Restoration of Well-Posedness of Infinite-dimensional Singular ODE's via Noise

TL;DR

This paper extends the theory of stochastic differential equations with singular drifts to infinite-dimensional spaces by employing fractal noise and Malliavin calculus, establishing well-posedness without classical PDE or semimartingale methods.

Contribution

It introduces a novel approach using fractal noise and Malliavin calculus to prove well-posedness of infinite-dimensional SDEs with singular drifts, bypassing traditional techniques.

Findings

Established existence of unique strong solutions in infinite dimensions.

Demonstrated regularization effect of fractal noise.

Provided a new framework for analyzing infinite-dimensional SDEs.

Abstract

In this paper we aim at generalizing the results of A. K. Zvonkin and A. Y. Veretennikov on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-H\"{o}lder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

Equations

$$\frac{d}{dt}{X}_t^x=b(t,X_t^x),X_0=x\in \mathcal{H},t\in [0,T]$$

$${\Vert b(t,x)\Vert }_{\mathcal{H}}\le C_1(1+{\Vert x\Vert }_{\mathcal{H}})$$

$${\Vert b(t,x)-b(t,y)\Vert }_{\mathcal{H}}\le C_2{\Vert x-y\Vert }_{\mathcal{H}}$$

$$d{X}_t^x=b(t,X_t^x)dt+\epsilon d{B}_t,t\in [0,T],X_0^x=x\in {\mathbb{R}}^d,$$

$$d{X}_t=A{X}_{t}dt+b(X_t)dt+\sqrt{Q}d{W}_t,t\in [0,T],X_0=x\in \mathcal{H}\text{,}$$

$$X_t^x=x+{\int }_0^{t}b(t,X_s^x)ds+{\mathbb{B}}_t,t\in [0,T],$$

$${\mathbb{B}}_t=\sum _{k\ge 1}{\lambda }_k{B}_t^{H_k}{e}_k.$$

$$H_k\searrow 0$$

$$\text{framebox}Weakexistence+\text{frameboxPathwise uniqueness}\Rightarrow \text{frameboxStrong uniqueness.}$$

$$\text{framebox}Strongexistence+\text{frameboxUniqueness in law}\Rightarrow \text{frameboxStrong uniqueness.}$$

$$X_t^{d,\epsilon }=x+{\int }_0^t{b}^{d,\epsilon }(X_s^{d,\epsilon })ds+{\mathbb{B}}_t,t\in [0,T].$$

$$D_s{X}_t^{d,\epsilon }={\int }_s^t{\left(b^{d,\epsilon }\right)}^{\prime }(X_u^{d,\epsilon })D_s{X}_u^{d,\epsilon }du+\sum _{n\ge 1}{\lambda }_n{K}_{H_n}(t,s){⟨e_n,\cdot ⟩}_{\mathcal{H}}{e}_n,s<t,$$

$${\int }_{t<s_1<...<s_n<u}\kappa (s)D^{\alpha }f({\mathbb{B}}_s^d)ds$$

$$=(-1{)}^{|\alpha |}{\int }_{{\mathbb{R}}^{dn}}f(z)D^{\alpha }{L}_{\kappa }^n(t,z)dz,$$

$$X_t^{d,\epsilon }⟶X_t\text{ in }{L}^2(\Omega )$$

$$X_t^{(k)}=x^{(k)}+{\int }_0^t{b}_k(s,X_s)ds+{\mathbb{B}}_t^{(k)},t\in [0,T],k\ge 1,$$

$${\pi }_{d}x:=\sum _{k=1}^d{x}^{(k)}{e}_k.$$

$$\tau x=\tau \sum _{k\ge 1}{x}^{(k)}{e}_k=\sum _{k\ge 1}{x}^{(k)}{\tilde{e}}_k,$$

$$\Vert X{\Vert }_{L^2(\mathcal{X};\mathcal{Y})}^2={\int }_{\mathcal{X}}\Vert X(\omega ){\Vert }_{\mathcal{Y}}^2\mu (d\omega ).$$

$${\Delta }_{s,t}^n:=\{(u_1,\dots ,u_n)\in [0,T{]}^n:s<u_1<\cdots <u_n<t\}.$$

$${\Delta }_{s,t}^m\times {\Delta }_{s,t}^n=\bigcup _{\sigma \in \mathcal{S}(m,n)}\{(w_1,\dots ,w_{m+n})\in [0,T{]}^{m+n}:w_{\sigma }\in {\Delta }_{s,t}^{m+n}\}\cup \mathfrak{N},$$

$${\int }_{{\Delta }_{s,t}^n}\prod _{j=1}^n{f}_j(u_j)du:={\int }_s^t{\int }_{u_1}^t\cdots {\int }_{u_{n-1}}^t\prod _{j=1}^n{f}_j(u_j)d{u}_n\cdots d{u}_{2}d{u}_1.$$

$${\int }_{{\Delta }_{s,t}^m}\prod _{j=1}^m{f}_j(u_j)du{\int }_{{\Delta }_{s,t}^n}\prod _{j=m+1}^{m+n}{f}_j(u_j)du=\sum _{\sigma \in \mathcal{S}(m,n)}{\int }_{{\Delta }_{s,t}^{m+n}}\prod _{j=1}^{m+n}{f}_{\sigma (j)}(w_j)dw.$$

$$I_{a^{+}}^{\alpha }f(x)=\frac{1}{\Gamma (\alpha )}{\int }_a^x(x-y{)}^{\alpha -1}f(y)dy,$$

$$I_{b^{-}}^{\alpha }g(x)=\frac{1}{\Gamma (\alpha )}{\int }_x^b(y-x{)}^{\alpha -1}g(y)dy,$$

$$D_{a^{+}}^{\alpha }f(x)=\frac{1}{\Gamma (1-\alpha )}\frac{d}{dx}{\int }_a^x\frac{f(y)}{(x-y{)}^{\alpha }}dy,$$

$$D_{b^{-}}^{\alpha }g(x)=\frac{1}{\Gamma (1-\alpha )}\frac{d}{dx}{\int }_x^b\frac{g(y)}{(y-x{)}^{\alpha }}dy,$$

$$D_{a^{+}}^{\alpha }f(x)=\frac{1}{\Gamma (1-\alpha )}\left(\frac{f(x)}{(x-a{)}^{\alpha }}+\alpha {\int }_a^x\frac{f(x)-f(y)}{(x-y{)}^{\alpha +1}}dy\right),$$

$$D_{b^{-}}^{\alpha }g(x)=\frac{1}{\Gamma (1-\alpha )}\left(\frac{g(x)}{(b-x{)}^{\alpha }}+\alpha {\int }_x^b\frac{g(x)-g(y)}{(y-x{)}^{\alpha +1}}dy\right).$$

$$I_{a^{+}}^{\alpha }(D_{a^{+}}^{\alpha }f)=f,$$

$$D_{a^{+}}^{\alpha }(I_{a^{+}}^{\alpha }g)=g,$$

Full text

Restoration of Well-Posedness of Infinite-dimensional Singular ODE’s via Noise

David Baños

D. Baños: Department of Mathematics, University of Oslo, Moltke Moes vei 35, P.O. Box 1053 Blindern, 0316 Oslo, Norway.

davidru@math.uio.no

, 

Martin Bauer

M. Bauer: Department of Mathematics, LMU, Theresienstr. 39, D-80333 Munich, Germany.

bauer@math.lmu.de

, 

Thilo Meyer-Brandis

T. Meyer-Brandis: Department of Mathematics, LMU, Theresienstr. 39, D-80333 Munich, Germany.

meyerbra@math.lmu.de

 and 

Frank Proske

F. Proske: Department of Mathematics, University of Oslo, Moltke Moes vei 35, P.O. Box 1053 Blindern, 0316 Oslo, Norway.

proske@math.uio.no

††The research is partially supported by the FINEWSTOCH (NFR-ISP) project.

Abstract. In this paper we aim at generalizing the results of A. K. Zvonkin [41] and A. Y. Veretennikov [39] on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

Keywords. Malliavin calculus; fractional Brownian motion; $L^{2}$-compactness criterion; strong solutions of SDEs; irregular drift coefficient

1. Introduction

The main objective of this paper is the construction of (unique) strong solutions of infinite-dimensional stochastic differential equations (SDEs) with a singular drift and additive noise. In fact, we want to derive our results from the perspective of a rather recently established theory of stochastic regularization (see [19] and the references therein) with respect to a new general method based on Malliavin calculus and another variational technique which can be applied to different types of SDEs and stochastic partial differential equations (SPDEs).

In order to explain the concept of stochastic regularization, let us consider the first-order ordinary differential equation (ODE)

[TABLE]

for a vector field $b:[0,T]\times\mathcal{H}\rightarrow\mathcal{H}$, where $\mathcal{H}$ is a separable Hilbert space with norm $\left\|\cdot\right\|_{\mathcal{H}}$.

Using Picard iteration, it is fairly straight forward to see that the ODE (1) has a unique (global) solution $(X_{t}^{x})_{t\in[0,T]}$, if the driving vector field $b$ satisfies a linear growth and Lipschitz condition, that is

[TABLE]

and

[TABLE]

for all $t,x$ and $y$ with constants $C_{1},C_{2}<\infty$.

However, well-posedness in the sense of existence and uniqueness of solutions may fail, if the vector field $b$ lacks regularity, that is if e.g. $b$ is not Lipschitz continuous. In this case, the ODE (1) may not even admit the existence of a solution in the case $\mathcal{H}=\mathbb{R}^{d}$.

On the other hand, the situation changes, if one integrates on both sides of the ODE (1) and adds a "regularizing" noise to the right hand side of the resulting integral equation.

More precisely, if $\mathcal{H}=\mathbb{R}^{d}$, well-posedness of the ODE (1) can be restored via regularization by a Brownian (additive) noise, that is by a perturbation of the ODE (1) given by the SDE

[TABLE]

where $(B_{t})_{t\in[0,T]}$ is a Brownian motion in $\mathbb{R}^{d}$ and $\varepsilon>0$.

If the vector field $b$ is merely bounded and measurable, it turns out that the SDE (2) – regardless how small $\varepsilon$ is – possesses a unique (global) strong solution, that is a solution $(X_{t}^{x})_{t\in[0,T]}$, which as a process is a measurable functional of the driving noise $(B_{t})_{t\in[0,T]}$. This surprising and remarkable result was first obtained by A. K. Zvonkin [41] in the one-dimensional case, whose proof, using PDE techniques, is based on a transformation ("Zvonkin-transformation"), that converts the SDE (2) into a SDE without drift part. Subsequently, this result was generalized by A. Y. Veretennikov [39] to the multi-dimensional case. Much later, that is 35 years later, Zvonkin’s and Veretennikov’s results were extended by G. Da Prato, F. Flandoli, E. Priola and M. Röckner [13] to the infinite-dimensional setting by using estimates of solutions of Kolmogorov’s equation on Hilbert spaces. In fact, the latter authors study mild solutions $(X_{t})_{t\in[0,T]}$ to the SDE

[TABLE]

where $(W_{t})_{t\in[0,T]}$ is a cylindrical Brownian motion on $\mathcal{H}$, $A:D(A)\rightarrow\mathcal{H}$ a negative self-adjoint operator with compact resolvent, $Q:\mathcal{H}\rightarrow\mathcal{H}$ a non-negative definite self-adjoint bounded operator and $b:\mathcal{H}\rightarrow\mathcal{H}$. Here, the authors prove for $b\in L^{\infty}(\mathcal{H};\mathcal{H})$ under certain conditions on $A$ and $Q$ the existence of a unique mild solution, which is adapted to a completed filtration generated by $(W_{t})_{t\in[0,T]}$. So restoration of well-posedness of the ODE (1) with a singular vector field is established via regularization by both the cylindrical Brownian noise $(W_{t})_{t\in[0,T]}$ and $A$, which cannot be chosen to be the zero operator.

