=t^{\prime}\,\,\,\mathrm{and}\,\,\,{x}(t\wedge\cdot)={x}^{\prime}(t^{\prime}\wedge\cdot)).

It is possible to endow this space with a metric structure, via the following distance function:

	$\displaystyle\displaystyle d_{\infty}((t,{x}),(t^{\prime},{x}^{\prime})):=\sup_{u\in[0,T]}\|{x}(u\wedge t)-{x}^{\prime}(u\wedge t^{\prime})\|+\|t-t^{\prime}\|$
	$\displaystyle\displaystyle=\|\|{x}-{x}^{\prime}\|\|_{\infty}+\|t-t^{\prime}\|.$

The space $\displaystyle(\Lambda_{T}^{d},d_{\infty})$ is then a complete metric space. Now, every map satisfying condition (6) can be viewed as a functional on the space of stopped paths.

Definition 3.1 (Non-anticipative functional).

A non-anticipative functional is a measurable map $\displaystyle F:(\Lambda_{T}^{d},d_{\infty})\rightarrow\mathbb{R}^{k}$ . We denote $\displaystyle\mathbb{C}^{0,0}(\Lambda_{T}^{d})$ the set of continuous maps $\displaystyle F:(\Lambda_{T}^{d},d_{\infty})\rightarrow\mathbb{R}^{k}$ .

$\displaystyle F\in\mathbb{C}^{0,0}(\Lambda_{T}^{d})$ implies joint continuity in $\displaystyle(t,{x})$ . We will also need some weaker notions of continuity [5].

Definition 3.2.

A non-anticipative functional $\displaystyle F$ is said to be:

•

continuous at fixed times if for any $\displaystyle t\in[0,T]$ , $\displaystyle F(t,\cdot)$ is continuous w.r.t. the uniform norm $\displaystyle||\cdot||_{\infty}$ , i.e., $\displaystyle\forall{x}\in D([0,T],\mathbb{R}^{d})$ , $\displaystyle\forall\epsilon>0$ , $\displaystyle\exists\nu>0$ such that $\displaystyle\forall{x}^{\prime}\in D([0,T],\mathbb{R}^{d})$ :

$\displaystyle\displaystyle||{x}_{t}-{x}^{\prime}_{t}||\infty<\nu\Rightarrow|F(t,{x})-F(t,{x}^{\prime})|<\epsilon,$

•

left-continuous if $\displaystyle\forall(t,{x})\in\Lambda_{T}^{d}$ , $\displaystyle\forall\epsilon>0$ , $\displaystyle\exists\nu>0$ such that $\displaystyle\forall(t^{\prime},{x}^{\prime})\in\Lambda_{T}^{d}$ :

\displaystyle\displaystyle t^{\prime}<t\,\,\,\mathrm{and}\,\,\,d_{\infty}((t,{x}),(t^{\prime},{x}^{\prime}))<\nu\Rightarrow|F(t,{x})-F(t^{\prime},{x}^{\prime})|<\epsilon.

We denote the set of left-continuous functionals by $\displaystyle\mathbb{C}^{0,0}_{l}(\Lambda_{T}^{d})$ .

We will also need a notion of local boundedness for functionals.

Definition 3.3.

A functional $\displaystyle F$ is called boundedness-preserving if for every compact subset $\displaystyle K$ of $\displaystyle\mathbb{R}^{d}$ , $\displaystyle\forall t_{0}\in[0,T]$ , $\displaystyle\exists C(K,t_{0})>0$ such that:

\displaystyle\displaystyle\forall t\in[0,t_{0}],\,\,\,\forall(t,{x})\in\Lambda_{T}^{d}:\,\,\,{x}([0,t])\subset K\Rightarrow|F(t,{x})|<C(K,t_{0}).

We denote the set of boundedness preserving functionals by $\displaystyle\mathbb{B}(\Lambda_{T}^{d})$ .

We now recall some definition of differentiability for non-anticipative functionals. Given $\displaystyle e\in\mathbb{R}^{d}$ and $\displaystyle{x}\in D([0,T],\mathbb{R}^{d})$ , we define the vertical perturbation $\displaystyle{x}_{t}^{e}$ of $\displaystyle(t,{x})$ as the càdlàg path obtained by adding a jump discontinuity to the path $\displaystyle{x}$ at time $\displaystyle t$ and of size $\displaystyle e$ , that is:

\displaystyle\displaystyle{x}_{t}^{e}:={x}_{t}+e\mathbbm{1}_{[t,T]}.

Definition 3.4.

A non-anticipative functional $\displaystyle F$ is said to be:

•

horizontally differentiable at $\displaystyle(t,{x})\in\Lambda_{T}^{d}$ if:

$\displaystyle\displaystyle\mathcal{D}F(t,{x})=\lim_{h\downarrow 0}\frac{F(t+h{x}_{t})-F(t,{x})}{h}$ (7)

exists. If $\displaystyle\mathcal{D}F$ exists for all $\displaystyle(t,{x})\in\Lambda_{T}^{d}$ , then $\displaystyle\mathcal{D}F$ defines a new non-anticipative functional, called the horizontal derivative of $\displaystyle F$ .

•

vertically differentiable at $\displaystyle(t,{x})\in\Lambda_{T}^{d}$ if the map:

	$\displaystyle\displaystyle g_{{}_{(t,{x})}}:\mathbb{R}^{d}$	$\displaystyle\displaystyle\rightarrow\mathbb{R}$
	$\displaystyle\displaystyle e$	$\displaystyle\displaystyle\mapsto F(t,{x}_{t}+e\ 1_{[t,T]})$

is differentiable at $\displaystyle 0$ . In that case, the gradient at $\displaystyle 0$ is called the Dupire derivative (or vertical derivative) of $\displaystyle F$ at $\displaystyle(t,{x})$ :

\displaystyle\displaystyle\nabla_{x}F(t,{x})=\nabla g_{{}_{(t,{x})}}(0)\in\mathbb{R}^{d},

(8)

that is, we have $\displaystyle\nabla_{x}F(t,{x})=(\partial_{i}F(t,{x}),i=1,...,d)$ with

\displaystyle\displaystyle\partial_{i}F(t,{x})=\lim_{h\rightarrow 0}\frac{F(t,{x}_{t}+he_{i}\mathbbm{1}_{[t,T]})-F(t,{x}_{t})}{h},

where $\displaystyle(e_{i},i=1,...,d)$ is the canonical basis of $\displaystyle\mathbb{R}^{d}$ . If $\displaystyle\nabla_{x}F$ exists for all $\displaystyle(t,{x})\in\Lambda_{T}^{d}$ , then $\displaystyle\nabla_{x}F:\Lambda_{T}^{d}\rightarrow\mathbb{R}^{d}$ defines a non-anticipative functional called the vertical derivative of $\displaystyle F$ .

Note that, since the objects that we obtain when computing these derivatives are still non-anticipative functionals, we can reiterate these operations and introduce higher order derivatives, such as $\displaystyle\nabla_{x}^{2}F$ . This leads to the definition of the following class of smooth functionals.

Definition 3.5.

We define $\displaystyle\mathbb{C}^{1,k}_{b}(\Lambda_{T}^{d})$ as the set of non-anticipative functionals $\displaystyle F:(\Lambda_{T}^{d},d_{\infty})\rightarrow\mathbb{R}$ which are:

•

horizontally differentiable, with $\displaystyle\mathcal{D}F$ continuous at fixed times;
•

$\displaystyle k$ times vertically differentiable, with $\displaystyle\nabla_{x}^{j}F\in\mathbb{C}_{l}^{0,0}(\Lambda_{T}^{d})$ for $\displaystyle j=0,\ldots,k$ ;
•

$\displaystyle\mathcal{D}F,\nabla_{x}F,\ldots,\nabla_{x}^{k}F\in\mathbb{B}(\Lambda_{T}^{d})$ .

Throughout the section, we will work with functionals satisfying the following assumption of Lipschitz continuity in the metric $\displaystyle d_{\infty}:$

Assumption 3.1.

$\displaystyle\exists K>0,\quad\forall(t,X),\,(t^{\prime},X^{\prime})\in\Lambda^{d}_{T},$

\displaystyle|F(t,X)-F(t^{\prime},X^{\prime})|\leq Kd_{\infty}((t,X),\,(t^{\prime},X^{\prime})).

The space of such functionals is denoted by $\displaystyle Lip(\Lambda_{T}^{d},d_{\infty}).$

We note that the above property implies the following Lipschitz continuity property

Assumption 3.2 (Uniformly Lipschitz continuity).

$\displaystyle\exists K>0,\quad\forall X,\,X^{\prime}\in D([0,T],{\mathbb{R}}^{d}),$

\displaystyle|F(t,X)-F(t,X^{\prime})|\leq K\|X_{t}-X^{\prime}_{t}\|_{\infty}.

We denote the space of such functionals by $\displaystyle Lip(\Lambda_{T}^{d},\|\cdot\|_{\infty}).$

Assumption 3.3 (Horizontal Lipschitz continuity).

$\displaystyle\exists K>0,\quad\forall X\in D([0,T],{\mathbb{R}}^{d}),$

\displaystyle|F(t,X)-F(t^{\prime},X)|\leq K|t^{\prime}-t|.

We denote the space of such functionals by $\displaystyle hLip(\Lambda_{T}^{d}).$

It is not hard to see that $\displaystyle Lip(\Lambda_{T}^{d},d_{\infty})=Lip(\Lambda_{T}^{d},\|\cdot\|_{\infty})\cap hLip(\Lambda_{T}^{d})$

3.2 Actions of functionals on rough paths

We are now ready to study actions of regular non-anticipative functionals on rough paths and controlled paths.

The following lemma is a particular case of [2][Lemma 5.11], which allows to approximate paths with finite variation by piece-wise affine paths.

Lemma 3.6.

For any path $\displaystyle X\in C^{p-var}([0,T],{\mathbb{R}}^{d})$ and an integer $\displaystyle N>1$ and interval $\displaystyle[t,s]$ there exists $\displaystyle X^{N}\in C([0,s];{\mathbb{R}}^{d})$ such that it is a piece-wise linear on $\displaystyle[t,s]$

•

$\displaystyle X^{N}$ coincides with $\displaystyle X$ on $\displaystyle[0,t]:$

$\displaystyle X^{N}_{t}=X_{t},$
•

$\displaystyle X\to X^{N}$ is a linear map and

$\displaystyle\|X^{N}\|_{\infty}\leq\|X\|_{\infty},\,\|X^{N}\|_{p,[t,s]}\leq\|X\|_{p,[t,s]},$
•

$\displaystyle X^{N}$ approximates $\displaystyle X$ :

$\displaystyle\|X-X^{N}\|_{\infty}\leq CN^{-\nu}\|X\|_{p,[t,s]},$
•

$\displaystyle X^{N}$ has a bounded variation on $\displaystyle[t,s]$ with the variation

$\displaystyle V_{1}(X^{N},t,s):=\int_{t}^{s}|dX^{N}|\leq CN^{1-\nu}\|X\|_{p,[t,s]},$

where $\displaystyle\nu:=p^{-1}.$

Next, we recall a corollary of [2][Theorem 5.12], which provides a connection between regular functionals and controlled paths.

Lemma 3.7.

Let $\displaystyle{p>2},\,X\in C^{p-var}([0,T],\mathbb{R}^{d})$ and $\displaystyle F\in\mathbb{C}^{0,1}_{b}(\Lambda^{d}_{T},\mathbb{R}^{n})$ with $\displaystyle F$ and $\displaystyle\nabla_{{x}}F$ in $\displaystyle Lip(\Lambda^{d}_{T},d_{\infty})$ . Define

\mathcal{R}^{F}_{t,s}(X):=F(s,X_{s})-F(t,X_{t})-\nabla_{{x}}F(t,X_{t})(X(s)-X(t)).

(9)

Then there exists a constant $\displaystyle C_{F,T}$ increasing in $\displaystyle T$ , which depends on the regularity properties of $\displaystyle F$ and its derivatives locally in a neighbourhood of $\displaystyle X$ , such that $\displaystyle{\mathcal{R}}^{F}(X)$ has bounded $\displaystyle q_{p}:=\frac{p^{2}}{p+1}$ -variation and

\displaystyle\|{\mathcal{R}}^{F}(X)\|_{q_{p},[t,s]}\leq C_{p,F,T}\left(|s-t|+\|X\|_{p,[t,s]}^{1+p^{-1}}\right).

Thus the pair $\displaystyle(F(\cdot,X),\nabla_{{x}}F(\cdot,X))$ is a controlled path:

\displaystyle(F(\cdot,X),\nabla_{{x}}F(\cdot,X))\in\mathcal{D}^{p,q_{p}}_{{}_{X}}([0,T],{{\mathbb{R}}^{n}}),\qquad q_{p}:=\frac{p^{2}}{p+1}.

