Time Reversal of Volterra Processes Driven Stochastic Differential Equations

Abstract

We consider stochastic differential equations driven by some Volterra processes. Under time reversal, these equations are transformed into past-dependent stochastic differential equations driven by a standard Brownian motion. We are then in position to derive existence and uniqueness of solutions of the Volterra driven SDE considered at the beginning.

1. Introduction

Fractional Brownian motion (fBm for short) of Hurst index 𝐻∈[0,1] is the Gaussian process which admits the following representation: for any 𝑡≥0, 𝐵^𝐻(𝑡)=^𝑡₀𝐾_𝐻(𝑡,𝑠)d𝐵(𝑠), (1) where 𝐵 is a one-dimensional Brownian motion and 𝐾_𝐻 is a triangular kernel, that is, 𝐾_𝐻(𝑡,𝑠)=0 for 𝑠>𝑡, the definition of which is given in (46). Fractional Brownian motion is probably the first process which is not a semimartingale and for which it is still interesting to develop a stochastic calculus. That means we want to define a stochastic integral and solve stochastic differential equations driven by such a process. From the very beginning of this program, two approaches do exist. One approach is based on the Hölder continuity or the finite 𝑝 variation of the fBm sample paths. The other way to proceed relies on the gaussianity of fBm. The former is mainly deterministic and was initiated by Zähle [1], Feyel and de la Pradelle [2], and Russo and Vallois [3, 4]. Then, came the notion of rough paths was introduced by Lyons [5], whose application to fBm relies on the work of Coutin and Qian [6]. These works have been extended in the subsequent works [7–17]. A new way of thinking came with the independent but related works of Feyel, de la Pradelle [18], and Gubinelli [19]. The integral with respect to fBm was shown to exist as the unique process satisfying some characterization (analytic in the case of [18], algebraic in [19]). As a byproduct, this showed that almost all the existing integrals throughout the literature were all the same as they all satisfy these two conditions. Behind each approach, but the last too, is a construction of an integral defined for a regularization of fBm, then the whole work is to show that, under some convenient hypothesis, the approximate integrals converge to a quantity which is called the stochastic integral with respect to fBm. The main tool to prove the convergence is either integration by parts in the sense of fractional deterministic calculus, or enrichment of the fBm by some iterated integrals proved to exist independently or by analytic continuation [20, 21].

In the probabilistic approach [22–30], the idea is also to define an approximate integral and then prove its convergence. It turns out that the key tool is here the integration by parts in the sense of Malliavin calculus.

In dimension greater than one, with the deterministic approach, one knows how to define the stochastic integral and prove existence and uniqueness of fBm-driven SDEs for fBm with Hurst index greater than 1/4. Within the probabilistic framework, one knows how to define a stochastic integral for any value of 𝐻 but one cannot prove existence and uniqueness of SDEs whatever the value of 𝐻. The primary motivation of this work is to circumvent this problem.