Other works in this direction in the infinite-dimensional setting based on different methods are e.g. A. S. Sznitman [38], A. Y. Pilipenko, M. V. Tantsyura [36] in connection with systems of McKean-Vlasov equations and G. Ritter, G. Leha [25] in the case of discontinuous drift vector fields of a rather specific form. We also refer to the references therein.

In this article, we aim at restoring well-posedness of singular ODE’s by using a certain non-Hölder continuous additive noise of fractal nature. More specifically, we want to analyze solutions to the following type of SDE:

[TABLE]

where the $\mathcal{H}-$valued regularizing noise $(\mathbb{B}_{t})_{t\in[0,T]}$ is a stationary Gaussian process with locally non-Hölder continuous paths given by

[TABLE]

Here $\{\lambda_{k}\}_{k\geq 1}\subset\mathbb{R}$, $\{e_{k}\}_{k\geq 1}$ is an orthonormal basis of $\mathcal{H}$ and $\{B_{\cdot}^{H_{k}}\}_{k\geq 1}$ are independent one-dimensional fractional Brownian motions with Hurst parameters $H_{k}\in(0,\frac{1}{2})$, $k\geq 1$, such that

[TABLE]

for $k\rightarrow\infty$.

Under certain (rather mild) growth conditions on the Fourier components $b_{k}$, $k\geq 1$, of the singular vector field $b:[0,T]\times\mathcal{H}\rightarrow\mathcal{H}$ (see (22) and (23)), which do not necessarily require that all $b_{k}$ are equal (compare e.g. to [38]), we show in this paper the existence of a unique (global) strong solution to the SDE (3) driven by the non-Markovian process $(\mathbb{B}_{t})_{t\in[0,T]}$.

Our approach for the construction of strong solutions to (3) relies on Malliavin calculus (see e.g. D. Nualart [32]) and another variational technique, which involves the use of spatial regularity of local time of finite-dimensional approximations of $\mathbb{B}_{t}$. In contrast to the above mentioned works (and most of other related works in the literature), the method in this paper is not based on PDE, Markov or semimartingale techniques. Furthermore, our technique corresponds to a construction principle, which is diametrically opposed to the commonly used Yamada-Watanabe principle (see e.g. [40]): Using the Yamada-Watanabe principle, one combines the existence of a weak solution to a SDE with pathwise uniqueness to obtain strong uniqueness of solutions. So

[TABLE]

This tool is in fact used by many authors in the literature. See e.g. the above mentioned authors or I. Gyöngy, T. Martinez [22], I. Gyöngy, N. V. Krylov [21], N. V. Krylov, M. Röckner [24] or S. Fang, T. S. Zhang [18], just to mention a few.

However, using our approach, verification of the existence of a strong solution, which is unique in law, provides strong uniqueness:

[TABLE]

See also H. J. Engelbert [17] in the finite-dimensional Brownian case regarding the latter construction principle.

In order to briefly explain our method in the case of time-homogeneous vector fields, we mention that we apply an infinite-dimensional generalization of a compactness criterion for square integrable Brownian functionals in $L^{2}(\Omega)$, which is originally due to G. Da Prato, P. Malliavin, and D. Nualart [32], to a double-sequence of strong solutions $\{(X_{t}^{d,\varepsilon})_{t\in[0,T]}\}_{d\geq 1,\varepsilon>0}$ associated with the following SDE’s

[TABLE]

Here $\{b^{d,\varepsilon}\}_{d\in\mathbb{N},\varepsilon>0}$ is an approximating double-sequence of vector fields of the singular drift $b$, which are smooth and live on $d-$dimensional subspaces of $\mathcal{H}$.

The application of the above mentioned compactness criterion (for each fixed $t$), however, requires certain (uniform) estimates with respect to the Malliavin derivative $D_{t}$ of $X_{t}^{d,\varepsilon}$ in the direction of a cylindrical Brownian motion. For this purpose, the Malliavin derivative $D_{\cdot}:\mathbb{D}^{1,2}(\mathcal{H})\longrightarrow L^{2}([0,T]\times\Omega)\otimes\mathcal{L}_{HS}(\mathcal{H},\mathcal{H})$ ($\mathbb{D}^{1,2}(\mathcal{H})$ is the space of $\mathcal{H}-$valued Malliavin differentiable random variables and $\mathcal{L}_{HS}(\mathcal{H},\mathcal{H})$ is the space of Hilbert-Schmidt operators from $\mathcal{H}$ to $\mathcal{H}$) in connection with a chain rule is applied to both sides of (4) and one obtains the following linear equation:

[TABLE]

where $\left(b^{d,\varepsilon}\right)^{\prime}$ is the derivative of $b^{d,\varepsilon}$, $\left\langle\cdot,\cdot\right\rangle_{\mathcal{H}}$ the inner product and $K_{H}$ a certain kernel function defined for Hurst parameters $H_{n}\in(0,\frac{1}{2})$.

We remark here that this type of linearization based on a stochastic derivative $D_{t}$ actually corresponds to the Nash-Moser principle, which is used for the construction of solutions of (non-linear) PDE’s by means of linearization of equations via classical derivatives. See e.g. J. Moser [31].

In a next step we then can derive a representation of $D_{s}X_{t}^{d,\varepsilon}$ (under a Girsanov change of measure) in (5) which is not based on derivatives of $b^{d,\varepsilon}$ by using Picard iteration and the following variational argument:

[TABLE]

where $\mathbb{B}_{s}^{d}:=(B_{s_{1}}^{H_{1}},...,B_{s_{1}}^{H_{d}},...,B_{s_{n}}^{H_{1}},...,B_{s_{n}}^{H_{d}})$ and $f:\mathbb{R}^{dn}\longrightarrow\mathbb{R}$ is a smooth function with compact support. Here $D^{\alpha}$ stands for a partial derivative of order $\left|\alpha\right|$ with respect a multi-index $\alpha$. Further, $L_{\kappa}^{n}(t,z)$ is a spatially differentiable local time of $\mathbb{B}_{\cdot}^{d}$ on a simplex scaled by non-negative integrable function $\kappa(s)=$ $\kappa_{1}(s)...\kappa_{n}(s)$.

Then, using the latter we can verify the required estimates for the Malliavin derivative of the approximating solutions in connection with the above mentioned compactness criterion and we finally obtain (under some additional arguments) that for each fixed $t$

[TABLE]

for $\varepsilon\searrow 0,d\longrightarrow\infty$, where $(X_{t})_{t\in[0,T]}$ is the unique strong solution to (3).

Finally, let us also mention a series of papers, from which our construction method gradually evolved: We refer to the works [27], [28], [29], [30] in the case of finite-dimensional Brownian noise. See [20] in the Hilbert space setting in connection with Hölder continuous drift vector fields. In the case of SDEs driven by Lévy processes we mention [23]. Other results can be found in [6], [1] with respect to SDEs driven by fractional Brownian motion and related noise. See also [7] in the case of "skew fractional Brownian motion", [5] with respect to singular delay equations and [8] in the case of Brownian motion driven mean-field equations.

We shall also point to the work of R. Catellier and M. Gubinelli [11], who prove existence and path by path uniqueness (in the sense of A. M. Davie [15]) of strong solutions of fractional Brownian motion driven SDEs with respect to (distributional) drift vector fields belonging to the Besov-Hölder space $B_{\infty,\infty}^{\alpha}$, $\alpha\in\mathbb{R}$. The approach of the authors is based inter alia on the theorem of Arzela-Ascoli and a comparison principle based on an average translation operator. In the distributional case, that is $\alpha<0$, the drift part of the SDE is given by a generalized non-linear Young integral defined via the topology of $B_{\infty,\infty}^{\alpha}$. See also D. Nualart, Y. Ouknine [33] in the one-dimensional case.

The structure of our article is as follows: In Section 2 we introduce the mathematical framework of this paper. Further, in Section 3 we discuss some properties of the process $\mathbb{B}_{\cdot}$ and weak solutions of the SDE (3). Section 4 is devoted to the construction of unique strong solutions to the SDE (3). Finally, in Section 5 examples of singular vector fields for which strong solutions exist are given.

Notation

For the sake of readability we assume throughout the paper that $1\leq T<\infty$ is a finite time horizon. We define $\mathcal{H}$ to be an infinite-dimensional separable real-valued Hilbert space with scalar product $\langle\cdot,\cdot\rangle_{\mathcal{H}}$ and orthonormal basis $\{e_{k}\}_{k\geq 1}$. Denote by $\|\cdot\|_{\mathcal{H}}$ the induced norm on $\mathcal{H}$ defined by $\|x\|_{\mathcal{H}}:=\langle x,x\rangle_{\mathcal{H}}^{\frac{1}{2}}$, $x\in\mathcal{H}$. For every $x\in\mathcal{H}$ and $k\geq 1$ we denote by $x^{(k)}:=\langle x,e_{k}\rangle_{\mathcal{H}}$ the projection onto the subspace spanned by $e_{k}$, $k\geq 1$. Loosely speaking we are referring to the subspace spanned by $e_{k}$, $k\geq 1$, as the $k$-th dimension. In line with this notation we denote the projection of the SDE (3) on the subspace spanned by $e_{k}$, $k\geq 1$, by $X^{(k)}:=\langle X,e_{k}\rangle_{\mathcal{H}}$. Moreover we can write the SDE (3) as an infinite dimensional system of real-valued stochastic differential equations, namely

[TABLE]

where $b_{k}$ and $\mathbb{B}^{(k)}$ are the projections on the subspace spanned by $e_{k}$, $k\geq 1$, of $b$ and $\mathbb{B}$, respectively. Note here that the function $b_{k}:[0,T]\times\mathcal{H}\to\mathbb{R}$ has still domain $[0,T]\times\mathcal{H}$. Furthermore, we define the truncation operator $\pi_{d}$, $d\geq 1$, which maps an element $x\in\mathcal{H}$ onto the first $d$ dimensions, by

[TABLE]

The truncated space $\pi_{d}\mathcal{H}$ is denoted by $\mathcal{H}_{d}$. We define the change of basis operator $\tau:\mathcal{H}\to\ell^{2}$ by

[TABLE]

where $\{\widetilde{e}_{k}\}_{k\geq 1}$ is an orthonormal basis of $\ell^{2}$. It is easily seen that the operator $\tau$ is a bijection and we denote its inverse by $\tau^{-1}:\ell^{2}\to\mathcal{H}$.

Further frequently used notation:

• •

Let $(\mathcal{X},\mathcal{A},\mu)$ denote a measurable space and $(\mathcal{Y},\|\cdot\|_{\mathcal{Y}})$ a normed space. Then $L^{2}(\mathcal{X};\mathcal{Y})$ denotes the space of square integrable functions $X$ over $\mathcal{X}$ taking values in $\mathcal{Y}$ and is endowed with the norm

[TABLE]

• •

The space $L^{2}(\Omega,\mathcal{F})$ denotes the space of square integrable random variables on the sample space $\Omega$ measurable with respect to the $\sigma$-algebra $\mathcal{F}$.