We omit the proof of this lemma as it is based on the same idea as the proof of the next result.

For our purposes, we would like to have a stability result for the estimate in the previous theorem in terms of the underlying path $\displaystyle Y.$ The following result allows us to control the error term $\displaystyle\mathcal{R}^{F}(Y)_{s,t}$ in $\displaystyle Y$ and will be useful in the proof of the existence of solutions to path-dependent RDEs.

Lemma 3.8 (Continuity of Control).

Let $\displaystyle Y_{1},Y_{2}\in C^{p-var}([0,T],\mathbb{R}^{d})$ for some $\displaystyle p\in[2,3)$ . Let $\displaystyle F\colon\Lambda^{d}_{T}\to V$ be a non-anticipative functional with values in a finite dimensional real vector space $\displaystyle V$ . Assume $\displaystyle F\in\mathbb{C}^{1,2}_{b}(\Lambda^{d}_{T}),\,\nabla F\in\mathbb{C}^{1,1}_{b}(\Lambda^{d}_{T})$ and $\displaystyle F,\mathcal{D}F,\,\mathcal{D}\nabla F,\,\nabla^{2}_{X}F\in Lip(\Lambda^{d}_{T},\|\cdot\|_{\infty})$ . Furthermore, if $\displaystyle R,M>0$ are such that

\displaystyle\|Y_{1}\|^{p}_{p,[0,T]}\,+\,\|Y_{2}\|^{p}_{p,[0,T]}\,+\,T\leq R.

and

\displaystyle\|Y_{1}-Y_{2}\|_{\infty,[0,T]}+\|Y_{1}-Y_{2}\|_{p,[0,T]}\leq M,

then for all $\displaystyle t\leq s\in[0,T]$

\displaystyle\|\mathcal{R}^{F}(Y_{1})-\mathcal{R}^{F}(Y_{2})\|_{q_{{}_{p}},[t,s]}\leq C_{F,M,R}\left(\|Y_{1}-Y_{2}\|^{\nu}_{\infty,[0,s]}+\|Y_{1}-Y_{2}\|_{p,[t,s]}^{\nu}\right),

where $\displaystyle q_{p}:=\frac{p^{2}}{p+1},\,\nu:=p^{-1}$ .

Proof.

We will prove only the case when the values of $\displaystyle F$ are scalar, i.e. $\displaystyle V={\mathbb{R}}$ , for the general case it is enough to use the result for each coordinate of $\displaystyle F$ . Let $\displaystyle\omega$ be a superadditive map on intervals of $\displaystyle[0,T]$ , given by

\displaystyle\omega([u,v]):=\|Y_{1}\|^{p}_{p,[u,v]}\,+\,\|Y_{2}\|^{p}_{p,[u,v]}\,+\,|u-v|.

We start by recalling the following result:

Lemma (see [5], Proposition 5.26).

Assume $\displaystyle G\in C^{1,1}_{b}(\mathcal{W}^{d}_{T})$ and $\displaystyle\lambda$ is a continuous path with finite variation on $\displaystyle[t,s]$ , then

G(s,\lambda_{s})-G(t,\lambda_{t})=\int_{t}^{s}\mathcal{D}G(u,\lambda_{u})du+\int_{t}^{s}\nabla G(u,\lambda_{u})d\lambda(u)

(*)

where the second integration is in the Riemann-Stieltjes sense.

Let us fix a Lipschitz continuous path $\displaystyle Y;[0,T]\to{\mathbb{R}}^{d}$ , using the above lemma repeatedly for $\displaystyle G=\nabla^{j}F,\,j=0,1,2$ , we will obtain a expression of $\displaystyle\mathcal{R}^{F}_{t,s}(Y)$ in terms of the derivatives $\displaystyle\mathcal{D}^{i}\nabla^{j}F$ . For the sake of convenience we denote by $\displaystyle\partial_{i}F$ and $\displaystyle Y^{i}$ respectively the $\displaystyle i$ -th coordinates of $\displaystyle\nabla F$ and $\displaystyle Y.$ We also use Einstein’s convention of summation in repeated indexes. Using (*) for $\displaystyle G=F$ , we have

\mathcal{R}^{F}_{t,s}(Y)=\int_{t}^{s}\mathcal{D}F(u,Y_{u})du+\int_{t}^{s}\left(\partial_{i}F(u,Y_{u})-\partial_{i}F(t,Y_{t})\right)dY^{i}(u)

(10)

For the second term on the right-hand side of the above identity we use the Lemma (*) with $\displaystyle G=\partial^{2}_{i}F$ and then Fubini’s theorem to get

	$\displaystyle\displaystyle\int_{t}^{s}\left(\partial_{i}F(u,Y_{u})-\partial_{i}F(t,Y_{t})\right)dY^{i}(u)=\int_{t}^{s}\int_{t}^{u}\mathcal{D}\partial_{i}F(r,Y_{r})drdY^{i}(u)$
	$\displaystyle\displaystyle+\int_{t}^{s}\int_{t}^{u}\partial^{2}_{ij}F(r,Y_{r})dY^{j}(r)dY^{i}(u)$
	$\displaystyle\displaystyle=\int_{t}^{s}\mathcal{D}\partial_{i}F(r,Y_{r})(Y^{i}(s)-Y^{i}(r))dr$
	$\displaystyle\displaystyle+\int_{t}^{s}\partial^{2}_{ij}F(r,Y_{r})(Y^{i}(s)-Y^{i}(r))\dot{Y}^{j}(r)dr.$		(11)

Combining 10, 3.2 we arrive to the formula

\displaystyle\displaystyle\mathcal{R}^{F}_{t,s}(Y)=\underbrace{\int_{t}^{s}\mathcal{D}F(u,Y_{u})du}_{:=I_{1}(Y)}+\underbrace{\int_{t}^{s}\mathcal{D}\partial_{i}F(r,Y_{r})Y^{i}_{r,s}dr}_{:=I_{2}(Y)}+\underbrace{\int_{t}^{s}\partial^{2}_{ij}F(r,Y_{r})Y^{i}_{r,s}\dot{Y}^{j}(r)dr}_{:=I_{3}(Y)}.

(12)

Let $\displaystyle Y^{N}$ be the piece-wise linear (affine) approximation of $\displaystyle Y$ given by Lemma 3.6. Using the estimates of that lemma, Lipschitz continuity of $\displaystyle\mathcal{D}F,\mathcal{D}\nabla F,\nabla^{2}F$ and the following consequences of triangle inequality

\displaystyle|a_{1}b_{1}-a_{2}b_{2}|\leq|a_{1}-a_{2}||b_{1}|+|a_{2}||b_{1}-b_{2}|,

\displaystyle|a_{1}b_{1}c_{1}-a_{2}b_{2}c_{2}|\leq|a_{1}-a_{2}||b_{1}||c_{1}|+|a_{2}||b_{1}-b_{2}||c_{1}|+|a_{2}||b_{2}||c_{1}-c_{2}|,

we obtain, for each term of representation (12), we have

\displaystyle|I_{1}(Y_{1}^{N})-I_{1}(Y^{N}_{2})|\leq C_{F}\|Y_{1}-Y_{2}\|_{\infty}|s-t|,

	$\displaystyle\displaystyle\|I_{2}(Y_{1}^{N})-I_{2}(Y^{N}_{2})\|\leq C_{F}\\|Y_{1}-Y_{2}\\|_{\infty}\\|Y_{1}\\|_{p,[t,s]}\|s-t\|$
	$\displaystyle\displaystyle+\\|\mathcal{D}\nabla F(\cdot,Y^{N}_{2})\\|_{\infty}\\|Y_{1}-Y_{2}\\|_{p,[t,s]}\|s-t\|$
	$\displaystyle\displaystyle\leq C_{F}\left(\\|Y_{1}-Y_{2}\\|_{\infty}\omega([t,s])^{\nu}\,+\,\\|Y_{1}-Y_{2}\\|_{p,[t,s]}\right)\omega([t,s]),$

and

	$\displaystyle\displaystyle\|I_{3}(Y_{1}^{N})-I_{3}(Y^{N}_{2})\|\leq C_{F}\\|Y_{1}-Y_{2}\\|_{\infty}N^{1-\nu}\\|Y_{1}\\|^{2}_{p,[t,s]}$
	$\displaystyle\displaystyle+\\|\nabla^{2}F(\cdot,Y^{N}_{2})\\|_{\infty}\\|Y_{1}-Y_{2}\\|_{p,[t,s]}(N^{1-\nu}\\|Y_{1}\\|_{p,[t,s]})$
	$\displaystyle\displaystyle+\\|\nabla^{2}F(\cdot,Y^{N}_{2})\\|_{\infty}\\|Y_{2}\\|_{p,[t,s]}(N^{1-\nu}\\|Y_{1}-Y_{2}\\|_{p,[t,s]})$
	$\displaystyle\displaystyle\leq N^{1-\nu}C_{F,R}\left(\\|Y_{1}-Y_{2}\\|_{\infty}\omega([t,s])^{\nu}\,+\,\\|Y_{1}-Y_{2}\\|_{p,[t,s]}\right)\omega([t,s])^{\nu}.$

From these

	$\displaystyle\displaystyle\|\mathcal{R}^{F}_{t,s}(Y^{N}_{1})-\mathcal{R}^{F}_{t,s}(Y^{N}_{2})\|\leq C_{F,R}(\\|Y_{1}-Y_{2}\\|_{\infty}+\\|Y_{1}-Y_{2}\\|_{p,[t,s]})\omega([t,s])$
	$\displaystyle\displaystyle+\,C_{F}N^{1-\nu}\left(\\|Y_{1}-Y_{2}\\|_{\infty}\omega([t,s])^{\nu}\,+\,\\|Y_{1}-Y_{2}\\|_{p,[t,s]}\right)\omega([t,s])^{\nu}$
	$\displaystyle\displaystyle\leq C_{F,T,R}(\\|Y_{1}-Y_{2}\\|_{\infty}\omega([t,s])^{\nu}+\\|Y_{1}-Y_{2}\\|_{p,[t,s]})$
	$\displaystyle\displaystyle\times(\omega([t,s])^{1-\nu}+N^{1-\nu}\omega([t,s])^{\nu}).$		(13)

On the other hand since $\displaystyle(Y^{N})_{t}=Y_{t}$ and $\displaystyle Y^{N}(s)=Y(s)$ we can replace $\displaystyle\mathcal{R}^{F}_{t,s}(Y^{N}_{1})$ and $\displaystyle\mathcal{R}^{F}_{t,s}(Y^{N}_{2})$ respectively with $\displaystyle\mathcal{R}^{F}_{t,s}(Y_{1})$ and $\displaystyle\mathcal{R}^{F}_{t,s}(Y_{2})$ with an error

	$\displaystyle\displaystyle\|\mathcal{R}^{F}_{t,s}(Y^{N}_{i})-\mathcal{R}^{F}_{t,s}(Y_{i})\|=\|F(s,Y^{N}_{i})-F(s,Y_{i})\|$
	$\displaystyle\displaystyle\leq C_{F}\\|Y^{N}_{i}-Y_{i}\\|_{\infty}\leq C_{F}\\|Y_{i}\\|_{p,[t,s]}N^{-\nu}\leq C_{F}N^{-\nu}\omega([t,s])^{\nu},\,i=1,\,2.$		(14)

Let

\displaystyle d_{\nu}(Y_{1},Y_{2}):=\|Y_{1}-Y_{2}\|_{\infty}\omega([t,s])^{\nu}+\|Y_{1}-Y_{2}\|_{p,[t,s]},

from (3.2) and (3.2) and triangle inequality

	$\displaystyle\displaystyle\|\mathcal{R}^{F}_{t,s}(Y_{1})-\mathcal{R}^{F}_{t,s}(Y_{2})\|\leq C_{F,R}\big{(}d_{\nu}(Y_{1},Y_{2})\omega([t,s])^{1-\nu}$
	$\displaystyle\displaystyle+N^{1-\nu}d_{\nu}(Y_{1},Y_{2})\omega([t,s])^{2\nu}+N^{-\nu}\omega([t,s])^{\nu}\big{)}.$

To optimize the above bound, we choose $\displaystyle N$ so that

\displaystyle N^{1-\nu}d_{\nu}(Y_{1},Y_{2})\omega([t,s])^{\nu}\approx N^{-\nu}\omega([t,s])^{\nu}

i.e. $\displaystyle N\approx d_{\nu}(Y_{1},Y_{2})^{-1}.$ Hence

	$\displaystyle\displaystyle\|\mathcal{R}^{F}_{t,s}(Y_{1})-\mathcal{R}^{F}_{t,s}(Y_{2})\|\leq C_{F,R}\,\left(d_{\nu}(Y_{1},Y_{2})\omega([t,s])^{1-\nu}+d_{\nu}(Y_{1},Y_{2})^{\nu}\omega([t,s])^{\nu}\right)$
	$\displaystyle\displaystyle\leq C_{F,M,R}\left(\\|Y_{1}-Y_{2}\\|^{\nu}_{\infty}\omega([t,s])^{\nu+\nu^{2}}+\\|Y_{1}-Y_{2}\\|_{p,[t,s]}^{\nu}\omega([t,s])^{\nu}\right).$

Using the inequality $\displaystyle(|a|+|b|)^{q}\leq 2^{q}(|a|^{q}+|b|^{q}),\forall q>0$ , we have

\displaystyle\displaystyle|\mathcal{R}^{F}_{t,s}(Y_{1})-\mathcal{R}^{F}_{t,s}(Y_{2})|^{q_{p}}\leq C_{F,M,R}\left(\|Y_{1}-Y_{2}\|^{p/(p+1)}_{\infty}\omega([t,s])+\|Y_{1}-Y_{2}\|_{p,[t,s]}^{p/(p+1)}\omega([t,s])^{p/(p+1)}\right).