In [26, 27], we defined stochastic integrals with respect to fBm as a “damped-Stratonovitch” integral with respect to the underlying standard Brownian motion. This integral is defined as the limit of Riemann-Stratonovitch sums, the convergence of which is proved after an integration by parts in the sense of Malliavin calculus. Unfortunately, this manipulation generates nonadaptiveness: formally the result can be expressed as ^𝑡₀𝑢(𝑠)∘d𝐵^𝐻(𝑠)=𝛿𝒦^∗_𝑡𝑢+trace𝒦^∗_𝑡∇𝑢, (2) where 𝒦 is defined by (𝑡)=𝑑𝒦𝑓𝑑𝑡^𝑡₀𝐾_𝐻(𝑡,𝑠)𝑓(𝑠)d𝑠 (3) and 𝒦^∗_𝑡 is the adjoint of 𝒦 in ℒ²([0,𝑡],𝐑). In particular, there exists 𝑘 such that 𝒦^∗_𝑡𝑓(𝑠)=^𝑡_𝑠𝑘(𝑡,𝑢)𝑓(𝑢)d𝑢 (4) for any 𝑓∈ℒ²([0,𝑡],𝐑) so that even if 𝑢 is adapted (with respect to the Brownian filtration), the process (𝑠↦𝒦^∗_𝑡𝑢(𝑠)) is anticipative. However, the stochastic integral process (𝑡↦∫^𝑡₀𝑢(𝑠)∘d𝐵^𝐻(𝑠)) remains adapted; hence, the anticipative aspect is, in some sense, artificial. The motivation of this work is to show that, up to time reversal, we can work with adapted process and Itô integrals. The time-reversal properties of fBm were already studied in [31] in a different context. It was shown there that the time reversal of the solution of an fBm-driven SDE of the form 𝑑𝑌(𝑡)=𝑢(𝑌(𝑡))d𝑡+d𝐵^𝐻(𝑡) (5) is still a process of the same form. With a slight adaptation of our method to fBm-driven SDEs with drift, one should recover the main theorem of [31].

In what follows, there is no restriction on the dimension, but we need to assume that any component of 𝐵^𝐻 is an fBm of Hurst index greater than 1/2. Consider that we want to solve the following equation: 𝑋_𝑡=𝑥+^𝑡₀𝜎𝑋_𝑠∘d𝐵^𝐻(𝑠),0≤𝑡≤𝑇, (6) where 𝜎 is a deterministic function whose properties will be fixed below. It turns out that it is essential to investigate the more general equations: 𝑋_𝑟,𝑡=𝑥+^𝑡_𝑟𝜎𝑋_𝑟,𝑠∘d𝐵^𝐻(𝑠),0≤𝑟≤𝑡≤𝑇. A^′ The strategy is then as follows. We will first consider the reciprocal problem: 𝑌_𝑟,𝑡=𝑥−^𝑡_𝑟𝜎𝑌_𝑠,𝑡∘d𝐵^𝐻(𝑠),0≤𝑟≤𝑡≤𝑇. B^′

The first critical point is that when we consider {𝑍_𝑟,𝑡∶=𝑌_{𝑡−𝑟,𝑡},𝑟∈[0,𝑡]}, this process solves an adapted, past-dependent, and stochastic differential equation with respect to a standard Brownian motion. Moreover, because 𝐾_𝐻 is lower-triangular and sufficiently regular, the trace term vanishes in the equation defining 𝑍. We have then reduced the problem to an SDE with coefficients dependent on the past, a problem which can be handled by the usual contraction methods. We do not claim that the results presented are new (for instance, see the brilliant monograph [32] for detailed results obtained via rough paths theory), but it seems interesting to have purely probabilistic methods which show that fBm driven SDEs do have strong solutions which are homeomorphisms. Moreover, the approach given here shows the irreducible difference between the case 𝐻<1/2 and 𝐻>1/2. The trace term only vanishes in the latter situation, so that such an SDE is merely a usual SDE with past-dependent coefficients. This representation may be fruitful, for instance, to analyze the support and prove the absolute continuity of solutions of (6).

This paper is organized as follows. After some preliminaries on fractional Sobolev spaces, often called Besov-Liouville space, we address, in Section 3, the problem of Malliavin calculus and time reversal. This part is interesting in its own since stochastic calculus of variations is a framework oblivious to time. Constructing such a notion of time is achieved using the notion of resolution of the identity as introduced in [33, 34]. We then introduce the second key ingredient which is the notion of strict causality or quasinilpotence; see [35] for a related application. In Section 4, we show that solving B^′ reduces to solve a past-dependent stochastic differential equation with respect to a standard Brownian motion; see (C) below. Then, we prove existence, uniqueness, and some properties of this equation. Technical lemmas are postponed to Section 5.