• •

We define $\mathbb{B}^{x}:=x+\mathbb{B}$.

• •

For any vector $u$ we denote its transposed by $u^{\top}$.

• •

We denote by $\operatorname*{Id}$ the identity operator.

• •

The Jacobian of a differentiable function is denoted by $\nabla$.

• •

For any multi-index $\alpha$ of length $d$ and any $d$-dimensional vector $u$ we define $u^{\alpha}:=\prod_{i=1}^{d}u_{i}^{\alpha_{i}}$.

• •

For two mathematical expressions $E_{1}(\theta),E_{2}(\theta)$ depending on some parameter $\theta$ we write $E_{1}(\theta)\lesssim E_{2}(\theta)$, if there exists a constant $C>0$ not depending on $\theta$ such that $E_{1}(\theta)\leq CE_{2}(\theta)$.

• •

Let $A$ be some countable set. Then we denote by $\#A$ its cardinality.

2. Preliminaries

2.1. Shuffles

Let $m$ and $n$ be two integers. We denote by $\mathcal{S}(m,n)$ the set of shuffle permutations, i.e. the set of permutations $\sigma:\{1,\dots,m+n\}\rightarrow\{1,\dots,m+n\}$ such that $\sigma(1)<\dots<\sigma(m)$ and $\sigma(m+1)<\dots<\sigma(m+n)$. Equivalently we denote for integers $k$ and $n$ by $\mathcal{S}(k;n)$ the set of shuffle permutations of $k$ sets of size $n$, i.e. the set of permutations $\sigma:\{1,\dots,k\cdot n\}\to\{1,\dots,k\cdot n\}$ such that $\sigma(m\cdot n+1)<\dots<\sigma((m+1)\cdot n)$ for all $m=0,\dots,k-1$. Furthermore the $n$-dimensional simplex $\Delta^{n}$ of the interval $(s,t)$ is defined by

[TABLE]

Note that the product of two simplices can be written as

[TABLE]

where the set $\mathfrak{N}$ has Lebesgue measure zero and $w_{\sigma}$ denotes the shuffled vector $(w_{\sigma(1)},\dots,w_{\sigma(m+n)})$. For the sake of readability we denote throughout the paper the integral over the simplex $\Delta_{s,t}^{n}$ of the product of integrable functions $f_{i}:[0,T]\rightarrow\mathbb{R}$, $i=1,\dots,n$, by

[TABLE]

Due to (8), we get for integrable functions $f_{i}:[0,T]\rightarrow\mathbb{R}$, $i=1,\dots,m+n$, that

[TABLE]

For a proof of a more general result we refer the reader to [6, Lemma 2.1].

2.2. Fractional Calculus

In the following we give some basic definitions and properties on fractional calculus. For more insights on the general theory we refer the reader to [34] and [37].

Let $a,b\in\mathbb{R}$ with $a<b$, $f,g\in L^{p}([a,b])$ with $p\geq 1$ and $\alpha>0$. We define the left- and right-sided Riemann-Liouville fractional integrals by

[TABLE]

and

[TABLE]

for almost all $x\in[a,b]$. Here $\Gamma$ denotes the gamma function.

Furthermore, for any given integer $p\geq 1$, let $I_{a^{+}}^{\alpha}(L^{p})$ and $I_{b^{-}}^{\alpha}(L^{p})$ denote the images of $L^{p}([a,b])$ by the operator $I_{a^{+}}^{\alpha}$ and $I_{b^{-}}^{\alpha}$, respectively. If $0<\alpha<1$ as well as $f\in I_{a^{+}}^{\alpha}(L^{p})$ and $g\in I_{b^{-}}^{\alpha}(L^{p})$, we define the left- and right-sided Riemann-Liouville fractional derivatives by

[TABLE]

and

[TABLE]

respectively. The left- and right-sided derivatives of $f$ and $g$ defined in (10) and (11) admit moreover the representations

[TABLE]

and

[TABLE]

Last, we get by construction that similar to the fundamental theorem of calculus

[TABLE]

for all $f\in I_{a^{+}}^{\alpha}(L^{p})$, and

[TABLE]

for all $g\in L^{p}([a,b])$. Equivalent results hold for $I_{b^{-}}^{\alpha}$ and $D_{b^{-}}^{\alpha}$.

2.3. Fractional Brownian motion

The one-dimensional fractional Brownian motion, in short fBm, $B^{H}=\left(B_{t}^{H}\right)_{t\in[0,T]}$ with Hurst parameter $H\in(0,\frac{1}{2})$ on a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$ is defined as a centered Gaussian process with covariance function

[TABLE]

Note that $\mathbb{E}\negthinspace\left[\left|B_{t}^{H}-B_{s}^{H}\right|^{2}\right]\negthinspace=|t-s|^{2H}$ and hence $B^{H}$ has stationary increments and almost surely Hölder continuous paths of order $H-\varepsilon$ for all $\varepsilon\in(0,H)$. However, the increments of $B^{H}$, $H\in(0,\frac{1}{2})$, are not independent and $B^{H}$ is not a semimartingale, see e.g. [32, Proposition 5.1.1].

Subsequently we give a brief outline of how a fractional Brownian motion can be constructed from a standard Brownian motion. For more details we refer the reader to [32].

Recall the following result (see [32, Proposition 5.1.3]) which gives the kernel of a fractional Brownian motion and an integral representation of $R_{H}(t,s)$ in the case of $H<\frac{1}{2}$.

Proposition 2.1

*Let $H<\frac{1}{2}$. The kernel *

[TABLE]

where $c_{H}=\sqrt{\frac{2H}{(1-2H)\beta\left(1-2H,H+\frac{1}{2}\right)}}$ and $\beta$ is the beta function, satisfies

[TABLE]

Subsequently, we denote by $W$ a standard Brownian motion on the complete filtered probability space $(\Omega,\mathcal{F},\mathbb{F}^{W},\mathbb{P})$, where $\mathbb{F}^{W}:=(\mathcal{F}_{t}^{W})_{t\in[0,T]}$ is the natural filtration of $W$ augmented by all $\mathbb{P}$-null sets. Using the kernel given in (14) it is well known that the fractional Brownian motion $B^{H}$ has a representation

[TABLE]

Note that due to representation (16) the natural filtration generated by $B^{H}$ is identical to $\mathbb{F}^{W}$. Furthermore, equivalent to the case of a standard Brownian motion, it exists a version of Girsanov’s theorem for fractional Brownian motion which is due to [16, Theorem 4.9]. In the following we state the version given in [33, Theorem 3.1].

But first let us define the isomorphism $K_{H}$ from $L^{2}([0,T])$ onto $I_{0+}^{H+\frac{1}{2}}(L^{2})$ (see [16, Theorem 2.1]) given by

[TABLE]

From (17) and the properties of the Riemann-Liouville fractional integrals and derivatives (12) and (13), the inverse of $K_{H}$ is given by

[TABLE]

It can be shown (see [33]) that if $\varphi$ is absolutely continuous

[TABLE]

where $\varphi^{\prime}$ denotes the weak derivative of $\varphi$.

Theorem 2.2** **(Girsanov’s theorem for fBm)

Let $u=\left(u_{t}\right)_{t\in[0,T]}$ be a process with integrable trajectories and set $\widetilde{B}_{t}^{H}=B_{t}^{H}+\int_{0}^{t}u_{s}ds,~{}t\in[0,T].$ Assume that

• (i)

$\int_{0}^{\cdot}u_{s}ds\in I_{0+}^{H+\frac{1}{2}}(L^{2}([0,T])$, $\mathbb{P}$-a.s., and

• (ii)

$\mathbb{E}\negthinspace\left[\mathcal{E}_{T}\right]\negthinspace=1$, where

[TABLE]

Then the shifted process $\widetilde{B}^{H}$ is an $\mathbb{F}^{W}$– fractional Brownian motion with Hurst parameter $H$ under the new probability measure $\widetilde{\mathbb{P}}$ defined by $\frac{d\widetilde{\mathbb{P}}}{d\mathbb{P}}=\mathcal{E}_{T}$.

Remark 2.3*.*

Theorem 2.2 can be extended to the multi- and infinite-dimensional cases, which will be considered in this paper primarily. Indeed, note first that the measure change in Girsanov’s theorem acts dimension-wise. In particular, consider the two dimensional shifted process

[TABLE]

where $B^{H_{1}}$ and $B^{H_{2}}$ are two fractional Brownian motions with Hurst parameters $H_{1}$ and $H_{2}$ generated by the independent standard Brownian motions $W^{(1)}$ and $W^{(2)}$, respectively, and $u^{(1)}$ and $u^{(2)}$ are two shifts fulfilling the conditions of Theorem 2.2. Then the measure change with respect to the stochastic exponential

[TABLE]

yields the two dimensional process

[TABLE]

Here, $\widetilde{B}^{H_{1}}$ is a fractional Brownian motions with respect to the measure $\widetilde{\mathbb{P}}$ defined by $\frac{d\widetilde{\mathbb{P}}}{d\mathbb{P}}=\mathcal{E}_{T}^{(1)}$. Note that $B^{H_{2}}$ is still a fractional Brownian motion under $\widetilde{\mathbb{P}}$, since $W^{(1)}$ and $W^{(2)}$ are independent. Applying Girsanov’s theorem again with respect to the stochastic exponential

[TABLE]

yields the two dimensional process

[TABLE]

where $\widetilde{B}^{H_{1}}$ and $\widetilde{B}^{H_{2}}$ are independent fractional Brownian motions with respect to the measure $\hat{\mathbb{P}}$ defined by

[TABLE]

Repeating iteratively yields the stochastic exponential – if well-defined –

[TABLE]

acting on infinite dimensions.

Finally, we give the property of strong local non-determinism of the fractional Brownian motion $B^{H}$ with Hurst parameter $H\in(0,\frac{1}{2})$ which was proven in [35, Lemma 7.1]. This property will essentially help us to overcome the limitations of not having independent increments of the underlying noise.