It remains to note that the right-hand side is superadditive function of the interval $\displaystyle[t,s]$ , thus summing up such inequalites over a partitions of $\displaystyle[t,s]$ , yields

\displaystyle\|\mathcal{R}^{F}_{t,s}(Y_{1})-\mathcal{R}^{F}_{t,s}(Y_{2})\|_{q_{p},[t,s]}\leq C_{F,M,R}\left(\|Y_{1}-Y_{2}\|^{\nu}_{\infty}\omega([t,s])^{\nu+\nu^{2}}+\|Y_{1}-Y_{2}\|_{p,[t,s]}^{\nu}\omega([t,s])^{\nu}\right),

hence the result. ∎

As a consequence of the previous theorem, we can control the $\displaystyle p$ -variation distance of the images of two paths under a regular functional:

Corollary 3.9.

Let $\displaystyle Y_{1},Y_{2}\in C^{p-var}([0,T],\mathbb{R}^{d})$ for some $\displaystyle p\in[2,3)$ and $\displaystyle F\colon\Lambda^{d}_{T}\to V$ . Assume $\displaystyle F\in\mathbb{C}^{1,2}_{b}(\Lambda^{d}_{T}),\,\nabla F\in\mathbb{C}^{1,1}_{b}(\Lambda^{d}_{T})$ and $\displaystyle\mathcal{D}F,\,\mathcal{D}\nabla F,\,\nabla F,\,\nabla^{2}_{X}F$ are in $\displaystyle Lip(\Lambda^{d}_{T},\|\cdot\|_{\infty})$ . Then

\displaystyle\|F(\cdot,Y_{1})-F(\cdot,Y_{2})\|_{p,[t,s]}\leq C_{F,M,R}\left(\|Y_{1}-Y_{2}\|_{\infty,[0,s]}^{\nu}\,+\,\|Y_{1}-Y_{2}\|_{p,[t,s]}^{\nu}\right),

provided

\displaystyle|Y_{1}(0)|+\|Y_{1}\|_{p,[0,T]},\,|Y_{2}(0)|+\|Y_{2}\|_{p,[0,T]}\leq R,

and

\displaystyle\|Y_{1}-Y_{2}\|_{\infty,[0,T]}+\|Y_{1}-Y_{2}\|_{p,[0,T]}\leq M.

Proof.

This follows immediately from Lemma 3.8 and following identity

	$\displaystyle\displaystyle(F(\cdot,Y_{1})-F(\cdot,Y_{2}))_{t,s}=(\nabla F(t,Y_{1})(Y_{1})_{t,s}-\nabla F(t,Y_{2})(Y_{2})_{t,s})$
	$\displaystyle\displaystyle+\mathcal{R}^{F}_{t,s}(Y_{1})-\mathcal{R}^{F}_{t,s}(Y_{2}).$

Indeed, we get

	$\displaystyle\displaystyle\|(F(\cdot,Y_{1})-F(\cdot,Y_{2}))_{t,s}\|\leq\|(\nabla F(t,Y_{1})-\nabla F(t,Y_{2}))(Y_{1})_{t,s}\|\,$
	$\displaystyle\displaystyle+\,\|\nabla F(t,Y_{2})(Y_{1}-Y_{2})_{t,s})\|\,+\,\|\mathcal{R}^{F}_{t,s}(Y_{1})-\mathcal{R}^{F}_{t,s}(Y_{2})\|$
	$\displaystyle\displaystyle\leq C_{F}\\|Y_{1}-Y_{2}\\|_{\infty}\\|Y_{1}\\|_{p,[t,s]}\,+\,C_{F}\\|Y_{1}-Y_{2}\\|_{p,[t,s]}+\,\|\mathcal{R}^{F}_{t,s}(Y_{1})-\mathcal{R}^{F}_{t,s}(Y_{2})\|.$

From this and the Lemma 3.8

	$\displaystyle\displaystyle\\|F(\cdot,Y_{1})-F(\cdot,Y_{2})\\|_{p,[t,s]}\leq C_{F}\\|Y_{1}-Y_{2}\\|_{\infty}\\|Y_{1}\\|_{p,[t,s]}\,+\,C_{F}\\|Y_{1}-Y_{2}\\|_{p,[t,s]}$
	$\displaystyle\displaystyle+C_{F,M,R}\Big{(}\\|Y_{1}-Y_{2}\\|_{\infty}^{\nu}+\\|Y_{1}-Y_{2}\\|_{p,[t,s]}^{\nu}\Big{)}$

∎

We will now use Lemma 3.7 to define rough integrals with regular non-anticipative integrands:

Theorem 3.10 (Rough integral for functionals).

Let $\displaystyle\bm{X}:=(X,\mathbb{X})\in\mathcal{C}^{p-var}([0,T],\mathbb{R}^{d})$ be a rough path for some $\displaystyle p\in[2,\sqrt{2}+1)$ . Assume $\displaystyle F\in\mathbb{C}^{0,1}_{b}(\Lambda^{d}_{T},\mathbb{R}^{n})$ with $\displaystyle F$ and $\displaystyle\nabla F$ are locally horizontally Lipschitz continuous and are in $\displaystyle Lip(\Lambda^{d}_{T},\|\cdot\|_{\infty})$ . Then the rough integral

\int_{0}^{u}F(s,X)d\bm{X}(s)

(15)

exists. Moreover,

	$\displaystyle\displaystyle\left\|\int_{t}^{s}Fd\bm{X}-F(t,X)(X(s)-X(t))-\nabla F(t,X)\mathbb{X}_{t,s}\right\|\lesssim$
	$\displaystyle\displaystyle\left(\\|X\\|_{p,[t,s]}\\|\mathcal{R}^{F}(X)\\|_{q_{p},[t,s]}+\\|\nabla F(\cdot,X_{\cdot})\\|_{p,[t,s]}\\|\mathbb{X}\\|_{p/2,[t,s]}\right).$

Proof.

By Lemma 3.7 the $\displaystyle(F(\cdot,X),\nabla F(\cdot,X))\in\mathscr{D}^{p,q_{p}}_{X}([0,T],{\mathbb{R}})$ (is controlled by $\displaystyle X$ in the sense of Definition 2.4). Thus the result follows by Theorem 2.5, one only needs to check that

for $\displaystyle p<\sqrt{2}+1$ , we have $\displaystyle p^{-1}+q_{p}^{-1}=\frac{2p+1}{p^{2}}>1$ . ∎

We continue to investigate the actions of regular functionals on controlled paths. The next result asserts the invariance of controlled paths under the action of regular functionals:

Theorem 3.11.

Let $\displaystyle X\in C^{p-var}([0,T],\mathbb{R}^{d})$ and $\displaystyle(Y,Y^{\prime})\in\mathscr{D}^{p,q_{p}}_{X}([0,T],{\mathbb{R}}^{k}),$ where $\displaystyle q_{p}=p^{2}/(p+1)$ for $\displaystyle p\in[2,\sqrt{2}+1)$ . Let $\displaystyle F\colon\Lambda^{d\times k}_{T}\to\mathcal{L}({\mathbb{R}}^{d},{\mathbb{R}}^{m})$ be a non-anticipative functional. Assume $\displaystyle\nabla F\in\mathbb{C}^{1,2}_{b}(\Lambda^{d\times k}_{T}),\nabla F\in\mathbb{C}^{1,1}_{b}(\Lambda^{d\times k}_{T})$ and $\displaystyle\mathcal{D}F,\,\mathcal{D}\nabla F,\,\nabla F,\,\nabla^{2}_{X}F$ are in $\displaystyle Lip(\Lambda^{d\times k}_{T},\|\cdot\|_{\infty})$ . Then

\displaystyle(F(\cdot,Y),F(\cdot,Y)^{\prime}):=(F(\cdot,Y),\nabla F(\cdot,Y)Y^{\prime})\in\mathscr{D}^{p,q_{p}}_{X}([0,T],{\mathbb{R}}^{m}).

Furthermore, assuming $\displaystyle 1+|Y^{\prime}_{0}|+\|Y,Y^{\prime}\|_{p,q_{p},X}\leq M$ , we have

\displaystyle\|F(\cdot,Y)^{\prime}\|_{p,[t,s]}\leq C_{F,M}(|s-t|+\|Y\|_{p,[t,s]}\,+\,\|Y^{\prime}\|_{p,[t,s]}).

and

\displaystyle\|R^{F(\cdot,Y)}\|_{q_{p},[t,s]}\leq C_{F,T}(\|R^{Y}\|_{q_{p},[t,s]}+\|Y\|_{p,[t,s]}^{1+\nu}+|s-t|).

Proof.

From Lipschitz continuity Assumptions on $\displaystyle F$ we obtain

	$\displaystyle\displaystyle\|F(s,Y_{s})-F(t,Y_{t})\|\leq\|F(s,Y_{s})-F(s,Y_{t})\|+\|F(s,Y_{t})-F(t,Y_{t})\|$
	$\displaystyle\displaystyle\leq C\\|Y_{s}-Y_{t}\\|_{\infty}+C\|s-t\|\leq C(\\|Y\\|_{p,[t,s]}+\|s-t\|),$

hence

\displaystyle\displaystyle\|F(\cdot,Y)\|_{p,[t,s]}\leq c_{F}(\|Y\|_{p,[t,s]}+|s-t|),

(16)

similarly

\displaystyle\displaystyle\|\nabla F(\cdot,Y)\|_{p,[t,s]}\leq c_{F}(\|Y\|_{p,[t,s]}+|s-t|).

(17)

From the last inequality and triangle inequality, we get

	$\displaystyle\displaystyle\|(F(\cdot,Y)^{\prime})_{t,s}\|=\|(\nabla F(\cdot,Y)Y^{\prime}(\cdot))_{t,s}\|$
	$\displaystyle\displaystyle\leq\|\nabla F(t,Y)(Y^{\prime})_{t,s}\|\,+\,\|(\nabla F(\cdot,Y))_{t,s}Y^{\prime}(s)\|$
	$\displaystyle\displaystyle\leq\Big{(}\\|\nabla F(\cdot,Y)\\|_{\infty}\\|Y^{\prime}\\|_{p,[t,s]}\,+\,\\|\nabla F(\cdot,Y)\\|_{p,[t,s]}\\|Y^{\prime}\\|_{\infty,[t,s]}\Big{)}.$

Plugging in (17)

\displaystyle\displaystyle\|F(\cdot,Y)^{\prime}\|_{p,[t,s]}\leq\|\nabla F(\cdot,Y)\|_{\infty}\|Y^{\prime}\|_{p,[t,s]}+C_{F,M}\|Y^{\prime}\|_{\infty}(|s-t|+\|Y\|_{p,[t,s]}).