2. Besov-Liouville Spaces

Let 𝑇>0 be fix real number. For a measurable function 𝑓∶[0,𝑇]→𝐑^𝑛, we define 𝜏_𝑇𝑓 by 𝜏_𝑇𝑓(𝑠)=𝑓(𝑇−𝑠)forany𝑠∈[0,𝑇]. (7) For 𝑡∈[0,𝑇], 𝑒_𝑡𝑓 will represent the restriction of 𝑓 to [0,𝑡], that is, 𝑒_𝑡𝑓=𝑓𝟏_[0,𝑡]. For any linear map 𝐴, we denote by 𝐴^∗_𝑇, its adjoint in ℒ²([0,𝑇];𝐑^𝑛). For 𝜂∈(0,1], the space of 𝜂-Hölder continuous functions on [0,𝑇] is equipped with the norm: 𝑓_Hol(𝜂)=sup_{0<𝑠<𝑡<𝑇}𝑓(𝑡)−𝑓(𝑠)|𝑡−𝑠|^𝜂+𝑓_∞. (8)

Its topological dual is denoted by Hol(𝜂)^∗. For 𝑓∈ℒ¹([0,𝑇];𝐑^𝑛;d𝑡) (denoted by ℒ¹ for short), the left and right fractional integrals of 𝑓 are defined by 𝐼^𝛾₀₊𝑓(𝑥)=1Γ𝛾^𝑥₀𝑓(𝑡)(𝑥−𝑡)^𝛾−1𝐼d𝑡,𝑥≥0,_𝛾𝑇−𝑓(𝑥)=1Γ𝛾^𝑇_𝑥𝑓(𝑡)(𝑡−𝑥)^𝛾−1d𝑡,𝑥≤𝑇, (9) where 𝛾>0 and 𝐼⁰₀⁺=𝐼⁰_𝑇⁻=Id. For any 𝛾≥0, 𝑝,𝑞≥1, any 𝑓∈ℒ^𝑝 and 𝑔∈ℒ^𝑞 where 𝑝⁻¹+𝑞⁻¹≤𝛾, we have ^𝑇₀𝑓(𝑠)𝐼^𝛾₀₊𝑔(𝑠)d𝑠=^𝑇₀𝐼^𝛾_𝑇−𝑓(𝑠)𝑔(𝑠)d𝑠. (10)

The Besov-Liouville space 𝐼^𝛾₀₊(ℒ^𝑝)∶=ℐ⁺_𝛾,𝑝 is usually equipped with the norm: 𝐼^𝛾₀₊𝑓_{ℐ+𝛾,𝑝}=𝑓_ℒ𝑝. (11)

Analogously, the Besov-Liouville space 𝐼^𝛾_𝑇−(ℒ^𝑝)∶=ℐ⁻_𝛾,𝑝 is usually equipped with the norm: 𝐼_𝑇^−𝛾₋𝑓_{ℐ−𝛾,𝑝}=𝑓_ℒ𝑝. (12)

We then have the following continuity results (see [2, 36]):

Proposition 1. Consider the following.(i) If 0<𝛾<1,1<𝑝<1/𝛾, then 𝐼^𝛾₀₊ is a bounded operator from ℒ^𝑝 into ℒ^𝑞 with 𝑞=𝑝(1−𝛾𝑝)⁻¹.(ii) For any 0<𝛾<1 and any 𝑝≥1,ℐ⁺_𝛾,𝑝 is continuously embedded in 𝐻𝑜𝑙(𝛾−1/𝑝) provided that 𝛾−1/𝑝>0.(iii) For any 0<𝛾<𝛽<1,Hol(𝛽) is compactly embedded in ℐ_𝛾,∞.(iv) For 𝛾𝑝<1, the spaces ℐ⁺_𝛾,𝑝 and ℐ⁻_𝛾,𝑝 are canonically isomorphic. We will thus use the notation ℐ_𝛾,𝑝 to denote any of these spaces.