Lemma 2.4

Let $B^{H}$ be a fractional Brownian motion with Hurst parameter $H\in(0,\frac{1}{2})$. Then there exists a constant $\mathfrak{K}_{H}$ dependent merely on $H$ such that for every $t\in[0,T]$ and $0<r\leq t$

[TABLE]

3. Cylindrical fractional Brownian motion and weak solutions

We start this section by defining the driving noise $(\mathbb{B}_{t})_{t\in[0,T]}$ in SDE (3). Let $\{W^{(k)}\}_{k\geq 1}$ be a sequence of independent one-dimensional standard Brownian motions on a joint complete probability space $(\Omega,\mathcal{F},\mathbb{P})$. We define the cylindrical Brownian motion $W$ taking values in $\mathcal{H}$ by

[TABLE]

and denote by $\mathbb{F}^{W}:=\left(\mathcal{F}_{t}^{W}\right)_{t\in[0,T]}$ its natural filtration augmented by the $\mathbb{P}$-null sets. Moreover, we define a sequence of Hurst parameters $H:=\{H_{k}\}_{k\geq 1}\subset\left(0,\frac{1}{2}\right)$ with the following properties:

1. (i)

$\sum_{k\geq 1}H_{k}<\frac{1}{6}$ 2. (ii)

$\sup_{k\geq 1}H_{k}<\frac{1}{12}$

Using $H$ we construct the sequence of fractional Brownian motions $\{B^{H_{k}}\}_{k\geq 1}$ associated to $\{W^{(k)}\}_{k\geq 1}$ by

[TABLE]

where the kernel $K_{H_{k}}(\cdot,\cdot)$ is defined as in (14). Note that the fractional Brownian motions $\{B^{H_{k}}\}_{k\geq 1}$ are independent by construction. Consequently, we define the cylindrical fractional Brownian motion $B^{H}$ with associated sequence of Hurst parameters $H$ by

[TABLE]

Nevertheless, the cylindrical fractional Brownian motion $B^{H}$ is not in the space $L^{2}(\Omega;\mathcal{H})$. That is why we consider the operator $Q:\mathcal{H}\to\mathcal{H}$ defined by

[TABLE]

for a given sequence of non-negative real numbers $\lambda:=\{\lambda_{k}\}_{k\geq 1}\in\ell^{2}$ such that $\frac{\lambda}{\sqrt{H}}:=\left\{\frac{\lambda_{k}}{\sqrt{H_{k}}}\right\}_{k\geq 1}\in\ell^{1}$. In particular, $Q$ is a self-adjoint operator and we have that the weighted cylindrical fractional Brownian motion

[TABLE]

lies in $L^{2}(\Omega;\mathcal{H})$ for every $t\in[0,T]$. Due to the following lemma the stochastic process $(\mathbb{B}_{t})_{t\in[0,T]}$ is continuous in time.

Lemma 3.1

The stochastic process $(\mathbb{B}_{t})_{t\in[0,T]}$ defined in (20) has almost surely continuous sample paths on $[0,T]$.

Proof.

Note first that due to [10][Theorem 1] for any fractional Brownian motion $B^{H}$ with Hurst parameter $H\in(0,\frac{1}{2})$ there exists a constant $C>0$ independent of $H$ such that

[TABLE]

Using monotone convergence and (21) we have that

[TABLE]

Thus, $(\sqrt{Q}B_{t}^{H})_{t\in[0,T]}$ is almost surely finite and $\{(\pi_{d}\sqrt{Q}B_{t}^{H})_{t\in[0,T]}\}_{d\geq 1}$ is a Cauchy sequence in $L^{1}(\Omega;\mathcal{C}([0,T];\mathcal{H}))$ which converges almost surely to $(\sqrt{Q}B_{t}^{H})_{t\in[0,T]}$. ∎

Before we come to the next result, let us recall the notion of a weak solution and uniqueness in law.

Definition 3.2

The sextuple $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P},\mathbb{B},X)$ is called a weak solution of stochastic differential equation (3), if

1. (i)

$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ is a complete filtered probability space, where $\mathbb{F}=\{\mathcal{F}_{t}\}_{t\in[0,T]}$ satisfies the usual conditions of right-continuity and completeness, 2. (ii)

$\mathbb{B}=(\mathbb{B}_{t})_{t\in[0,T]}$ is a weighted cylindrical fractional $(\mathbb{F},\mathbb{P})$-Brownian motion as defined in (20), and 3. (iii)

$X=(X_{t})_{t\in[0,T]}$ is a continuous, $\mathbb{F}$-adapted, $\mathcal{H}$-valued process satisfying $\mathbb{P}$-a.s.

[TABLE]

Remark 3.3*.*

For notational simplicity we refer solely to the process $X$ as a weak solution (or later on as a strong solution) in the case of an unambiguous stochastic basis $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P},B)$.

Definition 3.4

We say a weak solution $X^{1}$ with respect to the stochastic basis $(\Omega^{1},\mathcal{F}^{1},\mathbb{F}^{1},\mathbb{P}^{1},\mathbb{B}^{1})$ of the SDE (3) is weakly unique or unique in law, if for any other weak solution $X^{2}$ of (3) on a potential other stochastic basis $(\Omega^{2},\mathcal{F}^{2},\mathbb{F}^{2},\mathbb{P}^{2},\mathbb{B}^{2})$ it holds that

[TABLE]

whenever $\mathbb{P}^{1}_{X_{0}^{1}}=\mathbb{P}^{2}_{X_{0}^{2}}$.

Proposition 3.5

Let $b:[0,T]\times\mathcal{H}\to\mathcal{H}$ be a measurable and bounded function with $\left\|b_{k}\right\|_{\infty}\leq C_{k}\lambda_{k}<\infty$ for every $k\geq 1$ where $C:=\left\{C_{k}\right\}_{k\geq 1}\in\ell^{1}$. Then SDE (3) has a weak solution $(X_{t})_{t\in[0,T]}$ such that

[TABLE]

Moreover, the solution is unique in law.

Proof.

Let $\{W^{(k)}\}_{k\geq 1}$ be a sequence of independent standard Brownian motions on the filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{Q})$. Consider the cylindrical fractional Brownian motion $\widehat{B}^{H}$ generated by $\{W^{(k)}\}_{k\geq 1}$ as defined in (19) with associated sequence of Hurst parameters $H$. We define the stochastic exponential $\mathcal{E}$ by

[TABLE]

In order to show that the stochastic exponential $\mathcal{E}$ is well-defined we first have to verify that for every $k\geq 1$

[TABLE]

Due to (18) this property is fulfilled, if for all $k\geq 1$

[TABLE]

which holds since $\|b_{k}\|_{\infty}\leq C_{k}\lambda_{k}$. Furthermore, we can find a constant $C>0$ such that

[TABLE]

Hence, by Novikov’s criterion $\mathcal{E}_{t}$ is a martingale, in particular $\mathbb{E}\negthinspace\left[\mathcal{E}_{t}\right]\negthinspace=1$ for all $t\in[0,T]$. Consequently, under the probability measure $\mathbb{P}$, defined by $\frac{d\mathbb{P}}{d\mathbb{Q}}:=\mathcal{E}_{T}$, the process $B_{t}^{H}:=\widehat{B}_{t}^{H}-\int_{0}^{t}\sqrt{Q}^{-1}b\left(u,x+\sqrt{Q}\widehat{B}^{H}_{u}\right)du$, $t\in[0,T]$, is a cylindrical fractional Brownian motion due to Theorem 2.2 and Remark 2.3. Therefore, ${(\Omega,\mathcal{F},\mathbb{F},\mathbb{P},\sqrt{Q}B^{H},X)}$, where $X_{t}:=x+\sqrt{Q}\widehat{B}_{t}^{H}$, is a weak solution of SDE (3). Since the probability measures $\mathbb{Q}\approx\mathbb{P}$ are equivalent, the solution is unique in law. ∎

4. Strong Solutions and Malliavin Derivative

After establishing the existence of a weak solution, we investigate under which conditions SDE (3) has a strong solution. Therefore, let us first recall the notion of a strong solution and moreover the notion of pathwise uniqueness.

Definition 4.1

A weak solution $(\Omega,\mathcal{F},\mathbb{F}^{\mathbb{B}},\mathbb{P},\mathbb{B},X^{x})$ of the stochastic differential equation (3) is called strong solution, if $\mathbb{F}^{\mathbb{B}}$ is the filtration generated by the driving noise $\mathbb{B}$ and augmented with the $\mathbb{P}$-null sets.

Definition 4.2

We say a weak solution $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P},\mathbb{B},X^{1})$ of (3) is pathwise unique, if for any other weak solution $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P},\mathbb{B},X^{2})$ on the same stochastic basis,

[TABLE]

The cause of this paper is to establish the existence of strong solutions of stochastic differential equation (3) for singular drift coefficients $b$. More precisely, we define the class $\mathfrak{B}([0,T]\times\mathcal{H};\mathcal{H})$ of measurable functions $b:[0,T]\times\mathcal{H}\to\mathcal{H}$ for which there exist sequences $C\in\ell^{1}$ and $D\in\ell^{1}$ such that for every $k\geq 1$

[TABLE]

where $y=(y_{1},\dots,y_{d})$ and $\mathcal{K}:\mathcal{H}\to\mathcal{H}$ is the defined by

[TABLE]

for $\{\mathfrak{K}_{H_{k}}\}_{k\geq 1}$ being the local non-determinism constant of $\{B^{H_{k}}\}_{k\geq 1}$ as given in Lemma 2.4.

In order to prove the existence of a strong solution for drift coefficients of class $\mathfrak{B}([0,T]\times\mathcal{H};\mathcal{H})$ we proceed in the following way:

We define an approximating double-sequence $\{b^{d,\varepsilon}\}_{d\geq 1,\varepsilon>0}$ for drift coefficients of type (22) which merely act on $d$ dimensions and are sufficiently smooth 2. 2)

For every $d\geq 1$ and $\varepsilon>0$, we prove that the SDE

[TABLE]

has a unique strong solution which is Malliavin differentiable 3. 3)

We show that the double-sequence of strong solutions $X_{t}^{d,\varepsilon}$ converges weakly to $\mathbb{E}\left[X_{t}|\mathcal{F}_{t}^{W}\right]$, where $X_{t}$ is the unique weak solution of SDE (3) 4. 4)

Applying a compactness criterion based on Malliavin calculus, we prove that the double-sequence is relatively compact in $L^{2}(\Omega,\mathcal{F}_{t}^{W})$ 5. 5)

Last, we show that $X_{t}$ is adapted to the filtration $\mathbb{F}^{\mathbb{B}}$ and thus is a strong solution of SDE (3)

4.1. Approximating double-sequence

Recall the truncation operator $\pi_{d}$, $d\geq 1$, defined in (6) and the change of basis operator $\tau$ defined in (7). We define the operator $\widetilde{\pi}_{d}:\mathcal{H}\to\mathbb{R}^{d}$ as $\widetilde{\pi}_{d}:=\tau\circ\pi_{d}$. For every $k\geq 1$ let the function $\widetilde{b}^{d}:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ be defined by

[TABLE]

Let $\varphi_{\varepsilon}$, $\varepsilon>0$, be a mollifier on $\mathbb{R}^{d}$ such that for any locally integrable function $f:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ and for every $t\in[0,T]$ the convolution $f(t,\cdot)\ast\varphi_{\varepsilon}$ is smooth and

[TABLE]

almost everywhere with respect to the Lebesgue measure. Finally, we define for every $d\geq 1$ and $\varepsilon>0$ the double-sequence $b^{d,\varepsilon}:[0,T]\times\mathcal{H}\to\mathcal{H}$ by

[TABLE]

Analogously to (25), we define for $t\in[0,T]$ and $z\in\mathbb{R}^{d}$

[TABLE]

Due to the definition of the mollifier $\varphi_{\varepsilon}$ we have that for every $d\geq 1$

[TABLE]

for almost every $(t,z)\in[0,T]\times\mathbb{R}^{d}$ with respect to the Lebesgue measure. Thus, due to (28) and the canonical properties of the truncation operator we have that

[TABLE]

pointwise in $[0,T]\times\mathcal{H}$, where $b^{d}:=\pi_{d}b$. Due to the assumptions on $b$ we further get for every $p\geq 2$ using dominated convergence that

[TABLE]

Hence, we can speak of an approximating double-sequence $\{b^{d,\varepsilon}\}_{d\geq 1,\varepsilon>0}$ of the drift coefficient $b$. In line with the previously used notation we define

[TABLE]

Moreover, note that $b^{d,\varepsilon},b^{d}\in\mathfrak{B}([0,T]\times\mathcal{H};\mathcal{H})$.