(18)

which with $\displaystyle\|Y^{\prime}\|_{\infty,[0,T]}\leq|Y^{\prime}(0)|\,+\,\|Y^{\prime}\|_{p,[0,T]}\leq M$ implies the first inequality Next, for $\displaystyle R^{F}\equiv R^{F(\cdot,Y)}$ , we have

	$\displaystyle\displaystyle R^{F}_{t,s}=F(s,Y_{s})-F(t,Y_{t})-\nabla F(t,Y_{t})Y^{\prime}_{t}X_{t,s}=$
	$\displaystyle\displaystyle F(s,Y_{s})-F(t,Y_{t})-\nabla F(t,Y_{t})Y_{t,s}+\nabla F(t,Y_{t})R^{Y}_{t,s}$
	$\displaystyle\displaystyle=\mathcal{R}^{F}_{t,s}(Y)+\nabla F(t,Y_{t})R^{Y}_{t,s}.$		(19)

To estimate the above, note that from Lemma 3.7

\displaystyle\|\mathcal{R}^{F}(Y)\|_{q_{p},[t,s]}\leq C_{F,T}\left[|s-t|+\|Y\|_{p,[t,s]}^{1+\nu}\right]

thus (3.2) yields

	$\displaystyle\displaystyle\\|R^{F}\\|_{q_{p},[t,s]}\leq\\|\mathcal{R}^{F}(Y)\\|_{q_{p},[t,s]}+\\|\nabla F(\cdot,Y)\\|_{\infty}\\|R^{Y}\\|_{q_{p},[t,s]}$
	$\displaystyle\displaystyle\leq C_{F,M,T}(\\|R^{Y}\\|_{q_{p},[t,s]}+\\|Y\\|_{p,[t,s]}^{1+\nu}+\|s-t\|).$		(20)

∎

4 Path-dependent Differential Equations driven by rough paths

4.1 The setting of the problem

We now turn to our main objective: the study of path-dependent rough differential equations (RDEs). Let $\displaystyle(X,\mathbb{X})\in\mathcal{C}^{p-var}([0,T],\mathbb{R}^{d})$ be a given rough path. We are interested in the following differential equation

\begin{cases}dY(s)=b(s,Y_{s})ds+\sigma(s,Y_{s})dX(s),\\ Y_{t_{0}}=\xi_{t_{0}},\end{cases}

(21)

where $\displaystyle b\colon\Lambda_{T}^{k}\to{\mathbb{R}}^{k}$ and $\displaystyle\sigma\colon\Lambda_{T}^{k}\to\mathcal{L}({\mathbb{R}}^{d},{\mathbb{R}}^{k})$ are non-anticipative functionals. Here $\displaystyle\mathcal{L}(V,W)$ denotes the set of linear operators between linear spaces $\displaystyle V,W$ , by a slight abuse of notation, we identify $\displaystyle\mathcal{L}({\mathbb{R}}^{d},{\mathbb{R}}^{k})$ with the space of $\displaystyle d\times k$ matrices and the Euclidean space $\displaystyle{\mathbb{R}}^{d}\otimes{\mathbb{R}}^{k}\equiv{\mathbb{R}}^{d\times k}.$

To define solutions to this equation, we assume $\displaystyle\sigma$ satisfies the conditions of Theorem 3.11. Then

\displaystyle\displaystyle(Y,Y^{\prime})\in\mathscr{D}^{p,q_{p}}_{X}([0,T],{\mathbb{R}}^{k})\implies(\sigma(s,Y_{s}),\nabla\sigma(s,Y_{s})Y^{\prime}_{s})\in\mathscr{D}^{p,q_{p}}_{X}([0,T],\mathcal{L}({\mathbb{R}}^{d},{\mathbb{R}}^{k}))

and the equation 21 may be understood as a rough integral equation:

\displaystyle Y(t)=\xi_{t_{0}}\,+\,\int_{t_{0}}^{t}b(s,Y)\,{\rm d}s\,+\,\int_{t_{0}}^{t}\sigma(s,Y)d\bm{X}

where $\displaystyle\int_{t_{0}}^{t}\sigma(s,Y)d\bm{X}$ is the rough integral of the controlled path

\displaystyle\left(\Xi,\Xi^{\prime}\right):=(\sigma(s,Y),\nabla\sigma(s,Y)Y^{\prime}(s)).

More precisely, we have the following definition

Definition 4.1.

Let $\displaystyle\bm{X}=(X,\mathbb{X})\in\mathcal{C}^{p-var}([0,T],{\mathbb{R}}^{d})$ be rough path over $\displaystyle X$ , with $\displaystyle p\in[2,\sqrt{2}+1)$ . Assume $\displaystyle b\in\mathbb{C}^{0,0}_{b}(\Lambda^{k}_{T},\mathbb{R}^{k}),\,\sigma\in\mathbb{C}^{0,1}_{b}(\Lambda^{k}_{T},\mathbb{R}^{d\times k})$ with $\displaystyle\sigma$ and $\displaystyle\nabla\sigma$ are locally horizontally Lipschitz continuous and are in $\displaystyle Lip(\Lambda^{d}_{T},\|\cdot\|_{\infty})$ . A controlled path $\displaystyle(Y,Y^{\prime})\in\mathscr{D}^{p,q_{p}}_{X}([0,T],{\mathbb{R}}^{k})$ is called a solution to 21 if

Y(t)=\xi_{t_{0}}\,+\,\int_{t_{0}}^{t}b(s,Y)\,{\rm d}s\,+\,\int_{t_{0}}^{t}\sigma(s,Y)d\bm{X},

(22)

where the second integral is understood as the rough integral for the controlled path

\displaystyle s\in[0,T]\mapsto(\sigma(s,Y),\nabla\sigma(s,Y)Y^{\prime}(s))

(which exists due to Theorem 3.10).

Next we specify the assumptions on the coefficients in terms of regularity in Dupire’s sense [6, 13].

Assumption 4.1.

The functional $\displaystyle b\colon\Lambda_{T}^{k}\to{\mathbb{R}}^{k}$ is Lipschitz continuous in $\displaystyle d_{\infty}$ ; $\displaystyle b\in Lip(\Lambda^{d}_{T},d_{\infty})$

Assumption 4.2.

For the vector field $\displaystyle\sigma\colon\Lambda_{T}^{k}\to\mathcal{L}({\mathbb{R}}^{d},{\mathbb{R}}^{k})$ , we assume

•

$\displaystyle\sigma\in\mathbb{C}^{1,2}_{b}(\Lambda^{d}_{T},\mathcal{L}({\mathbb{R}}^{d},{\mathbb{R}}^{k})),\nabla\sigma\in\mathbb{C}^{1,1}_{b}(\Lambda^{d}_{T},\mathcal{L}({\mathbb{R}}^{k},\mathcal{L}({\mathbb{R}}^{d},{\mathbb{R}}^{k})))$
•

The derivatives $\displaystyle\sigma,\,\mathcal{D}\sigma,\,\nabla\sigma,\,\mathcal{D}\nabla\sigma,\,\nabla^{2}\sigma$ are Lipschitz continuous in $\displaystyle d_{\infty}$ .

The pioneering work of B. Dupire [13] and the works by R. Cont and D.A. Fourniér [6], [7], [8] have a number of examples of regular functional in the sense of Dupire derivatives. Some further examples are discussed in [10] and [22]. Here we modify some of these examples to present functionals which satisfy the above assumptions.

Example 4.2.

Running Maximum: One of the basic examples of path-dependent functionals is the running maximum. Let $\displaystyle z\colon[0,T]\to{\mathbb{R}}_{+}$ define

\displaystyle m(t,z):=\max_{s\in[0,t]}z(s)

One can easily check that $\displaystyle m\in Lip(\Lambda^{1}_{T},d_{\infty})$ , moreover $\displaystyle m$ is boundedness preserving and horizontally differentiable with $\displaystyle\mathcal{D}m=0.$ However, in general this functional may fail to be vertically differentiable at the point of the maximum of $\displaystyle z$ . Following Dupire [13] we consider the following approximation of the running maximum

\displaystyle M_{\varepsilon,h}(t,z):=\begin{cases}m(t,z)-\varepsilon,\,&0\leq z(t)\leq m(t,z)-2\varepsilon,\\ m(t,z)-\varepsilon\,+\,h(z(t)-(m(t,z)-2\varepsilon)),\,&m(t,z)-2\varepsilon\leq z(t)\leq m(t,z),\\ z(t),\,&z(t)\geq m(t,z).\end{cases}

As shown in [13] for $\displaystyle h(z)=z^{2}/(4\varepsilon)$ the functional $\displaystyle M_{\varepsilon,h}$ is twice vertically differentiable. More generally, if we take $\displaystyle h$ to be $\displaystyle C^{2}$ function with

\displaystyle h(0)=h^{\prime}(0)=h^{\prime\prime}(0)=0\text{ and }h(2\varepsilon)=\varepsilon,h^{\prime}(2\varepsilon)=1,h^{\prime\prime}(2\varepsilon)=0

then $\displaystyle M_{\varepsilon,h}$ is $\displaystyle\mathbb{C}^{1,2}$ functional with

\displaystyle\nabla_{x}M_{\varepsilon,h}(t,z):=\begin{cases}0\,&0\leq z(t)\leq m(t,z)-2\varepsilon,\\ h^{\prime}(z(t)-(m(t,z)-2\varepsilon)),\,&m(t,z)-2\varepsilon\leq z(t)\leq m(t,z),\\ 1,\,&z(t)\geq m(t,z),\end{cases}

and

\displaystyle\nabla^{2}_{x}M_{\varepsilon,h}(t,z):=\begin{cases}0\,&0\leq z(t)\leq m(t,z)-2\varepsilon,\\ h^{\prime\prime}(z(t)-(m(t,z)-2\varepsilon)),\,&m(t,z)-2\varepsilon\leq z(t)\leq m(t,z),\\ 0,\,&z(t)\geq m(t,z).\end{cases}

Furthermore, if $\displaystyle h^{\prime\prime}$ is Lipschitz continuous $\displaystyle M_{\varepsilon,h}$ satisfies the conditions of Assumption 4.2, note however that the functional $\displaystyle M_{\varepsilon,h}$ is not Fréchet differentiable.

The above functionals can be adapted for multidimensional paths. Let $\displaystyle\varphi\colon{\mathbb{R}}^{d}\to{\mathbb{R}}_{+}$ be a Lipschitz continuous functional, consider the following non-anticipative functional:

\displaystyle b(t,x):=m(t,\,\varphi\circ x).

Under the assumptions on $\displaystyle\varphi$ one can easily check that $\displaystyle b$ is boundedness preserving and $\displaystyle b\in Lip(\Lambda^{d}_{T},d_{\infty})$ . If the function $\displaystyle\varphi\in C^{2}$ with Lipschitz continuous derivatives then the functional

\displaystyle\sigma(t,x):=M_{\varepsilon,h}(t,\,\varphi\circ x)

satisfies Assumption 4.2.

Discrete time dependence: Let $\displaystyle t_{1}<\ldots<t_{m}$ be given time-points in $\displaystyle[0,T]$ and let $\displaystyle\phi\colon[0,T]\times({\mathbb{R}}^{k})^{m}\to{\mathbb{R}}^{N}$ be a Lipschitz continuous function. Define a functional $\displaystyle\sigma\Lambda_{T}^{k}\to{\mathbb{R}}^{N}$ as follows

\displaystyle\sigma(t,x):=\phi(t,\,x(t\wedge t_{1}),\,\ldots,\,x(t_{m}\wedge t)).

Furthermore, if $\displaystyle\phi\in C^{1,2}(\colon[0,T]\times({\mathbb{R}}^{k})^{m},\,{\mathbb{R}}^{N})$ and $\displaystyle\nabla\phi\in C^{1,1}$ with Lipschitz continuous derivatives, then $\displaystyle\sigma$ satisfies the regularity properties of Assumption 4.2. Indeed, it follows from the following formula for the vertical derivatives

\displaystyle\nabla_{{x}}\sigma(t,x)=\sum_{i\colon t_{i}\geq t}\nabla_{x_{i}}\phi(t,\,x(t\wedge t_{1}),\,\ldots,\,x(t_{m}\wedge t)),

and

\displaystyle\nabla^{2}_{x}\sigma(t,x)=\sum_{i,j\colon t_{i},t_{j}\geq t}\nabla^{2}_{x_{i}x_{j}}\phi(t,\,x(t\wedge t_{1}),\,\ldots,\,x(t_{m}\wedge t)).

Integral dependence: Let $\displaystyle\psi\colon[0,T]\times D([0,T],{\mathbb{R}}^{d})\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{N}$ be a Lipschitz continuous functional, then

\displaystyle F(t,x)=\int_{0}^{t}\psi(s,x_{s},x(t))ds,

is in $\displaystyle Lip(\Lambda^{d}_{T},d_{\infty})$ and is horizontally differentiable with $\displaystyle\mathcal{D}F(t,x)=\psi(t,x_{t},x(t))$ . If furthermore $\displaystyle\phi$ is twice differentiable in the last variable then $\displaystyle F$ is of class $\displaystyle\mathbb{C}^{1,2}$ with the corresponding derivates

\displaystyle\nabla_{{x}}F(t,x)=\int_{0}^{t}\nabla_{y}\psi(s,x_{s},x(t))ds,\,\,\text{ and }\,\,\nabla^{2}_{{x}}F(t,x)=\int_{0}^{t}\nabla^{2}_{y}\psi(s,x_{s},x(t))ds,

where $\displaystyle\nabla_{y}$ denotes the derivative in the last variable of $\displaystyle\psi$ . In particular, if $\displaystyle\nabla_{y}\psi,\nabla^{2}_{y}\psi$ are Lipschitz continuous then $\displaystyle F$ satisfies the regularity properties of Assumption 4.2. Note that we do not require any differentiability for $\displaystyle\psi$ in the path $\displaystyle x_{s}$ , thus in general $\displaystyle F$ is not Fréchet differentiable in the path.