3. Malliavin Calculus and Time Reversal

Our reference probability space is Ω=𝒞₀([0,𝑇],𝐑^𝑛), the space of 𝐑^𝑛-valued, continuous functions, null at time 0. The Cameron-Martin space is denoted by 𝐇 and is defined as 𝐇=𝐼¹₀⁺(ℒ²([0,𝑇])). In what follows, the space ℒ²([0,𝑇]) is identified with its topological dual. We denote by 𝜅 the canonical embedding from 𝐇 into Ω. The probability measure 𝐏 on Ω is such that the canonical map 𝑊∶𝜔↦(𝜔(𝑡),𝑡∈[0,𝑇]) defines a standard 𝑛-dimensional Brownian motion. A mapping 𝜙 from Ω into some separable Hilbert space ℌ is called cylindrical if it is of the form 𝜙(𝑤)=∑^𝑑_𝑖=1𝑓_𝑖(⟨𝑣_𝑖,1,𝑤⟩,…,⟨𝑣_𝑖,𝑛,𝑤⟩)𝑥_𝑖, where for each 𝑖,𝑓_𝑖∈𝒞^∞₀(𝐑^𝑛,𝐑) and (𝑣_𝑖,𝑗,𝑗=1,…,𝑛) is a sequence of Ω^∗. For such a function we define ∇^W𝜙 as ∇^W𝜙(𝑤)=_𝑖,𝑗=1𝜕_𝑗𝑓_𝑖𝑣_𝑖,1,𝑤,…,𝑣_𝑖,𝑛,𝑤𝑣_𝑖,𝑗⊗𝑥_𝑖, (13) where 𝑣̃ is the image of 𝑣∈Ω^∗ by the map (𝐼¹₀⁺∘𝜅)^∗. From the quasi-invariance of the Wiener measure [37], it follows that ∇^W is a closable operator on 𝐿^𝑝(Ω;ℌ), 𝑝≥1, and we will denote its closure with the same notation. The powers of ∇^W are defined by iterating this procedure. For 𝑝>1,𝑘∈𝐍, we denote by 𝔻_𝑝,𝑘(ℌ) the completion of ℌ-valued cylindrical functions under the following norm: 𝜙_𝑝,𝑘=^𝑘_𝑖=0(∇^W)^𝑖𝜙_{𝐿𝑝(Ω;ℌ⊗ℒ𝑝([0,1])⊗𝑖)}. (14)

We denote by 𝕃_𝑝,1 the space 𝔻_𝑝,1(ℒ^𝑝([0,𝑇];𝐑^𝑛)). The divergence, denoted as 𝛿^W, is the adjoint of ∇^W: 𝑣 belongs to Dom_𝑝𝛿^W whenever, for any cylindrical 𝜙, 𝐄^𝑇₀𝑣_𝑠∇^W_𝑠𝜙d𝑠≤𝑐𝜙_𝐿𝑝 (15) and, for such a process 𝑣, 𝐄^𝑇₀𝑣_𝑠∇^W_𝑠𝜙d𝑠=𝐄𝜙𝛿^W𝑣. (16)

We introduced the temporary notation 𝑊 for standard Brownian motion to clarify the forthcoming distinction between a standard Brownian motion and its time reversal. Actually, the time reversal of a standard Brownian is also a standard Brownian motion, and thus, both of them “live” in the same Wiener space. We now precise how their respective Malliavin gradient and divergence are linked. Consider 𝐵=(𝐵(𝑡),𝑡∈[0,𝑇]) an 𝑛-dimensional standard Brownian motion and 𝐵̆^𝑇=(𝐵(𝑇)−𝐵(𝑇−𝑡),𝑡∈[0,𝑇]) its time reversal. Consider the following map: Θ_𝑇̆(𝑇)∶Ω→Ω𝜔↦𝜔=𝜔−𝜏_𝑇𝜔, (17) and the commutative diagram: (18)