Remark 4.3*.*

Note that we needed to truncate and shift the domain of the function $b$ to $\mathbb{R}^{d}$ merely in order to apply mollification.

4.2. Malliavin differentiable strong solutions for regular drifts

In the following proposition we establish the existence of a unique strong solution for a class of drift coefficients which contains the approximating sequence $\{b^{d,\varepsilon}\}_{d\geq 1,\varepsilon>0}$. More specifically, we consider drift coefficients $b\in\mathfrak{B}([0,T]\times\mathcal{H};\mathcal{H})$ such that for all $k\geq 1$ and all $t\in[0,T]$

[TABLE]

where $L\in\ell^{2}$. We denote the space of such functions by $\mathfrak{L}([0,T]\times\mathcal{H};\mathcal{H})$.

Proposition 4.4

Let $b\in\mathfrak{L}([0,T]\times\mathcal{H};\mathcal{H})$. Then SDE (3) has a pathwise unique strong solution.

Proof.

In order to prove the existence of a strong solution we use Picard iteration and proceed similar to the well-known case of finite dimensional SDEs. More precisely, we define inductively the sequence $Y^{0}:=x+\mathbb{B}$ and for all $n\geq 1$

[TABLE]

We show next that $\{Y^{n}\}_{n\geq 0}$ is a Cauchy sequence in $L^{2}([0,T]\times\Omega)$. Indeed, due to monotone convergence we get for every $n\geq 1$ and $t\in[0,T]$

[TABLE]

and

[TABLE]

By induction we obtain for every $n\geq 0$ a constant $A$ depending on $C$, $\lambda$ and $L$ such that

[TABLE]

Hence, for every $m,n\geq 0$

[TABLE]

Since $B(n,m)$ is bounded by $T^{\frac{1}{2}}e^{AT}$, the series converges and

[TABLE]

Therefore $\{Y^{n}\}_{n\geq 0}$ is a Cauchy sequence in $L^{2}([0,T]\times\Omega;\mathcal{H})$. Define

[TABLE]

as the $L^{2}([0,T]\times\Omega;\mathcal{H})$ limit of $\{Y^{n}\}_{n\geq 0}$. Then $X_{t}$ is $\mathcal{F}^{\mathbb{B}}_{t}$ adapted for all $t\in[0,T]$ since this holds for all $Y_{t}^{n}$, $n\geq 0$. We prove that $X_{t}$ solves SDE (3):

We have for all $n\geq 0$ and $t\in[0,T]$ that

[TABLE]

Using the Lipschitz continuity of $b$, we get

[TABLE]

Hence, $(X_{t})_{t\in[0,T]}$ is a strong solution of SDE (3).

In order to show pathwise uniqueness, let $X$ and $Y$ be two strong solutions on the same stochastic basis $(\Omega,\mathcal{F},\mathbb{P},\mathbb{B})$ with the same initial condition. Then for all $t\in[0,T]$ we get similar to (30) that

[TABLE]

Using Grönwall’s inequality yields that $\mathbb{E}\negthinspace\left[\left\|X_{t}-Y_{t}\right\|_{\mathcal{H}}^{2}\right]\negthinspace=0$ for all $t\in[0,T]$, and therefore $X_{t}=Y_{t}$ $\mathbb{P}$-a.s. for all $t\in[0,T]$. But since $X$ and $Y$ are almost surely continuous we get

[TABLE]

∎

Next we investigate under which conditions the unique strong solution is Malliavin differentiable. But let us start with a definition of Malliavin differentiability of a random variable in the space $\mathcal{H}$.

Definition 4.5

Let $X$ be an $\mathcal{H}$-valued square integrable functional of the cylindrical Brownian motion $(W_{t})_{t\in[0,T]}$. We define the operator $D^{m}$, $m\geq 1$, such that

[TABLE]

as the Malliavin derivative in the direction of the $m$-th Brownian motion $W^{(m)}$. Here, $D^{m}X^{(k)}$, $m,k\geq 1$, is the (standard) Malliavin derivative with respect to the Brownian motion $W^{(m)}$ of the square integrable random variable $X^{(k)}$ taking values in $\mathbb{R}$. We say a random variable $X$ with values in $\mathcal{H}$ is in the space $\mathbb{D}^{1,2}(\mathcal{H})$ of Malliavin differentiable functions in $L^{2}(\Omega)$ if and only if

[TABLE]

Moreover, a stochastic process $(X_{t})_{t\in[0,T]}$ with values in $\mathcal{H}$ is said to be in the space $\mathbb{D}^{1,2}([0,T]\times\mathcal{H})$ if and only if for every $t\in[0,T]$

[TABLE]

By means of Definition 4.5 we extend the well-known chain rule in Malliavin Calculus, cf. [32, Proposition 1.2.4], to Malliavin differentiable random variables taking values in $\mathcal{H}$. But first we define the class $\mathcal{L}_{0}(\mathcal{H})$ of Lipschitz continuous functions on $\mathcal{H}$ with vanishing Lipschitz constants.

We say a function $f:\mathcal{H}\to\mathcal{H}$ is in the space $\mathcal{L}_{0}(\mathcal{H})$ if there exist sequences of constants $L,M\in\ell^{2}$ such that for all $k\geq 1$ and $x,y\in\mathcal{H}$

[TABLE]

Lemma 4.6

Let $f\in\mathcal{L}_{0}(\mathcal{H})$ with associated Lipschitz sequences $L,M\in\ell^{2}$ and $Y\in\mathbb{D}^{1,2}(\mathcal{H})$. Then, $f(Y)\in\mathbb{D}^{1,2}(\mathcal{H})$ and there exists a double-sequence $\{G_{i}^{(k)}\}_{k,i\geq 1}$ of random variables with $G_{i}^{(k)}\leq L_{k}\cdot M_{i}$ $\mathbb{P}$-a.s. for all $k,i\geq 1$ such that for every $m\geq 1$

[TABLE]

Moreover,

[TABLE]

Proof.

First, consider the case $f:\mathbb{R}^{d}\to\mathbb{R}^{d}$ for some $d\geq 1$, where $Y$ is taking values in $\mathbb{R}^{d}$. Using the chain rule, see [32, Proposition 1.2.4], and the notion of Malliavin Differentiability in Definition 4.5, there exists a double-sequence $\{G_{i}^{(k)}\}_{1\leq k,i\leq d}$ of random variables with $G_{i}^{(k)}\leq L_{k}\cdot M_{i}$ $\mathbb{P}$-a.s. for all $1\leq k,i\leq d$ such that for every $m\geq 1$

[TABLE]

Recall the change of basis operator $\tau:\mathcal{H}\to\ell^{2}$ defined in (7). Let now $f:\mathcal{H}_{d}\to\mathcal{H}_{d}$, where $Y$ is taking values in $\mathcal{H}_{d}$. Define $g:\mathbb{R}^{d}\to\mathbb{R}^{d}$ by $g:=\tau\circ f\circ\tau^{-1}$. Then $g$ is Lipschitz continuous in the sense of (31) with associated Lipschitz sequences $L,M\in\ell^{2}$ and due to equality (33) we get the identity

[TABLE]

Thus, equation (32) holds for $f:\mathcal{H}_{d}\to\mathcal{H}_{d}$. Let finally $f:\mathcal{H}\to\mathcal{H}$, where $Y$ is taking values in $\mathcal{H}$. Recall the truncation operator $\pi_{d}:\mathcal{H}\to\mathcal{H}_{d}$ defined in (6). Since $f$ is Lipschitz continuous, $f(\pi_{d}Y)$ converges to $f(Y)$ in $L^{2}(\Omega)$. Furthermore, we have for every $d\geq 1$ that

[TABLE]

Note that the double-sequence $\{G_{i}^{d,(k)}\}_{i\geq 1,k\geq 1}$ depends on $d\geq 1$. Nevertheless, $\left\|\pi_{d}f(\pi_{d}Y)\right\|_{\mathbb{D}^{1,2}(\mathcal{H})}$ is uniformly bounded in $d\geq 1$. Thus, due to [32, Lemma 1.2.3] and dominated convergence we have $f(Y)\in\mathbb{D}^{1,2}(\mathcal{H})$ and $D^{m}(\pi_{d}f(\pi_{d}Y))$ converges weakly to $D^{m}f(Y)$ for every $m\geq 1$. Moreover, the sequence $\{G_{i}^{d,(k)}\}_{d\geq 1}$ is bounded by $L_{k}\cdot M_{i}$ for every $k,i\geq 1$. Hence, for every $k,i\geq 1$ there exists a subsequence $\{G_{i}^{d_{n},(k)}\}_{n\geq 1}$ which converges weakly to some random variable $\widetilde{G}_{i}^{(k)}$ which is bounded by $L_{k}\cdot M_{i}$. Summarizing we get that in $L^{2}([0,T]\times\Omega;\mathcal{H})$

[TABLE]

where the last equality holds due to (34) and dominated convergence. ∎

Define the class $\mathfrak{L}_{0}([0,T]\times\mathcal{H};\mathcal{H})$ by

[TABLE]

and note that $f(t,\cdot)\in\mathcal{L}_{0}(\mathcal{H})$ uniformly in $t\in[0,T]$ implies $f_{k}(t,\cdot)\in\text{Lip}_{L_{k}}(\mathcal{H};\mathbb{R})$, $k\geq 1$, uniformly in $t\in[0,T]$ for some sequence $L\in\ell^{2}$. Thus, $\mathfrak{L}_{0}([0,T]\times\mathcal{H};\mathcal{H})\subset\mathfrak{L}([0,T]\times\mathcal{H};\mathcal{H})$.

Proposition 4.7

Let $b\in\mathfrak{L}_{0}([0,T]\times\mathcal{H};\mathcal{H})$. Then the unique strong solution $(X_{t})_{t\in[0,T]}$ of (3) is Malliavin differentiable.

Proof.

Recall the Picard iteration defined in (29)

[TABLE]

and $Y^{0}=x+\mathbb{B}$. We denote the $k$-th dimension of the infinite dimensional system (35) by $Y^{n,(k)}:=\langle Y^{n},e_{k}\rangle_{\mathcal{H}}$.

Using the Picard iteration (35), we show that for every step $n\geq 0$ the process $Y^{n}$ is Malliavin differentiable. We prove this using induction. For $n=0$ we have that for all $t\in[0,T]$ using (15)

[TABLE]

Now suppose that $\left\|Y_{t}^{n}\right\|_{\mathbb{D}^{1,2}(\mathcal{H})}<\infty$ for $n\geq 0$. Due to Lemma 4.6 $b(t,Y_{t}^{n})$ is in $\mathbb{D}^{1,2}(\mathcal{H})$ and we have for every $t\in[0,T]$ that

[TABLE]

for some $L,M\in\ell^{2}$ independent of $n\geq 0$. Moreover, $\int_{0}^{T}b(r,Y_{r}^{n})dr$ is Malliavin differentiable admitting for all $0\leq s\leq T$ the representation

[TABLE]

Thus, we get for $Y^{n+1}$ that

[TABLE]

Hence, $Y^{n+1}$ is Malliavin differentiable in the sense of Definition 4.5. Moreover, we can find a positive constant $A$ depending on $L,M,\lambda$ and $T$ such that

[TABLE]

Consequently, $\left\|Y_{t}^{n}\right\|_{\mathbb{D}^{1,2}(\mathcal{H})}^{2}$ is uniformly bounded in $n\geq 0$ and therefore, since $Y^{n}\rightarrow X$ in $L^{2}([0,T]\times\Omega)$ and the Malliavin derivative is a closable operator, also $X$ is Malliavin differentiable in the sense of Definition 4.5. ∎

Let us finally put the previous results together and show that SDE (24) has a unique Malliavin differentiable strong solution.