Remark 4.3.

It is worth to mention that with minor technical modifications, the results of the article would hold if we replace the Lipschitz continuity assumption with an assumption of Hőlder continuity in the metric $\displaystyle d_{\infty}$ (as in [10] and [22]). However, we chose to work in the Lipschitz continuous setting to avoid unnecessary complications.

4.2 Proof of the main result

Theorem 4.4 (Existence of solutions).

Let $\displaystyle\bm{X}=(X,\mathbb{X})\in\mathcal{C}^{p-var}([0,T],{\mathbb{R}}^{d})$ be rough path over $\displaystyle X$ , with $\displaystyle p\in[2,\sqrt{2}+1)$ . Assume Assumptions 4.1 and 4.2 hold. Then for any $\displaystyle\xi\in C^{p-var}([0,t_{0}],{\mathbb{R}}^{k})$ there exist $\displaystyle(Y,Y^{\prime})\in\mathscr{D}^{p,q_{p}}_{X}([0,T],{\mathbb{R}}^{k})$ a solution to (21) in the sense of Definition 4.1.

Our proof follows an argument similar to the ones in [14] for the Hőlder setting, however unlike them, instead of a contraction argument, we use the Schauder fixed point theorem ([17, Theorem 11.1]).

Proof.

Without loss of generality we assume $\displaystyle t_{0}=0.$ We will prove that there exists a small enough time $\displaystyle T_{0}$ (depending only on $\displaystyle b,\,\sigma$ and $\displaystyle X$ ), such that the solutions exists on $\displaystyle[0,T_{0}],$ then one can apply the result on the intervals $\displaystyle[T_{0},2T_{0}],\,[2T_{0},3T_{0}],\ldots$ until it reaches $\displaystyle T$ . For any $\displaystyle(Y,Y^{\prime})\in\mathscr{D}^{p,q_{p}}_{X}([0,T_{0}],{\mathbb{R}}^{k})$ , let us denote

\displaystyle(\Xi,\Xi^{\prime})=(\sigma(\cdot,Y),\nabla\sigma(\cdot,Y)Y^{\prime}),

and define a mapping $\displaystyle\mathcal{M}_{T_{0}}\colon\mathscr{D}^{p,q_{p}}_{X}([0,T_{0}],{\mathbb{R}}^{k})\to\mathscr{D}^{p,q_{p}}_{X}([0,T_{0}],{\mathbb{R}}^{k})$ by

\displaystyle\mathcal{M}_{T_{0}}(Y,Y^{\prime}):=\left(\xi_{0}+\int_{0}^{\cdot}b(t,Y)dt+\int_{0}^{\cdot}\Xi\cdot d\bm{X},\,\,\,\Xi\right).

The statement of the theorem is equivalent to the fact that $\displaystyle\mathcal{M}_{T_{0}}$ has a fixed point. To be able to use a compactness argument we will prove the existence first in a larger space; take $\displaystyle r,p^{\prime}$ such that $\displaystyle p<r<p^{\prime}<\sqrt{2}+1$ and $\displaystyle q_{p^{\prime}}<p.$ We denote $\displaystyle\kappa=r^{-1},\nu=p^{-1}$ , $\displaystyle\nu^{\prime}=p^{\prime-1}$ and $\displaystyle\beta_{\kappa}=q_{r}^{-1},\beta_{\nu}=q_{p}^{-1}$ , $\displaystyle\beta_{\nu^{\prime}}=q_{p^{\prime}}^{-1}.$

We will prove the existence of a solution in $\displaystyle\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}$ and then argue that it is also in the initial space $\displaystyle\mathscr{D}^{p,q_{p}}_{X}([0,T_{0}],{\mathbb{R}}^{k})$ . We consider the subspace $\displaystyle A$ in $\displaystyle\mathscr{D}^{r,q_{r}}_{X}$ , in the neighbourhood of the controlled path with constant Gubinelli derivative:

\displaystyle t\mapsto(\underbrace{\xi+b(0,\xi)t+\sigma(0,\xi)X(t)}_{:=\bar{\xi}(t)},\,\sigma(0,\xi)),

with $\displaystyle Y_{0}=\xi,\quad Y^{\prime}_{0}=\sigma(0,\xi)$ . To be precise, we introduce the following superadditive function on the intervals of $\displaystyle[0,T]$ :

\displaystyle\rho_{{}_{\mathbf{X}}}([t,s]):=|s-t|+\|X\|_{p,[t,s]}^{p}+\|\mathbb{X}\|_{\frac{p}{2},[t,s]}^{\frac{p}{2}}.

Now, we can define the following Hőlder seminorm associated to $\displaystyle\rho=\rho_{{}_{\mathbf{X}}}:$

\displaystyle\|(Z,Z^{\prime})\|_{\kappa,\beta,\rho}:=\|Z^{\prime}\|_{\kappa,\rho,[0,T]}\,+\|R^{Z}\|_{\beta,\rho,[0,T]},

where $\displaystyle\|W\|_{\gamma,\rho,[0,T]}:=\sup_{0\leq t<s\leq T}\frac{|W_{t,s}|}{\rho([t,s])^{\gamma}},\,\gamma\in(0,1]$ note that

\displaystyle\|W\|_{p,[t,s]}\leq\|W\|_{{}_{\nu,\rho,[t,s]}}\,\rho([t,s])^{\nu},\,\text{ with }\nu=p^{-1}.

Define the following subset of $\displaystyle\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}$ :

A=\{(Y,Y^{\prime})\in\mathscr{D}^{r,q_{r}}_{X}\colon Y_{0}=\xi,\quad Y^{\prime}_{0}=\sigma(0,\xi),\quad\|(Y-b(0,\xi)t,Y^{\prime})\|_{\kappa,\beta_{\kappa},\rho}\leq 1\}\subset\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X},\quad

(23)

where $\displaystyle\kappa=r^{-1}$ and $\displaystyle\beta_{\kappa}=q_{r}^{-1}$ .

It is easily checked that $\displaystyle A$ is a closed, convex subset of $\displaystyle\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}$ . Moreover, by the following proposition $\displaystyle A$ is compact in $\displaystyle\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}$ .

Proposition 4.5.

Let $\displaystyle\alpha,\beta\in(0,1)$ and $\displaystyle p,q>1$ be such that $\displaystyle p>\frac{1}{\alpha},\,q>\frac{1}{\beta},$ and $\displaystyle\rho$ be a continuous superadditive function. Then for any $\displaystyle M>0$ , the set

\displaystyle\left\{(Y,Y^{\prime})\in C([0,T],{\mathbb{R}}^{d}\times{\mathbb{R}}^{k\times d})\colon|Y(0)|\,+\,|Y^{\prime}(0)|\,+\,\|Y,Y^{\prime}\|_{\alpha,\beta,\rho}\leq M\right\},

is compact in $\displaystyle\mathscr{D}^{p,q}_{X}.$

We divide the proof in two steps, where we check that the assumptions of Schauder theorem hold for $\displaystyle\mathcal{M}_{T}$ on the set $\displaystyle A\subset\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}$ and for small enough $\displaystyle T$ . We already noted that $\displaystyle A$ defined above is compact and convex in $\displaystyle\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}$ , thus it remains to check the properties of $\displaystyle\mathcal{M}_{T}$ .

Property 1 (Invariance).

There exist $\displaystyle\delta\in(0,1)$ (depending only on global properties of $\displaystyle\xi,b,\text{ and }\sigma$ ) such that if $\displaystyle\rho([0,T_{0}])<\delta$ , then the set $\displaystyle A$ defined by (23) is invariant under $\displaystyle\mathcal{M}_{T_{0}}$ :

\displaystyle\mathcal{M}_{T_{0}}(A)\subset A.

Let $\displaystyle(Y,Y^{\prime})\in A$ , we need to prove that $\displaystyle\mathcal{M}_{T_{0}}(Y,Y^{\prime})\in A.$ First note that by definition of $\displaystyle A$ and the path $\displaystyle Y$

	$\displaystyle\displaystyle\|Y^{\prime}_{0}\|+\\|(Y,Y^{\prime})\\|_{\kappa,\beta_{\kappa},\rho}\leq\|Y^{\prime}_{0}\|+\\|b(0,\xi)t\\|_{\kappa,\rho}+\\|(Y-b(0,\xi)t,Y^{\prime})\\|_{\kappa,\beta_{\kappa},\rho}$
	$\displaystyle\displaystyle\leq\\|\sigma(0,\cdot)\\|_{\infty}+\\|b(0,\cdot)\\|_{\infty}+1:=M.$

We obviously have

\displaystyle\mathcal{M}_{T_{0}}(Y,Y^{\prime})(0)=\xi,\,\mathcal{M}_{T_{0}}(Y,Y^{\prime})^{\prime}(0)=\Xi_{0}=\sigma(0,\xi),

thus it remains to check that for small $\displaystyle\delta$ , we have $\displaystyle\left\|\mathcal{M}_{T_{0}}(Y,Y^{\prime})-(b(0,\xi)t,0)\right\|_{\kappa,\beta_{\kappa},\rho}\leq 1$ . For that note

\displaystyle\displaystyle\left\|\mathcal{M}_{T_{0}}(Y,Y^{\prime})-(b(0,\xi)t,0)\right\|_{\kappa,\beta_{\kappa},\rho}\leq\left\|\int_{0}^{\cdot}b(t,Y)-b(0,\xi)dt\right\|_{q_{r},\rho,[0,T_{0}]}\,+\,\left\|\int_{0}^{\cdot}\Xi d\bm{X},\Xi\right\|_{\kappa,\beta_{\kappa},\rho}.

For the first term on the right-hand side

\displaystyle\left\|\int_{0}^{\cdot}b(t,Y)-b(0,\xi)dt\right\|_{\beta_{\kappa},\rho,[0,T_{0}]}\leq T_{0}^{1-\beta_{\kappa}}\|b(\cdot,Y)-b(0,\xi)\|_{\infty}\leq C_{b,M}\delta^{1-\beta_{\kappa}}.

where the last inequality follows from $\displaystyle d_{\infty}$ Lipschitz continuity of $\displaystyle b$ and $\displaystyle\|Y-\xi\|_{\infty}\leq\|Y\|_{r,[0,T_{0}]}\leq C_{M}$ . To estimate the second term, let $\displaystyle(Z,Z^{\prime}):=\left(\int_{0}^{\cdot}\Xi d\bm{X},\Xi\right)$ , then the estimate 2 from the proof of Theorem 2.5 implies:

	$\displaystyle\displaystyle\|R^{Z}_{t,s}\|\leq\\|\Xi^{\prime}\\|_{\infty}\\|\mathbb{X}\\|_{r/2,[t,s]}+C\left(\\|X\\|_{r,[t,s]}\\|R^{\Xi}\\|_{q_{r},[t,s]}+\\|\mathbb{X}\\|_{r/2,[t,s]}\\|\Xi^{\prime}\\|_{r,[t,s]}\right)$
	$\displaystyle\displaystyle\leq C\left(\\|\Xi^{\prime}\\|_{\infty}+\\|\Xi^{\prime}\\|_{r,[0,T_{0}]}\right)\\|\mathbb{X}\\|_{r/2,[t,s]}+C\\|R^{\Xi}\\|_{q_{r},[t,s]}\\|X\\|_{r,[t,s]}.$		(24)

To estimate the first term in (4.2) note that from the first inequality of Theorem 3.11

	$\displaystyle\displaystyle\\|\Xi^{\prime}\\|_{\infty}+\\|\Xi^{\prime}\\|_{r,[0,T_{0}]}\leq\|\Xi^{\prime}_{0}\|+2\\|\Xi^{\prime}\\|_{r,[0,T_{0}]}\leq$
	$\displaystyle\displaystyle\|\Xi^{\prime}_{0}\|+C_{M,{\sigma}}(T_{0}+\\|Y\\|_{r,[0,T_{0}]}\,+\,\\|Y^{\prime}\\|_{r,[0,T_{0}]})\leq$
	$\displaystyle\displaystyle\|\nabla\sigma(0,\xi)\sigma(0,\xi)\|+C_{M,{\sigma}}\left(1+\|Y^{\prime}_{0}\|+\\|(Y,Y^{\prime})\\|_{\kappa,\beta_{\kappa},\rho}\right)\leq C_{{}_{\xi,\sigma,M}},$		(25)

For the second term in (4.2), we use the second inequality of Theorem 3.11, which gives

	$\displaystyle\displaystyle\\|R^{\Xi}\\|_{q_{r},[t,s]}\leq C_{\sigma,M}(\\|R^{Y}\\|_{q_{r},[t,s]}+\\|Y\\|_{r,[t,s]}^{1+\kappa}+\|s-t\|)$
	$\displaystyle\displaystyle\leq C_{\sigma,M}(\\|R^{Y}\\|_{{}_{\beta_{\kappa}},\rho}\rho([t,s])^{\beta_{\kappa}}+\\|Y\\|_{{}_{\kappa,\rho}}^{1+\kappa}\rho([t,s])^{\beta_{\kappa}}+\rho([t,s]))\leq C_{{}_{\sigma,M}}\,\rho([t,s])^{\beta_{\kappa}}.$		(26)

Consequently, from (4.2)

\displaystyle\displaystyle|R^{Z}_{t,s}|\leq C_{{}_{\xi,\sigma,M}}\|\mathbb{X}\|_{r/2,[t,s]}+C_{{}_{\sigma,M}}\,\|X\|_{r,[t,s]}\rho([t,s])^{\beta_{\kappa}}\leq C_{{}_{\xi,\sigma,M}}\left(\rho([t,s])^{2\kappa}+\rho([t,s])^{\kappa+\beta_{\kappa}}\right),

\displaystyle\displaystyle\|R^{Z}\|_{\beta_{\kappa},\rho}\leq C_{{}_{\xi,\sigma,M}}\delta^{2\kappa-\beta_{\kappa}}.