Corollary 4.8

Let $b^{d,\varepsilon}:[0,T]\times\mathcal{H}\to\mathcal{H}$ be defined as in (26). Then, SDE (24) has a unique strong solution $\left(X_{t}^{d,\varepsilon}\right)_{t\in[0,T]}$ which is Malliavin differentiable. Furthermore, the Malliavin derivative $D_{s}^{m}X_{t}^{d,\varepsilon}$ has for $0\leq s<t\leq T$ a.s. the representation

[TABLE]

where $\eta_{n}=m$ and $\widetilde{b}^{d,\varepsilon}:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ is defined as in (27).

Proof.

If the drift function $b^{d,\varepsilon}$ is in the class $\mathfrak{L}_{0}([0,T]\times\mathcal{H},\mathcal{H})$, then SDE (24) has a unique Malliavin differentiable strong solution by Proposition 4.4 and Proposition 4.7. Thus we merely need to show that $b^{d,\varepsilon}(t,\cdot)\in\mathcal{L}_{0}(\mathcal{H})$ uniformly in $t\in[0,T]$. Let $t\in[0,T]$ and $y,z\in\mathcal{H}$. Then, using the triangular inequality and the mean-value theorem we get for all $1\leq k\leq d$ that

[TABLE]

Note that we can find sequences $\{L_{k}\}_{1\leq k\leq d}$ and $\{M_{i}\}_{1\leq i\leq d}$ such that for all $1\leq k,i\leq d$ we have $\sup_{\xi\in\mathbb{R}^{d}}|\partial_{i}\widetilde{b}_{k}^{d,\varepsilon}(t,\xi)|\leq L_{k}\cdot M_{i}$. Hence, $b^{d,\varepsilon}\in\mathfrak{L}_{0}([0,T]\times\mathcal{H};\mathcal{H})$.

It is left to show that representation (36) holds. First note that due to the definition of the Malliavin derivative of a random variable $Y$ with values in $\mathcal{H}$, see Definition 4.5, we have that $D^{m}(\tau Y)=\tau D^{m}Y$, for all $m\geq 1$. Consequently, we get for $0\leq s<t\leq T$ using Lemma 4.6 that the Malliavin derivative $D_{s}^{m}X_{t}^{d,\varepsilon}$ can be written as

[TABLE]

Iterating this step yields

[TABLE]

Further note that

[TABLE]

Thus, we get for every $n\geq 1$

[TABLE]

where $\eta_{0}=k$ and $\eta_{n}=l$ and consequently, representation (36) holds. ∎

4.3. Weak convergence

In this step we show that the sequence of unique strong solutions $\{X^{d,\varepsilon}\}_{d\geq 1,\varepsilon>0}$ of the approximating SDEs (24) converge weakly to the weak solution of (3) where $b\in\mathfrak{B}([0,T]\times\mathcal{H};\mathcal{H})$.

Lemma 4.9

Let $b\in\mathfrak{B}([0,T]\times\mathcal{H};\mathcal{H})$. Furthermore, let $(X_{t})_{t\in[0,T]}$ be the weak solution of (3). Consider the approximating sequence of strong solutions $\{(X_{t}^{d,\varepsilon})_{t\in[0,T]}\}_{d\geq 1,\varepsilon>0}$ of SDEs (24), where $b^{d,\varepsilon}:[0,T]\times\mathcal{H}\to\mathcal{H}$ is defined as in (26). Then, for every $t\in[0,T]$ and for any bounded continuous function $\phi:\mathcal{H}\to\mathbb{R}$

[TABLE]

weakly in $L^{2}(\Omega,\mathcal{F}_{t}^{W})$.

Proof.

Using the Wiener transform

[TABLE]

of some random variable $Z\in L^{2}(\Omega,\mathcal{F}_{T}^{W})$ in $f\in L^{2}([0,T];\mathcal{H})$, it suffices to show for any arbitrary $f\in L^{2}([0,T];\mathcal{H})$ that

[TABLE]

So, let $f\in L^{2}([0,T];\mathcal{H})$ be arbitrary, then by using Girsanov’s theorem we get

[TABLE]

Using the inequality

[TABLE]

we get

[TABLE]

where

[TABLE]

For every $k\geq 1$, we get with representation (18) that

[TABLE]

which is bounded by

[TABLE]

Consequently, we get for every $d\geq 1$ using the Burkholder-Davis-Gundy inequality that

[TABLE]

Hence, by dominated convergence

[TABLE]

Equivalently, we have

[TABLE]

Thus, again by dominated convergence

[TABLE]

Similarly, one can show that $A_{d,\varepsilon}(f)$ vanishes for every $f\in L^{2}([0,T];\mathcal{H})$ as $\varepsilon\to 0$ and $d\to\infty$. Consequently, \phi(X_{t}^{d,\varepsilon})\xrightarrow[d\to\infty,\varepsilon\to 0]{}\mathbb{E}\negthinspace\left[\phi(X_{t})\big{|}\mathcal{F}_{t}^{W}\right]\negthinspace weakly in $L^{2}(\Omega,\mathcal{F}_{t}^{W})$. ∎

4.4. Application of the compactness criterion

Theorem 4.10

The double-sequence $\left\{X_{t}^{d,\varepsilon}\right\}_{d\geq 1,\varepsilon>0}$ of strong solutions of SDE (24) is relatively compact in $L^{2}(\Omega,\mathcal{F}_{t}^{W})$.

Proof.

We are aiming at applying the compactness criterion given in Theorem A.3. Therefore, let $0<\alpha_{m}<\beta_{m}<\frac{1}{2}$ and $\gamma_{m}>0$ for all $m\geq 1$ and define the sequence $\mu_{s,m}=2^{-i\alpha_{m}}\gamma_{m}$, if $s=2^{i}+j$, $i\geq 0$, $0\leq j\leq 2^{i},$ $m\geq 1$ where $\mu_{s,m}\longrightarrow 0$ for $s,m\longrightarrow\infty$. We have to check that there exists a uniform constant $C$ such that for all $\{X_{t}^{d,\varepsilon}\}_{d\geq 1,\varepsilon>0}$

[TABLE]

[TABLE]

and

[TABLE]

Note first that (38) is fulfilled due to the uniform boundedness of $\{b^{d,\varepsilon}\}_{d\geq 1,\varepsilon>0}$ and the definition of the process $\left(\mathbb{B}_{t}\right)_{t\in[0,T]}$, see (20).

Next we show uniform boundedness of (39). Note first that under the assumption $u\leq s$ we have

[TABLE]

Using iteration we obtain the representation

[TABLE]

where by Corollary 4.8

[TABLE]

Consequently, we get due to (37) that

[TABLE]

where

[TABLE]

In the following we consider each $\mathfrak{I}_{i}$, $i=1,2,3$, separately starting with the first. Due to Lemma B.3 there exists $\beta_{1}\in\left(0,\frac{1}{2}\right)$ and a constant $K_{1}>0$ such that

[TABLE]

Consider now $\mathfrak{I}_{2}$. Define the density $\mathcal{E}_{t}^{d}$ by

[TABLE]

Then applying Girsanov’s theorem 2.2, monotone convergence and noting that $\sup_{d\geq 1}\sup_{t\in[0,T]]}\|\mathcal{E}_{t}^{d}\|_{L^{4}(\Omega)}<\infty$ yields

[TABLE]

Using equation (9) yields that

[TABLE]

can be written as

[TABLE]

where for $j=1,\dots,n$

[TABLE]

Repeating the application of (9) yields

[TABLE]

Defining $f_{j}^{d,\varepsilon}(t,\widetilde{y}):=\widetilde{b}_{\eta_{j-1}}^{d,\varepsilon}\left(t,\sqrt{Q}\sqrt{\mathcal{K}}\widetilde{y}\right)$ permits the use of Proposition B.2 with $\sum_{j=1}^{4n}\varepsilon_{j}=4$, $|\alpha_{j}|=1$ for all $1\leq j\leq 4n$ and thus $|\alpha|=4n$. Consequently, we get using the assumptions on $H$ and $b$ that

[TABLE]

where

[TABLE]

For $n\geq 1$ we have due to the assumptions on $H$ that

[TABLE]

Thus, we have for $n$ sufficiently large that

[TABLE]

and therefore by the approximations in Remark B.7

[TABLE]

where $C>0$ is a constant which may in the following vary from line to line. Using Stirling’s formula we have moreover that

[TABLE]

Consequently, we have for $S_{n}$ that

[TABLE]

Furthermore, using Lemma C.4 we have for every $n\geq 1$ that

[TABLE]

Moreover, due to the assumptions on $H$ there exists a finite constant $K>0$ which is independent of $d$ and $H$ such that $K_{d,H}\leq K$, cf. (62). Consequently, there exists a constant $C>0$ independent of $d$, $\varepsilon$ and $n$ such that for $n$ sufficiently large

[TABLE]

and thus due to the comparison test

[TABLE]

Hence, there exists a constant $C_{2}>0$ independent of $d$ and $\varepsilon$ such that

[TABLE]

and thus we can find a $\beta_{2}\in\left(0,\frac{1}{2}\right)$ sufficiently small such that

[TABLE]

Equivalently, we can show for $\mathfrak{I}_{3}$ that there exists a $\beta_{3}\in\left(0,\frac{1}{2}\right)$ such that

[TABLE]

where $C_{H_{m},T}=C\cdot c_{H_{m}}$ due to Lemma B.4. Here, $c_{H_{m}}$ is the constant in (14). Thus, we can find a constant $\tilde{C}>0$ independent of $H_{m}$ such that $\sup_{H\in(0,\frac{1}{6})}C_{H,T}\leq C<\infty$. Finally, we get with $\beta_{m}:=\min\{\beta_{1},\beta_{2},\beta_{3}\}$ that we can find $\gamma_{m}$, $m\geq 1$, such that

[TABLE]

uniformly in $d\geq 1$ and $\varepsilon>0$. Similarly, we can show that

[TABLE]

uniformly in $d\geq 1$ and $\varepsilon>0$ and consequently the compactness criterion Theorem A.3 yields the result. ∎

4.5. $\mathbb{F}^{\mathbb{B}}$ adaptedness and strong solution

Finally, we can state and prove the main statement of this paper

Theorem 4.11

Let $b\in\mathfrak{B}([0,T]\times\mathcal{H};\mathcal{H})$. Then SDE (3) has a unique Malliavin differentiable strong solution.

Proof.

Let $(X_{t})_{t\in[0,T]}$ be a weak solution of SDE (3) which is unique in law due to Proposition 3.5. Due to Lemma 4.9 we know that for every bounded globally Lipschitz continuous function $\phi:\mathcal{H}\to\mathbb{R}$

[TABLE]

weakly in $L^{2}(\Omega,\mathcal{F}_{t}^{W})$. Furthermore, by Theorem 4.10 there exist subsequences $\{d_{k}\}_{k\geq 1}$ and $\{\varepsilon_{n}\}_{n\geq 1}$ such that

[TABLE]

strongly in $L^{2}(\Omega,\mathcal{F}_{t}^{W})$. Uniqueness of the limit yields that $X_{t}$ is $\mathcal{F}_{t}^{W}$–measurable for all $t\in[0,T]$. Since $\mathbb{F}^{W}=\mathbb{F}^{\mathbb{B}}$, we get that $(X_{t})_{t\in[0,T]}$ is a unique strong solution of SDE (3). Malliavin differentiability follows by (40) and noting that the estimate holds also for $\gamma_{m}\equiv 1$. ∎

5. Example

In this section we give an example of a drift function $b\in\mathfrak{B}([0,T]\times\mathcal{H};\mathcal{H})$ to show that the class does not merely contain the null function.

Let $f_{k}\in L^{1}(\ell^{2};L^{\infty}([0,T];\ell^{2}))$, $k\geq 1$, i.e. for all $k\geq 1$ we have for all $z\in\ell^{2}$

[TABLE]

such that $C^{f},D^{f}\in\ell^{1}$ and define for every $k\geq 1$ an operator $A_{k}:\mathcal{H}\to\mathcal{H}$ which is invertible on $A_{k}\mathcal{H}$ such that for all $k\geq 1$

[TABLE]

where $\mathcal{D}^{A}\in\ell^{1}$. Then, we define

[TABLE]

This yields

[TABLE]

Due to the definition $C^{f}\in\ell^{1}$ and $D^{f}\cdot\mathcal{D}^{A}\in\ell^{1}$ and thus $b\in\mathfrak{B}([0,T]\times\mathcal{H};\mathcal{H})$.

A possible choice for $f$ is

[TABLE]

where $a,b\in\mathbb{R}$ and $A\subset\mathcal{H}$, which obviously fulfills the assumptions (41). The operator $A_{k}$, $k\geq 1$, can for example be chosen such that there exists a finite subset $N_{k}\subset\mathbb{N}$ such that for all $k\geq 1$

[TABLE]

and we have for every $x\in\mathcal{H}$

[TABLE]

Then $A_{k}$ is invertible on $A_{k}\mathcal{H}$ for every $k\geq 1$ and

[TABLE]

Appendix A Compactness Criterion

The following result which is originally due to [14] in the finite dimensional case and which can be e.g. found in [9], provides a compactness criterion of square integrable cylindrical Wiener processes on a Hilbert space.

Theorem A.1

Let $(B_{t})_{t\in[0,T]}$ be a cylindrical Wiener process on a separable Hilbert space $\mathcal{H}$ with respect to a complete probability space $(\Omega,\mathcal{F},\mu)$, where $\mathcal{F}$ is generated by $(B_{t})_{t\in[0,T]}$. Further, let $\mathcal{L}_{HS}(\mathcal{H},\mathbb{R})$ be the space of Hilbert-Schmidt operators from $\mathcal{H}$ to $\mathbb{R}$ and let $D:\mathbb{D}^{1,2}\longrightarrow L^{2}(\Omega;L^{2}([0,T])\otimes\mathcal{L}_{HS}(\mathcal{H},\mathbb{R}))$ be the Malliavin derivative in the direction of $(B_{t})_{t\in[0,T]}$, where $\mathbb{D}^{1,2}$ is the space of Malliavin differentiable random variables in $L^{2}(\Omega).\newline$ Suppose that $C$ is a self-adjoint compact operator on $L^{2}([0,T])\otimes\mathcal{L}_{HS}(\mathcal{H},\mathbb{R})$ with dense image. Then for any $c>0$ the set

[TABLE]

is relatively compact in $L^{2}(\Omega)$.

In this paper we aim at using a special case of the previous theorem, which is more suitable for explicit estimations. To this end we need the following auxiliary result from [14].

Lemma A.2

Denote by $v_{s},s\geq 0$, with $v_{0}=1$ the Haar basis of $L^{2}([0,1])$. Define for any $0<\alpha<\frac{1}{2}$ the operator $A_{\alpha}$ on $L^{2}([0,1])$ by

[TABLE]

and

[TABLE]

Then for $\alpha<\beta<\frac{1}{2}$ we have that

[TABLE]

Theorem A.3

Let $D^{k}$ be the Malliavin derivative in the direction of the $k$-th component of $(B_{t})_{t\in[0,T]}$. In addition, let $0<\alpha_{k}<\beta_{k}<\frac{1}{2}$ and $\gamma_{k}>0$ for all $k\geq 1$. Define the sequence $\mu_{s,k}=2^{-i\alpha_{k}}\gamma_{k}$, if $s=2^{i}+j$, $i\geq 0$, $0\leq j\leq 2^{i},$ $k\geq 1$. Assume that $\mu_{s,k}\longrightarrow 0$ for $s,k\longrightarrow\infty$. Let $c>0$ and $\mathcal{G}$ the collection of all $G\in\mathbb{D}^{1,2}$ such that

[TABLE]

[TABLE]

and

[TABLE]

Then $\mathcal{G}$ is relatively compact in $L^{2}(\Omega)$.

Proof.

As before denote by $v_{s},s\geq 0$, with $v_{0}=1$ the Haar basis of $L^{2}([0,1])$ and by $e_{k}^{\ast}=\langle e_{k},\cdot\rangle_{H},k\geq 1$, an orthonormal basis of $\mathcal{L}_{HS}(\mathcal{H},\mathbb{R})$, where $e_{k},k\geq 0$, is an orthonormal basis of $\mathcal{H}$. Define a self-adjoint compact operator $C$ on $L^{2}([0,1])\otimes\mathcal{L}_{HS}(\mathcal{H},\mathbb{R})$ with dense image by

[TABLE]

Then it follows for $G\in\mathbb{D}^{1,2}$ from Lemma A.2 that

[TABLE]

for a constant $M<\infty$. So using Theorem A.1 we obtain the result. ∎

Appendix B Integration by parts formula

In this section we derive an integration by parts formula similar to [6] which is used in the proof of Theorem 4.10 to verify the conditions of the compactness criterion Theorem A.3. Before stating the integration by parts formula, we start by giving some definitions and notations frequently used during the course of this section.

Let $n$ be a given integer. We consider the function $f:[0,T]^{n}\times(\mathbb{R}^{d})^{n}\rightarrow\mathbb{R}$ of the form

[TABLE]

where $f_{j}:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}$, $j=1,\dots,n$, are compactly supported smooth functions. Further, we deal with the function $\varkappa:[0,T]^{n}\rightarrow\mathbb{R}$ which is of the form

[TABLE]

with integrable factors $\varkappa_{j}:[0,T]\rightarrow\mathbb{R}$, $j=1,\dots,n$.

Let $\alpha_{j}$ be a multi-index and $D^{\alpha_{j}}$ its corresponding differential operator. For $\alpha:=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}_{0}^{d\times n}$ we define the norm $|\alpha|=\sum_{j=1}^{n}\sum_{k=1}^{d}\alpha_{j}^{(k)}$ and write

[TABLE]

Let $k$ be an arbitrary integer. Given $(s,z)=(s_{1},\dots,s_{kn},z_{1},\dots,z_{n})\in[0,T]^{kn}\times(\mathbb{R}^{d})^{n}$ and a shuffle permutation $\sigma\in\mathcal{S}(n,n)$ we define the shuffled functions

[TABLE]

and

[TABLE]

where $[j]$ is equal to $(j-in)$ if $(in+1)\leq j\leq(i+1)n$, $i=0,\dots,(k-1)$. For a multi-index $\alpha$, we define

[TABLE]

and

[TABLE]

where for any $a,b\in\mathbb{R}$

[TABLE]

Theorem B.1

Suppose the functions $\Psi_{\alpha}^{f}(\theta,t,z,H,d)$ and $\Psi_{\alpha}^{\varkappa}(\theta,t,H,d)$ defined in (44) and (45), respectively, are finite. Then,

[TABLE]

where $\widehat{B}_{t}^{d,H}:=\left(\frac{B_{t}^{H_{1}}}{\sqrt{\mathfrak{K}_{H_{1}}}},\dots,\frac{B_{t}^{H_{d}}}{\sqrt{\mathfrak{K}_{H_{d}}}}\right)^{\top}$ and $\mathfrak{K}_{H_{k}}$ is the constant in Lemma 2.4, is a square integrable random variable in $L^{2}(\Omega)$ and

[TABLE]

Furthermore,

[TABLE]

and the integration by parts formula

[TABLE]

holds.

Proof.

For notational simplicity we consider merely the case $\theta=0$ and write $\Lambda_{\alpha}^{f}(t,z):=\Lambda_{\alpha}^{f}(0,t,z)$. For any integrable function $g:(\mathbb{R}^{d})^{n}\longrightarrow\mathbb{C}$ we have that

[TABLE]

where the change of variables $(u_{n+1},...,u_{2n})\longmapsto(-u_{n+1},...,-u_{2n})$ was applied in the last equality. Thus,

[TABLE]

where we applied shuffling in the sense of (9). Taking the expectation on both sides together with the independence of the fractional Brownian motions $B^{H_{k}}$, $k=1,...,d$, yields that

[TABLE]

where $u_{\sigma}^{(k)}=\left(u_{\sigma(1)}^{(k)},\dots,u_{\sigma(2n)}^{(k)}\right)^{\top}$ and

[TABLE]

Moreover, we obtain for every $\sigma\in\mathcal{S}(n,n)$ that

[TABLE]

where $u^{(k)}:=\left(u_{1}^{(k)},\dots,u_{2n}^{(k)}\right)^{\top}.$ For every $1\leq k\leq d$ we have by using substitution that

[TABLE]

Considering a standard Gaussian random vector $Z\sim\mathcal{N}(0,\operatorname*{Id}_{2n})$, we get that

[TABLE]

Using a Brascamp-Lieb type inequality which is due to Lemma C.1, we further get that

[TABLE]

where $\left|\alpha^{(k)}\right|:=\sum_{j=1}^{n}\alpha_{j}^{(k)}$ and $\operatorname*{perm}\negthinspace\left(A_{k}\right)\negthinspace$ is the permanent of the covariance matrix $A_{k}=(a_{i,j}^{(k)})_{1\leq i,j\leq 2\left|\alpha^{(k)}\right|}$ of the Gaussian random vector

[TABLE]

and $S_{m}$ denotes the permutation group of size $m$. Using an upper bound for the permanent of positive semidefinite matrices which is due to [3], we find that

[TABLE]

Now let $\sum_{l=1}^{j-1}\alpha_{[\sigma(l)]}^{(k)}+1\leq i\leq\sum_{l=1}^{j}\alpha_{[\sigma(l)]}^{(k)}$ for some fixed $j\in\{1,...,2n\}$. Then

[TABLE]

Substitution gives moreover that

[TABLE]

Applying Lemma C.2 we get

[TABLE]

where $\sigma_{j}^{2}:=\operatorname*{Var}\negthinspace\left(B_{s_{j}}^{H_{k}}\left|B_{s_{1}}^{H_{k}},...,B_{s_{2n}}^{H_{k}}\text{ without }B_{s_{j}}^{H_{k}}\right.\right)\negthinspace.$

Subsequently, we aim at the application of the strong local non-determinism property of the fractional Brownian motions, cf. Lemma 2.4, i.e. for all $0<r<t\leq T$ exists a constant $\mathfrak{K}_{H_{k}}$ depending on $H_{k}$ and $T$ such that

[TABLE]

Hence, we get due to Lemma C.5 and Lemma C.6 that

[TABLE]

and

[TABLE]

Thus,

[TABLE]

Concluding from (54), (55), (B), and (58) we have that

[TABLE]

Consequently,

[TABLE]

Therefore we get from (B), (B), (52), (53), and (57) that

[TABLE]

Since $\sup_{k\geq 1}\mathfrak{K}_{H_{k}}\in(0,1)$, inequality (47) holds.