(27)

Using Lemma 3.7, as in the proof of (17), we get

\displaystyle|\Xi_{t,s}|\leq\|\Xi\|_{r,[t,s]}=\|{\sigma}(\cdot,Y)\|_{r,[t,s]}\leq C_{{}_{\sigma,M}}\Big{(}|s-t|+\|Y\|_{r,[t,s]}\Big{)}.

From the identity $\displaystyle(Y)_{t,s}=Y^{\prime}(t)(X)_{t,s}\,+\,R^{Y}_{t,s}$ through the chain of inequalities:

	$\displaystyle\displaystyle\\|Y\\|_{r,[t,s]}\leq\\|Y^{\prime}\\|_{\infty}\\|X\\|_{p,[t,s]}+\\|R^{Y}\\|_{q_{r},[t,s]}\leq\rho([t,s])^{\min\{\nu,\beta_{\kappa}\}}\Big{(}\\|Y^{\prime}\\|_{\infty}\,+\,\\|R^{Y}\\|_{\beta_{\kappa},\rho}\Big{)}$
	$\displaystyle\displaystyle\leq\Big{(}\|Y^{\prime}_{0}\|+\\|(Y,Y^{\prime})\\|_{\kappa,\beta_{\kappa},\rho}\Big{)}\rho([t,s])^{\min\{\nu,\beta_{\kappa}\}}\leq C_{{}_{\sigma,M}}\rho([t,s])^{\min\{\nu,\beta_{\kappa}\}}.$

The previous two inequalities yield

\displaystyle\displaystyle\|\Xi\|_{\kappa,\rho}\leq C_{{}_{\sigma,M}}\delta^{\min\{\nu,\beta_{\kappa}\}-\kappa}

(28)

Finally (27) and (28) give

	$\displaystyle\displaystyle\left\\|\mathcal{M}_{T_{0}}(Y,Y^{\prime})-(b(0,\xi)t,0)\right\\|_{\kappa,\beta_{\kappa},\rho}\leq\left\\|\int_{0}^{\cdot}b(t,Y)-b(0,\xi)dt\right\\|_{\beta_{\kappa},\rho}\,+\,\left\\|\int_{0}^{\cdot}\Xi d\bm{X},\Xi\right\\|_{\kappa,\beta_{\kappa},\rho}$
	$\displaystyle\displaystyle\leq C_{{}_{b,M}}\delta^{1-\beta_{\kappa}}+C_{{}_{\xi,\sigma,M}}\delta^{\min\{\nu,\beta_{\kappa}\}-\kappa}.$

It remains to take $\displaystyle T_{0}$ small enough so that $\displaystyle C_{{}_{b,M}}\delta^{1-\beta_{\kappa}}+C_{{}_{\xi,\sigma,M}}\delta^{\min\{\nu,\beta_{\kappa}\}-\kappa}\leq 1,$ to get $\displaystyle\mathcal{M}_{T_{0}}(Y,Y^{\prime})\in A$ .

Property 2 (Continuity).

The map

\displaystyle\mathcal{M}_{T_{0}}\colon\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}\to\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}.

is continuous.

Let $\displaystyle(Y_{1},Y_{1}^{\prime}),\,(Y_{2},Y_{2}^{\prime})\in\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}$ , and $\displaystyle\Delta(s)=\sigma(s,Y_{1})-\sigma(s,Y_{2}),\,\Delta^{\prime}(s)=\nabla\sigma(s,Y_{1})-\nabla\sigma(s,Y_{2})$ . Let $\displaystyle R,M>0$ be such that

\displaystyle|Y_{1}(0)|+\|Y_{1}\|_{p^{\prime},[0,T_{0}]}+\|R^{Y_{1}}\|_{q_{p^{\prime}},[0,T_{0}]},\,|Y_{2}(0)|+\|Y_{2}\|_{p^{\prime},[0,T_{0}]}\leq R,

and

\displaystyle\|Y_{1}-Y_{2}\|_{\infty}+\|Y_{1}-Y_{2}\|_{p,[0,T_{0}]}\leq M.

We can estimate the distance between the values $\displaystyle\mathcal{M}_{T_{0}}(Y_{1},Y_{1}^{\prime})$ and $\displaystyle\mathcal{M}_{T_{0}}(Y_{2},Y_{2}^{\prime})$ by Theorem 2.5

	$\displaystyle\displaystyle\left\\|\mathcal{M}_{T_{0}}(Y_{1},Y_{1}^{\prime})-\mathcal{M}_{T_{0}}(Y_{2},Y_{2}^{\prime})\right\\|_{p^{\prime},q_{p^{\prime}},X}=\left\\|\int_{0}^{\cdot}\Delta d\mathbf{X},\Delta\right\\|_{p^{\prime},q_{p^{\prime}},X}\leq$
	$\displaystyle\displaystyle\\|\Delta\\|_{p^{\prime},[0,T_{0}]}+C\left(\\|X\\|_{p^{\prime}}\\|R^{\Delta}\\|_{q_{p^{\prime}},[0,T_{0}]}+\\|\Delta^{\prime}\\|_{p^{\prime},[0,T_{0}]}\\|\mathbb{X}\\|_{p^{\prime}/2,[0,T_{0}]}\right)$
	$\displaystyle\displaystyle\leq\\|\Delta\\|_{p^{\prime}}+C_{\mathbf{X}}\left(\|\Delta^{\prime}_{0}\|+\\|(\Delta,\Delta^{\prime})\\|_{p^{\prime},q_{p^{\prime}},X}\right)$		(29)

For the first term in 4.2, we use the Corollary 3.9 for $\displaystyle\sigma$

\displaystyle\displaystyle\|\Delta\|_{p^{\prime},[0,T_{0}]}=\|{\sigma}(\cdot,Y_{1})-{\sigma}(\cdot,Y_{2})\|_{p^{\prime},[0,T_{0}]}\leq C_{\sigma,M,R}\left(\|Y_{1}-Y_{2}\|_{\infty}^{\nu^{\prime}}\,+\,\|Y_{1}-Y_{2}\|_{p^{\prime},[0,T_{0}]}^{\nu^{\prime}}\right).

(30)

For $\displaystyle\nabla\sigma$ the same corollary provides

\displaystyle\displaystyle\|\nabla{\sigma}(\cdot,Y_{1})-\nabla{\sigma}(\cdot,Y_{2})\|_{p^{\prime},[0,T_{0}]}\leq C_{\nabla\sigma,M,R}\left(\|Y_{1}-Y_{2}\|_{\infty}^{\nu^{\prime}}\,+\,\|Y_{1}-Y_{2}\|_{p^{\prime},[t,s]}^{\nu^{\prime}}\right)

Also by $\displaystyle\sigma\in Lip(\Lambda_{T}^{d},d_{\infty})$

\displaystyle\displaystyle\|\nabla\sigma(\cdot,Y_{1})\|_{\infty}\leq|\nabla\sigma(0,\xi)|+C_{\sigma}\|Y_{1}-\xi\|_{\infty}\leq C_{\xi,\sigma,R},

From previous two inequalities, we conclude

	$\displaystyle\displaystyle\\|\Delta^{\prime}\\|_{p^{\prime},[0,T_{0}]}=\\|\nabla{\sigma}(\cdot,Y_{1})Y^{\prime}_{1}-\nabla{\sigma}(\cdot,Y_{2})Y^{\prime}_{2}\\|_{p^{\prime},[0,T_{0}]}$
	$\displaystyle\displaystyle\leq\\|\nabla\sigma(\cdot,Y_{1})\\|_{\infty}\\|Y^{\prime}_{1}-Y^{\prime}_{2}\\|_{p^{\prime},[0,T_{0}]}+\\|\nabla{\sigma}(\cdot,Y_{1})-\nabla{\sigma}(\cdot,Y_{2})\\|_{p^{\prime},[0,T_{0}]}\\|Y^{\prime}_{2}\\|_{\infty}$
	$\displaystyle\displaystyle\leq C_{\xi,\sigma,M,R}\Big{(}\\|Y_{1}-Y_{2}\\|_{\infty}^{\nu^{\prime}}\,+\,\\|Y^{\prime}_{1}-Y^{\prime}_{2}\\|_{p^{\prime},[0,T_{0}]}\,+\,\left\\|Y_{1}-Y_{2}\right\\|_{p^{\prime},[0,T_{0}]}^{{\nu^{\prime}}}\Big{)}$
	$\displaystyle\displaystyle\leq C_{\xi,\sigma,M,R}\,\Big{(}\|(Y_{1}-Y_{2})(0)\|^{\nu^{\prime}}+\left\\|Y_{1}-Y_{2},Y^{\prime}_{1}-Y^{\prime}_{2}\right\\|^{\nu^{\prime}}_{p^{\prime},q_{p^{\prime}},X}\Big{)}$		(31)

where we have used

\displaystyle\|Y_{1}-Y_{2}\|_{p^{\prime},[0,T_{0}]}\leq C_{T_{0},X}\Big{(}|(Y_{1}-Y_{2})(0)|+\left\|Y_{1}-Y_{2},Y^{\prime}_{1}-Y^{\prime}_{2}\right\|_{p^{\prime},q_{p^{\prime}},X}\Big{)}.

Finally, from

\displaystyle R^{\Delta}_{t,s}=\mathcal{R}^{\sigma,Y_{1}}_{t,s}-\mathcal{R}^{\sigma,Y_{2}}_{t,s}+\nabla{\sigma}(t,Y_{1})R_{t,s}^{Y_{1}}-\nabla{\sigma}(t,Y_{2})R_{t,s}^{Y_{2}}

we have

\displaystyle\displaystyle|R^{\Delta}_{t,s}|\leq|\mathcal{R}^{\sigma,Y_{1}}_{t,s}-\mathcal{R}^{\sigma,Y_{2}}_{t,s}|\,+\,|\nabla{\sigma}(t,Y_{1})-\nabla{\sigma}(t,Y_{2})|\,|R_{t,s}^{Y_{1}}|\,+\,|\nabla{\sigma}(t,Y_{1})|\,|R_{t,s}^{Y_{1}}-R_{t,s}^{Y_{2}}|

from Lipschitz continuity of $\displaystyle\nabla\sigma$ , the inequality $\displaystyle\|\nabla\sigma(\cdot,Y_{1})\|_{\infty}\leq C_{\xi,R}$ (obtained above) and Lemma 3.8

	$\displaystyle\displaystyle\\|R^{\Delta}\\|_{q_{\nu^{\prime}}}\leq C_{\sigma,T_{0},R}(\\|Y_{1}-Y_{2}\\|^{\nu^{\prime}}_{\infty}+\\|Y_{1}-Y_{2}\\|^{\nu^{\prime}}_{p^{\prime},[0,T_{0}]})\,+\,C_{\sigma,T_{0}}\\|Y_{1}-Y_{2}\\|_{\infty}\\|R^{Y_{1}}\\|_{q_{\nu^{\prime}}}\,$
	$\displaystyle\displaystyle+\,C_{\sigma}\left\\|Y_{1}-Y_{2},Y^{\prime}_{1}-Y^{\prime}_{2}\right\\|_{p^{\prime},q_{p^{\prime}},X}\leq C_{\xi,\sigma,T_{0},R}\,\,\left(\left\\|Y_{1}-Y_{2},Y^{\prime}_{1}-Y^{\prime}_{2}\right\\|^{\nu^{\prime}}_{p^{\prime},q_{p^{\prime}},X}\right).$		(32)