Next we prove the estimate (48). With inequality (47), we get that

[TABLE]

Taking the supremum over $[0,T]$ with respect to each function $f_{j}$, i.e.

[TABLE]

yields that

[TABLE]

Finally, we show the integration by parts formula (49). Note that a priori one cannot interchange the order of integration in (46), since e.g. for $m=1$, $f\equiv 1$ one gets an integral of the Donsker-Delta function which is not a random variable in the usual sense. Therefore, we define for $R>0$,

[TABLE]

where $B(0,R):=\{v\in(\mathbb{R}^{d})^{n}:\,|v|<R\}$. This yields

[TABLE]

for a sufficient constant $C_{R}$. Under the assumption that the above right-hand side is integrable over $(\mathbb{R}^{d})^{n}$, similar computations as above show that $\Lambda^{f}_{\alpha,R}(\theta,t,z)\to\Lambda^{f}_{\alpha}(\theta,t,z)$ in $L^{2}(\Omega)$ as $R\to\infty$ for all $\theta,t$ and $z$. By Lebesgue’s dominated convergence theorem and the fact that the Fourier transform is an automorphism on the Schwarz space, we obtain

[TABLE]

which is exactly the integration by parts formula (49). ∎

Applying Theorem B.1 we obtain the following crucial estimate (compare [1], [2], [6], and [7]):

Proposition B.2

Let the functions $f$ and $\varkappa$ be defined as in (42) and (43), respectively. Further, let $0\leq\theta^{\prime}<\theta<t\leq T$ and for some $m\geq 1$

[TABLE]

for every $j=1,...,n$ with $(\varepsilon_{1},...,\varepsilon_{n})\in\{0,1\}^{n}$. Let $\alpha\in(\mathbb{N}_{0}^{d})^{n}$ be a multi-index. Assume there exists $\delta$ such that

[TABLE]

for all $j=1,\dots n$ and $d\geq 1$, where $\gamma_{m}\in(0,H_{m})$ is sufficiently small. Then there exist constants $C_{T}$ (depending on $T$) and $K_{d,H}$ (depending on $d$ and $H$), such that for any $0\leq\theta<t\leq T$ we have

[TABLE]

In order to prove this result we need the following two auxiliary results.

Lemma B.3

Let $H\in\left(0,\frac{1}{2}\right)$ and $t\in[0,T]$ be fixed. Then, there exists $\beta\in\left(0,\frac{1}{2}\right)$ and a constant $C>0$ independent of $H$ such that

[TABLE]

Proof.

Let $0\leq\theta^{\prime}<\theta\leq t$ be fixed. Write

[TABLE]

where $f_{t}(\theta):=\left(\frac{t}{\theta}\right)^{H-\frac{1}{2}}(t-\theta)^{H-\frac{1}{2}}$ and $g_{t}(\theta):=\int_{\theta}^{t}\frac{f_{u}(\theta)}{u}du$.

We continue with the estimation of $K_{H}(t,\theta)-K_{H}(t,\theta^{\prime})$. First, observe that there exists a constant $0<C<1$ such that

[TABLE]

for every $0<y<x<\infty$ and $\alpha:=(\frac{1}{2}-H)\in\left(0,\frac{1}{2}\right)$ as well as $0<\gamma<\frac{1}{2}-\alpha$. Indeed, rewriting (60) yields using the substitution $z:=\frac{x}{y}$, $z\in(1,\infty)$,

[TABLE]

Furthermore, since $\alpha+\gamma<1$ we get that

[TABLE]

and

[TABLE]

Moreover, for $2\leq z\leq\infty$ we get the upper bound

[TABLE]

and for $1<z<2$ we have that

[TABLE]

This shows inequality (60) which then implies for $0<\gamma<H$ that

[TABLE]

Further,

[TABLE]

Consequently, we get for $\gamma\in(0,H)$, $0<\theta^{\prime}<\theta<t\leq T$, that

[TABLE]

where $C>0$ is a constant merely depending on $T$. Thus

[TABLE]

for sufficiently small $\gamma$ and $\beta$. On the other hand, we have that

[TABLE]

Therefore,

[TABLE]

∎

Lemma B.4

Let $H\in\left(0,\frac{1}{2}\right)$, $0\leq\theta<t\leq T$ and $(\varepsilon_{1},\dots,\varepsilon_{n})\in\{0,1\}^{n}$ be fixed. Assume $w_{j}+\left(H-\frac{1}{2}-\gamma\right)\varepsilon_{j}>-1$ for all $j=1,\dots,n$. Then there exists a finite constant $C_{H,T}>0$ depending only on $H$ and $T$ such that for $\gamma\in(0,H)$

[TABLE]

where

[TABLE]

Proof.

Recall, that for given exponents $a,b>-1$ and some fixed $s_{j+1}>s_{j}$ we have

[TABLE]

Due to Lemma B.3 we have that for every $\gamma\in(0,H)$, $0<\theta^{\prime}<\theta<s_{j}\leq T$,

[TABLE]

for $C_{H,T}:=C\cdot c_{H}$, where $c_{H}$ is the constant in (14) and $C>0$ is some constant merely depending on $T$. Consequently, we get that

[TABLE]

where

[TABLE]

Noting that

[TABLE]

and iterative integration yields the desired formula. ∎

Finally, we are able to give the proof of Proposition B.2.

Proof of Proposition B.2.

The integration by parts formula (49) yields that

[TABLE]

Taking the expectation and applying Theorem B.1 we get that

[TABLE]

where

[TABLE]

Under the assumption $-\sum_{k=1}^{d}H_{k}(1+\alpha_{[\sigma(j)]}^{(k)}+\alpha_{[\sigma(j-1)]}^{(k)})+(H_{m}-\frac{1}{2}-\gamma_{m})\varepsilon_{[\sigma(j)]}>-1$ for all $j=1,...,2n$, we can apply Lemma B.4 and thus get

[TABLE]

where $\Pi_{\gamma}(2n)$ is defined as in (61). We define the constant $K_{d,H}$ by

[TABLE]

and thus an upper bound of $\Pi_{\gamma}(2n)$ is given by

[TABLE]

Note that $\sum_{j=1}^{2n}\varepsilon_{[\sigma(j)]}=2\sum_{j=1}^{n}\varepsilon_{j}$ and

[TABLE]

Hence, it follows that

[TABLE]

∎

Proposition B.5

Let the functions $f$ and $\varkappa$ be defined as in (42) and (43), respectively. Let $0\leq\theta<t\leq T$ and

[TABLE]

for every $j=1,\dots,n$ with $(\varepsilon_{1},\dots,\varepsilon_{n})\in\{0,1\}^{n}$. Let $\alpha\in(\mathbb{N}_{0}^{d})^{n}$ be a multi-index and suppose that there exists $\delta$ such that

[TABLE]

for all $j=1,\dots,n$ and $d\geq 1$. Then there exist constants $C_{T}$ (depending on $T$) and $K_{d,H}$ (depending on $d$ and $H$) such that for any $0\leq\theta<t\leq T$ we have

[TABLE]

The proof of Proposition B.5 is similar to the one of Proposition B.2 by using the subsequent lemma instead of Lemma B.4 and thus it is omitted in this manuscript.

Lemma B.6

Let $H\in\left(0,\frac{1}{2}\right)$, $0\leq\theta<t\leq T$ and $(\varepsilon_{1},\dots,\varepsilon_{n})\in\{0,1\}^{n}$ be fixed. Assume $w_{j}+\left(H-\frac{1}{2}\right)\varepsilon_{j}>-1$ for all $j=1,\dots,n$. Then there exists a finite constant $C_{H,T}>0$ depending only on $H$ and $T$ such that

[TABLE]

where $\Pi_{0}$ is defined in (61).

Proof.

Using similar arguments as in the proof of Lemma B.3 we get the following estimate

[TABLE]

for every $0<\theta<s_{j}<T$ and $C_{H,T}:=C\cdot c_{H}$, where $c_{H}$ is the constant in (14) and $C>0$ is some constant merely depending on $T$. Thus,

[TABLE]

Proceeding similar to the proof of Lemma B.4 yields the desired estimate. ∎

Remark B.7*.*

Note that

[TABLE]

Indeed, since for $n\geq 1$ sufficiently large we have by Stirling’s formula that

[TABLE]

we get by assuming without loss of generality that $|\alpha^{(k)}|\geq 1$ for all $1\leq k\leq d$, that

[TABLE]

Appendix C Technical Results

The following technical result can be found in [26].

Lemma C.1

Assume that $X_{1},...,X_{n}$ are real centered jointly Gaussian random variables, and $\Sigma=(\mathbb{E}[X_{j}X_{k}])_{1\leq j,k\leq n}$ is the covariance matrix, then

[TABLE]

where $\operatorname*{perm}\negthinspace\left(A\right)\negthinspace$ is the permanent of a matrix $A=(a_{ij})_{1\leq i,j\leq n}$ defined by

[TABLE]

for the symmetric group $\mathcal{S}_{n}$.

The next lemma corresponds to [12, Lemma 2]:

Lemma C.2

Let $Z_{1},...,Z_{n}$ be mean zero Gaussian random variables which are linearly independent. Then for any measurable function $g:\mathbb{R}\longrightarrow\mathbb{R}_{+}$ we have that

[TABLE]

where $\sigma_{1}^{2}:=\operatorname*{Var}\negthinspace\left(Z_{1}\left|Z_{2},...,Z_{n}\right.\right)\negthinspace$.

Remark C.3*.*

Note that here linearly independence is meant in the sense that $\det\operatorname*{Cov}\negthinspace\left(Z_{1},...,Z_{n}\right)\negthinspace\neq 0$.

Lemma C.4

Let $a\in\ell^{p}$, $1\leq p<\infty$. Then, for every $n\geq 1$ and $d\geq 1$

[TABLE]

and

[TABLE]

Proof.

We proof equation (63) by induction. For $n=1$ the result holds. Therefore we assume that (63) holds for $n$ and we show that it also holds for $n+1$. Thus, we get by the induction hypothesis that

[TABLE]

Equation (64) is an immediate consequence of (63) and the continuity of the function $f(x)=x^{n}$ for fixed $n\geq 1$. ∎

The subsequent lemmas are due to [4].

Lemma C.5

Let $(X_{1},\dots,X_{n})$ be a mean-zero Gaussian random vector. Then,

[TABLE]

Lemma C.6

For any square integrable random variable $X$ and $\sigma$-algebras $\mathcal{G}_{1}\subset\mathcal{G}_{2}$

[TABLE]