Combining (4.2) and (4.2) and using

\displaystyle\|Y_{1}-Y_{2}\|_{\infty}+\|Y_{1}-Y_{2}\|_{p^{\prime},[0,T_{0}]}\leq C_{T_{0}}\left\|Y_{1}-Y_{2},Y^{\prime}_{1}-Y^{\prime}_{2}\right\|_{p^{\prime},q_{p^{\prime}},X}

we get

	$\displaystyle\displaystyle\\|(\Delta,\Delta^{\prime})\\|_{p^{\prime},q_{p^{\prime}},X}=\\|\Delta^{\prime}\\|_{p^{\prime},[0,T_{0}]}+\\|R^{\Delta}\\|_{q_{\nu^{\prime}}}$
	$\displaystyle\displaystyle\leq C_{\xi,\sigma,T_{0},R}\quad\Big{(}\|(Y_{1}-Y_{2})(0)\|^{\nu^{\prime}}+\left\\|Y_{1}-Y_{2},Y^{\prime}_{1}-Y^{\prime}_{2}\right\\|^{\nu^{\prime}}_{p^{\prime},q_{p^{\prime}},X}\Big{)}.$

Thus (4.2) and (30) yield

	$\displaystyle\displaystyle\left\\|\mathcal{M}_{T_{0}}(Y_{1},Y_{1}^{\prime})-\mathcal{M}_{T_{0}}(Y_{2},Y_{2}^{\prime})\right\\|_{p^{\prime},q_{p^{\prime}},X}$
	$\displaystyle\displaystyle\leq C_{\xi,\sigma,T_{0},R}\quad\Big{(}\|(Y_{1}-Y_{2})(0)\|^{\nu^{\prime}}+\left\\|Y_{1}-Y_{2},Y^{\prime}_{1}-Y^{\prime}_{2}\right\\|^{\nu^{\prime}}_{p^{\prime},q_{p^{\prime}},X}\Big{)}$
	$\displaystyle\displaystyle\leq C_{\xi,\sigma,T_{0},R}\left\\|Y_{1}-Y_{2},Y^{\prime}_{1}-Y^{\prime}_{2}\right\\|^{\nu^{\prime}}_{\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}}.$

hence $\displaystyle\mathcal{M}_{T_{0}}$ is continuous in $\displaystyle\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}$ .

We have proved that $\displaystyle\mathcal{M}_{T_{0}}\colon\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}\to\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}$ is continuous, $\displaystyle\mathcal{M}_{T_{0}}(A)\subset A$ , and $\displaystyle A\subset\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}$ is a compact, convex subset. Thus by Schauder fixed point theorem, $\displaystyle\mathcal{M}_{T_{0}}$ has a fixed point $\displaystyle(Y,Y^{\prime})\in\mathscr{D}^{p^{\prime},q_{p^{\prime}}}_{X}$ .

To conclude the proof of the theorem it remains to prove that $\displaystyle(Y_{s},Y^{\prime}_{s})\in\mathscr{D}^{p,q_{p}}_{X}$ . Indeed, from the representation $\displaystyle Y_{t,s}=Y^{\prime}_{t}X_{t,s}+R^{Y}_{t,s}$ , $\displaystyle X\in C^{p-var}([0,T],{\mathbb{R}})$ and $\displaystyle q_{p^{\prime}}<p$ it follows that

\displaystyle\|Y\|_{p,[0,T_{0}]}\leq\|Y^{\prime}\|_{\infty}\|X\|_{p,[0,T_{]}}\,+\,\|R^{Y}_{t,s}\|_{q_{p^{\prime}},[0,T_{0}]}<+\infty,

hence $\displaystyle Y\in C^{p-var}([0,T_{0}],{\mathbb{R}})$ . Now, using the fixed point property

\displaystyle(Y,Y^{\prime})=\left(\xi_{0}+\int_{0}^{\cdot}b(t,Y)dt+\int_{0}^{\cdot}\Xi\cdot d\bm{X},\,\,\,\Xi\right),\,(\Xi,\Xi^{\prime})=(\sigma(\cdot,Y),\nabla\sigma(\cdot,Y)Y^{\prime}),

and Corollary 3.9, we get $\displaystyle Y^{\prime}=\sigma(\cdot,Y)\in C^{p-var}$ . Next, by Theorem 2.5

\displaystyle\left|R^{Y}_{t,s}\right|\leq\|b\|_{\infty}|s-t|\,+\,\|\Xi^{\prime}\|_{\infty}|\mathbb{X}_{t,s}|\,+\,C\left(\|{R}^{\Xi}\|_{q_{p}^{\prime},[t,s]}\|X\|_{p^{\prime},[t,s]}+\|\Xi^{\prime}\|_{p^{\prime},[t,s]}\|{\mathbb{X}}\|_{p^{\prime}/2,[t,s]}\right).

Using that $\displaystyle 1/p^{\prime}+1/q_{p^{\prime}}>1$ , $\displaystyle p^{\prime}<3$ and the properties of superadditive functions the above yields

\displaystyle\left|R^{Y}_{t,s}\right|\leq\|\Xi^{\prime}\|_{\infty}|\mathbb{X}_{t,s}|\,+\,\omega([t,s]),

where $\displaystyle\omega$ is a superadditive interval function. Since $\displaystyle p/2<q_{p}$ and $\displaystyle\|X\|_{p,[0,T]}<+\infty$ , we conclude that $\displaystyle\|R^{Y}\|_{q_{p},[0,T_{0}]}<+\infty$ , therefore $\displaystyle(Y,Y^{\prime})\in\mathscr{D}^{p,q_{p}}_{X}.$ ∎

Remark 4.6.

Our proofs suggest that the results would still hold under the following regularity assumption on the coefficient $\displaystyle\sigma$ :

Assumption 4.3.

There exist a non-anticipative functional $\displaystyle\sigma^{\prime}\colon\Lambda^{k}_{T}\to\mathcal{L}({\mathbb{R}}^{k},\mathcal{L}({\mathbb{R}}^{d},{\mathbb{R}}^{k}))$ such that

•

$\displaystyle\sigma,\sigma^{\prime}$ are continuous in the p-variation norm; there exist a modulus of continuity $\displaystyle\rho\colon{\mathbb{R}}_{+}\to{\mathbb{R}}_{+},\,\rho(r)\xrightarrow[r\to 0]{}0$ :

	$\displaystyle\displaystyle\\|\sigma(\cdot,Y_{1})-\sigma(\cdot,Y_{2})\\|_{{C}^{p-var}}\leq C_{\sigma,T}\,\,\rho\left(\\|Y_{1}-Y_{2})\\|_{{C}^{p-var}}\right),$
	$\displaystyle\displaystyle\\|\sigma^{\prime}(\cdot,Y_{1})-\sigma^{\prime}(\cdot,Y_{2})\\|_{{C}^{p-var}}\leq C_{\sigma,T}\,\,\rho\left(\\|Y_{1}-Y_{2})\\|_{{C}^{p-var}}\right).$

•
$\displaystyle R^{\sigma,\sigma^{\prime}}_{t,s}(Y):=\sigma(s,Y)\,-\,\sigma(t,Y)\,-\,\sigma^{\prime}(t,Y)(Y(s)-Y(t))$ satisfies
- a)
  
  $\displaystyle\|R^{\sigma,\sigma^{\prime}}(Y)\|_{q_{p},[t,s]}\leq C_{\sigma,T}\,\,\rho\left(\|Y\|_{\infty,[0,s]}\,+\,\|Y\|_{p,[t,s]}\right),\forall Y\in C^{p-var}([0,T],{\mathbb{R}}^{k})$
- b)
  
  $\displaystyle\|R^{\sigma,\sigma^{\prime}}(Y_{1})\,-\,R^{\sigma,\sigma^{\prime}}(Y_{2})\|_{q_{p},[t,s]}\leq C_{\sigma,T}\,\,\rho\left(\|Y_{1}\,-\,Y_{2}\|_{{C}^{p-var}}\right),\forall Y_{1},\,Y_{2}\in C^{p-var}([0,T],{\mathbb{R}}^{k})$
for some $\displaystyle q_{p}\in(0,1)$ with $\displaystyle q_{p}^{-1}+p^{-1}>1$ .

Appendix: Proof of Proposition 4.5

In this section we present the proof of the Proposition 4.5:

Proof.

It is enough to show that any sequence $\displaystyle(Y_{n},Y^{\prime}_{n})\in C([0,T],{\mathbb{R}}^{d}\times{\mathbb{R}}^{k\times d})$ satisfying

\displaystyle|Y_{n}(0)|\,+\,|Y_{n}^{\prime}(0)|\,+\,\|Y_{n},Y^{\prime}_{n}\|_{\alpha,\beta,\rho}\leq M

has a convergent subsequence in $\displaystyle\mathscr{D}^{p,q}_{X}.$ For this note that since

	$\displaystyle\displaystyle\|Y_{n}(s)-Y_{n}(t)\|\leq C_{M,T}\rho([s,t])^{\alpha}$
	$\displaystyle\displaystyle\|Y^{\prime}_{n}(s)-Y^{\prime}_{n}(t)\|\leq M\rho([s,t])^{\alpha}$		(33)

by Arzela-Ascoli theorem we can assume that $\displaystyle(Y_{n},Y^{\prime}_{n})\to(Y,Y^{\prime})\in C([0,T],{\mathbb{R}}^{d}\times{\mathbb{R}}^{k\times d})$ uniformly. From (Proof.) and since also

\displaystyle\displaystyle|R^{Y_{n}}_{t,s}|\leq M\rho([s,t])^{\beta},

we conclude $\displaystyle\|\tilde{Y}^{\prime}\|_{\alpha,\rho}\leq M,\,\|R^{{Y}}\|_{\beta,\rho}\leq M.$ Let now $\displaystyle\tilde{Y}_{n}:=Y_{n}-Y,\,\tilde{Y}^{\prime}_{n}:=Y^{\prime}_{n}-Y^{\prime}$ and $\displaystyle\alpha^{\prime}:=p^{-1}<\alpha,\,\beta^{\prime}:=q^{-1}<\beta$ . We have that $\displaystyle\|\tilde{Y}_{n}\|_{\infty},\|\tilde{Y}_{n}^{\prime}\|_{\infty}\to 0$ and consequently from $\displaystyle\|R^{{Y}}\|_{\infty}:=Y_{t,s}-Y^{\prime}(t)X_{t,s}$

\displaystyle\|R^{\tilde{Y}_{n}}\|_{\infty}\leq C\left(\|\tilde{Y}_{n}\|_{\infty}\,+\,\|\tilde{Y}^{\prime}_{n}\|_{\infty}\|X\|_{\infty}\right)\to 0.

Since

\displaystyle\|\tilde{Y}_{n}^{\prime}\|_{\alpha,\rho}\leq\|Y_{n}^{\prime}\|_{\alpha,\rho}+\|\tilde{Y}^{\prime}\|_{\alpha,\rho}\leq 2M,\,\|R^{\tilde{Y}_{n}}\|_{\beta,\rho}\leq\|R^{{Y}_{n}}\|_{\beta,\rho}\,+\,\|R^{{Y}}\|_{\beta,\rho}\leq 2M,

we get

\displaystyle\|\tilde{Y}_{n}^{\prime}\|_{\alpha^{\prime},\rho}\leq\left(2\|\tilde{Y}_{n}^{\prime}\|_{\infty}\right)^{1-\frac{\alpha^{\prime}}{\alpha}}\left(\|\tilde{Y}_{n}^{\prime}\|_{\alpha,\rho}\right)^{\frac{\alpha^{\prime}}{\alpha}}\leq C_{M}\left(\|\tilde{Y}_{n}^{\prime}\|_{\infty}\right)^{1-\frac{\alpha^{\prime}}{\alpha}}\to 0,

\displaystyle\|R^{\tilde{Y}_{n}}\|_{\alpha^{\prime},\rho}\leq\left(\|R^{\tilde{Y}_{n}}\|_{\infty}\right)^{1-\frac{\beta^{\prime}}{\beta}}\left(\|R^{\tilde{Y}_{n}}\|_{\beta,\rho}\right)^{\frac{\beta^{\prime}}{\beta}}\leq C_{M}\left(\|R^{\tilde{Y}_{n}}\|_{\infty}\right)^{1-\frac{\beta^{\prime}}{\beta}}\to 0,

Convergence in $\displaystyle\mathscr{D}^{p,q}_{X}$ follows from the inequalities

\displaystyle\|Y^{\prime}\|_{p,[0,T]}\leq\|Y^{\prime}\|_{{}_{\alpha^{\prime},\rho}}\,\rho([0,T])^{\alpha^{\prime}},\,\|R^{Y}\|_{q,[0,T]}\leq\|R^{Y}\|_{{}_{\beta^{\prime},\rho}}\,\rho([0,T])^{\beta^{\prime}}.

∎

References

[1] S Aida. Rough differential equations containing path-dependent bounded variation terms. Working paper, 2019.
[2] Anna Ananova. Pathwise Integration and Functional Calculus for paths with finite quadratic variation. PhD thesis, Imperial College London, December 2018.
[3] Ismaël Bailleul. Flows driven by Banach space-valued rough paths. In Catherine Donati-Martin, Antoine Lejay, and Alain Rouault, editors, Séminaire de Probabilités XLVI, pages 195–205. Springer, 2014.
[4] Ismael Bailleul. Flows driven by rough paths. Rev. Mat. Iberoamericana, 31(3):901–934, 2015.
[5] Rama Cont. Functional Ito Calculus and functional Kolmogorov equations. In Stochastic Integration by Parts and Functional Ito Calculus (Lecture Notes of the Barcelona Summer School in Stochastic Analysis, July 2012), Advanced Courses in Mathematics, pages 115–208. Birkhauser Basel, 2016.
[6] Rama Cont and David-Antoine Fournié. A functional extension of the Ito formula. Comptes Rendus Mathématique Académie des Sciences, 348(1-2):57–61, 2009.
[7] Rama Cont and David-Antoine Fournié. Change of variable formulas for non-anticipative functionals on path space. J. Funct. Anal., 259(4):1043–1072, 2010.
[8] Rama Cont and David-Antoine Fournié. Functional Ito calculus and stochastic integral representation of martingales. Annals of Probability, 41(1):109–133, 2013.
[9] Rama Cont and Alexander Kalinin. On the support of solutions to stochastic differential equations with path-dependent coefficients. Stochastic Processes and their Applications, 2019.
[10] Rama Cont and Alexander Kalinin. On the support of solutions to stochastic differential equations with path-dependent coefficients. Stochastic Processes and their Applications, 2019.
[11] A. M. Davie. Differential Equations Driven by Rough Paths: An Approach via Discrete Approximation. Applied Mathematics Research eXpress, 2008, 01 2008.
[12] Aurelien Deya, Massimiliano Gubinelli, Martina Hofmanova, and Samy Tindel. One-dimensional reflected rough differential equations. Stochastic Processes and their Applications, 129(9):3261 – 3281, 2019.
[13] Bruno Dupire. Functional Ito calculus. Quantitative Finance, 19(5):721–729, 2019.
[14] Peter K. Friz and Martin Hairer. A course on rough paths. Springer, 2014.
[15] Peter K. Friz and Atul Shekhar. General rough integration, Lévy rough paths and a Lévy-Kintchine-type formula. Ann. Probab., 45(4):2707–2765, 2017.
[16] Peter K. Friz and Nicolas B. Victoir. Multidimensional stochastic processes as rough paths, volume 120 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2010.
[17] David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
[18] Massimiliano Gubinelli. Controlling rough paths. J. Funct. Anal., 216(1):86–140, 2004.
[19] Massimiliano Gubinelli. Ramification of rough paths. Journal of Differential Equations, 248(4):693 – 721, 2010.
[20] Martin Hairer. A theory of regularity structures. Inventiones Mathematicae, 198(2):269–504, 2014.
[21] Kiyoshi Itô and Makiko Nisio. On stationary solutions of a stochastic differential equation. J. Math. Kyoto Univ., 4:1–75, 1964.
[22] Alexander Kalinin. Support characterization for regular path-dependent stochastic volterra integral equations, 2019.
[23] Harold J. Kushner. On the stability of processes defined by stochastic difference-differential equations. J. Differential Equations, 4:424–443, 1968.
[24] Terry J. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana, 14(2):215–310, 1998.
[25] Terry J. Lyons, Michael Caruana, and Thierry Lévy. Differential equations driven by rough paths, volume 1908 of Lecture Notes in Mathematics. Springer, 2007.
[26] Terry J. Lyons and Zhongmin Qian. System control and rough paths. Oxford Mathematical Monographs. Oxford University Press, 2002.
[27] Terry J. Lyons and Nicolas Victoir. An extension theorem to rough paths. Ann. Inst. H. Poincaré Anal. Non Linéaire, 24(5):835–847, 2007.
[28] Salah-Eldin A. Mohammed. Stochastic differential systems with memory: theory, examples and applications. In Stochastic analysis and related topics, VI (Geilo, 1996), volume 42 of Progress in Probability, pages 1–77. Birkhäuser, 1998.
[29] Andreas Neuenkirch, Ivan Nourdin, and Samy Tindel. Delay equations driven by rough paths. Electron. J. Probab., 13:2031–2068, 2008.
[30] Giuseppe Peano. Démonstration de l’intégrabilité des équations différentielles ordinaires. Mathematische Annalen, 37(2):182–228, 1890.

	$\displaystyle\displaystyle\|I_{3}(Y_{1}^{N})-I_{3}(Y^{N}_{2})\|\leq C_{F}\\|Y_{1}-Y_{2}\\|_{\infty}N^{1-\nu}\\|Y_{1}\\|^{2}_{p,[t,s]}$
	$\displaystyle\displaystyle+\\|\nabla^{2}F(\cdot,Y^{N}_{2})\\|_{\infty}\\|Y_{1}-Y_{2}\\|_{p,[t,s]}(N^{1-\nu}\\|Y_{1}\\|_{p,[t,s]})$
	$\displaystyle\displaystyle+\\|\nabla^{2}F(\cdot,Y^{N}_{2})\\|_{\infty}\\|Y_{2}\\|_{p,[t,s]}(N^{1-\nu}\\|Y_{1}-Y_{2}\\|_{p,[t,s]})$
	$\displaystyle\displaystyle\leq N^{1-\nu}C_{F,R}\left(\\|Y_{1}-Y_{2}\\|_{\infty}\omega([t,s])^{\nu}\,+\,\\|Y_{1}-Y_{2}\\|_{p,[t,s]}\right)\omega([t,s])^{\nu}.$

	$\displaystyle\displaystyle\|\mathcal{R}^{F}_{t,s}(Y^{N}_{1})-\mathcal{R}^{F}_{t,s}(Y^{N}_{2})\|\leq C_{F,R}(\\|Y_{1}-Y_{2}\\|_{\infty}+\\|Y_{1}-Y_{2}\\|_{p,[t,s]})\omega([t,s])$
	$\displaystyle\displaystyle+\,C_{F}N^{1-\nu}\left(\\|Y_{1}-Y_{2}\\|_{\infty}\omega([t,s])^{\nu}\,+\,\\|Y_{1}-Y_{2}\\|_{p,[t,s]}\right)\omega([t,s])^{\nu}$
	$\displaystyle\displaystyle\leq C_{F,T,R}(\\|Y_{1}-Y_{2}\\|_{\infty}\omega([t,s])^{\nu}+\\|Y_{1}-Y_{2}\\|_{p,[t,s]})$
	$\displaystyle\displaystyle\times(\omega([t,s])^{1-\nu}+N^{1-\nu}\omega([t,s])^{\nu}).$		(13)

	$\displaystyle\displaystyle\\|\Xi^{\prime}\\|_{\infty}+\\|\Xi^{\prime}\\|_{r,[0,T_{0}]}\leq\|\Xi^{\prime}_{0}\|+2\\|\Xi^{\prime}\\|_{r,[0,T_{0}]}\leq$
	$\displaystyle\displaystyle\|\Xi^{\prime}_{0}\|+C_{M,{\sigma}}(T_{0}+\\|Y\\|_{r,[0,T_{0}]}\,+\,\\|Y^{\prime}\\|_{r,[0,T_{0}]})\leq$
	$\displaystyle\displaystyle\|\nabla\sigma(0,\xi)\sigma(0,\xi)\|+C_{M,{\sigma}}\left(1+\|Y^{\prime}_{0}\|+\\|(Y,Y^{\prime})\\|_{\kappa,\beta_{\kappa},\rho}\right)\leq C_{{}_{\xi,\sigma,M}},$		(25)

	$\displaystyle\displaystyle\left\\|\mathcal{M}_{T_{0}}(Y_{1},Y_{1}^{\prime})-\mathcal{M}_{T_{0}}(Y_{2},Y_{2}^{\prime})\right\\|_{p^{\prime},q_{p^{\prime}},X}=\left\\|\int_{0}^{\cdot}\Delta d\mathbf{X},\Delta\right\\|_{p^{\prime},q_{p^{\prime}},X}\leq$
	$\displaystyle\displaystyle\\|\Delta\\|_{p^{\prime},[0,T_{0}]}+C\left(\\|X\\|_{p^{\prime}}\\|R^{\Delta}\\|_{q_{p^{\prime}},[0,T_{0}]}+\\|\Delta^{\prime}\\|_{p^{\prime},[0,T_{0}]}\\|\mathbb{X}\\|_{p^{\prime}/2,[0,T_{0}]}\right)$
	$\displaystyle\displaystyle\leq\\|\Delta\\|_{p^{\prime}}+C_{\mathbf{X}}\left(\|\Delta^{\prime}_{0}\|+\\|(\Delta,\Delta^{\prime})\\|_{p^{\prime},q_{p^{\prime}},X}\right)$		(29)

	$\displaystyle\displaystyle\\|\Delta^{\prime}\\|_{p^{\prime},[0,T_{0}]}=\\|\nabla{\sigma}(\cdot,Y_{1})Y^{\prime}_{1}-\nabla{\sigma}(\cdot,Y_{2})Y^{\prime}_{2}\\|_{p^{\prime},[0,T_{0}]}$
	$\displaystyle\displaystyle\leq\\|\nabla\sigma(\cdot,Y_{1})\\|_{\infty}\\|Y^{\prime}_{1}-Y^{\prime}_{2}\\|_{p^{\prime},[0,T_{0}]}+\\|\nabla{\sigma}(\cdot,Y_{1})-\nabla{\sigma}(\cdot,Y_{2})\\|_{p^{\prime},[0,T_{0}]}\\|Y^{\prime}_{2}\\|_{\infty}$
	$\displaystyle\displaystyle\leq C_{\xi,\sigma,M,R}\Big{(}\\|Y_{1}-Y_{2}\\|_{\infty}^{\nu^{\prime}}\,+\,\\|Y^{\prime}_{1}-Y^{\prime}_{2}\\|_{p^{\prime},[0,T_{0}]}\,+\,\left\\|Y_{1}-Y_{2}\right\\|_{p^{\prime},[0,T_{0}]}^{{\nu^{\prime}}}\Big{)}$
	$\displaystyle\displaystyle\leq C_{\xi,\sigma,M,R}\,\Big{(}\|(Y_{1}-Y_{2})(0)\|^{\nu^{\prime}}+\left\\|Y_{1}-Y_{2},Y^{\prime}_{1}-Y^{\prime}_{2}\right\\|^{\nu^{\prime}}_{p^{\prime},q_{p^{\prime}},X}\Big{)}$		(31)

Rough differential equations with path-dependent coefficients

Abstract

1 Introduction

Outline

2 Rough paths and rough integration

Definition 2.1 (p-variation paths).

Definition 2.2 (Superadditive interval functions).

Proposition.

Definition 2.3 (Space of p\displaystyle p-rough paths).

Definition 2.4 (Controlled paths).

Theorem 2.5 (c.f. [14][Theorem 4.10], [18][Theorem 1]).

Proof.

3 Non-anticipative functionals of rough paths

3.1 Non-anticipative functionals

Definition 3.1 (Non-anticipative functional).

Definition 3.2.

Definition 3.3.

Definition 3.4.

Definition 3.5.

Assumption 3.1.

Assumption 3.2 (Uniformly Lipschitz continuity).

Assumption 3.3 (Horizontal Lipschitz continuity).

3.2 Actions of functionals on rough paths

Lemma 3.6.

Lemma 3.7.

Lemma 3.8 (Continuity of Control).

Proof.

Lemma (see [5], Proposition 5.26).

Corollary 3.9.

Proof.

Theorem 3.10 (Rough integral for functionals).

Proof.

Theorem 3.11.

Proof.

4 Path-dependent Differential Equations driven by rough paths

4.1 The setting of the problem

Definition 4.1.

Assumption 4.1.

Assumption 4.2.

Example 4.2.

Remark 4.3.

4.2 Proof of the main result

Theorem 4.4 (Existence of solutions).

Proof.

Proposition 4.5.

Property 1 (Invariance).

Property 2 (Continuity).

Remark 4.6.

Assumption 4.3.

Appendix: Proof of Proposition 4.5

Proof.

References

Rough differential equations with
path-dependent coefficients

Definition 2.3 (Space of $\displaystyle p$ -rough paths).