Note that Θ_𝑇⁻¹=Θ_𝑇 since 𝜔(0)=0. For a function 𝑓∈𝒞^∞_𝑏(𝐑^𝑛𝑘), we define the following: ∇_𝑟𝑓𝜔𝑡₁,…,𝜔𝑡_𝑘=^𝑘_𝑗=1𝜕_𝑗𝑓𝜔𝑡₁,…,𝜔𝑡_𝑘𝟏_[0,𝑡𝑗](𝑟),∇̆_𝑟𝑓𝜔̆𝑡₁,…,𝜔̆𝑡_𝑘=^𝑘_𝑗=1𝜕_𝑗𝑓𝜔̆𝑡₁,…,𝜔̆𝑡_𝑘𝟏_[0,𝑡𝑗](𝑟). (19)

The operator ∇=∇^𝐵 (resp., ∇̆=∇^̆𝐵) is the Malliavin gradient associated with a standard Brownian motion (resp., its time reversal). Since 𝑓𝜔̆𝑡₁,…,𝜔̆𝑡_𝑘=𝑓𝜔(𝑇)−𝜔𝑇−𝑡₁,…,𝜔(𝑇)−𝜔𝑇−𝑡_𝑘, (20) we can consider 𝑓(𝜔̆(𝑡₁),…,𝜔̆(𝑡_𝑘)) as a cylindrical function with respect to the standard Brownian motion. As such its gradient is given by ∇_𝑟𝑓𝜔̆𝑡₁,…,𝜔̆𝑡_𝑘=^𝑘_𝑗=1𝜕_𝑗𝑓𝜔̆𝑡₁,…,𝜔̆𝑡_𝑘𝟏_{[𝑇−𝑡𝑗,𝑇]}(𝑟). (21)

We thus have, for any cylindrical function 𝐹, ∇𝐹∘Θ_𝑇(𝜔)=𝜏_𝑇∇̆𝐹(𝜔̆). (22)

Since Θ^∗_𝑇𝐏=𝐏 and 𝜏_𝑇 is continuous from ℒ^𝑝 into itself for any 𝑝, it is then easily shown that the spaces 𝔻_𝑝,𝑘 and 𝔻̆_𝑝,𝑘 (with obvious notations) coincide for any 𝑝,𝑘 and that (22) holds for any element of one of these spaces. Hence we have proved the following theorem.

Theorem 2. For any 𝑝≥1 and any integer 𝑘, the spaces 𝔻_𝑝,𝑘 and 𝔻̆_𝑝,𝑘 coincide. For any 𝐹∈𝔻_𝑝,𝑘  for some 𝑝,𝑘, ∇𝐹∘Θ_𝑇=𝜏_𝑇∇̆𝐹∘Θ_𝑇,𝐏𝑎.𝑠. (23)

By duality, an analog result follows for divergences.

Theorem 3. A process 𝑢 belongs to the domain of 𝛿 if and only if 𝜏_𝑇𝑢 belongs to the domain of 𝛿̆, and, then, the following equality holds: 𝛿̆(𝑢(𝜔̆))(𝜔̆)=𝛿𝜏_𝑇𝑢(𝜔̆)(𝜔)=𝛿𝜏_𝑇𝑢∘Θ_𝑇(𝜔). (24)

Proof. For 𝑢∈ℒ², for cylindrical 𝐹, we have on the one hand: 𝐄𝐹(𝜔̆)𝛿̆𝑢(𝜔̆)=𝐄∇̆𝐹(𝜔̆),𝑢_ℒ2, (25) and on the other hand, 𝐄∇̆𝐹(𝜔̆),𝑢_ℒ2=𝐄𝜏_𝑇∇𝐹∘Θ_𝑇(𝜔),𝑢_ℒ2=𝐄∇𝐹∘Θ_𝑇(𝜔),𝜏_𝑇𝑢_ℒ2=𝐄𝐹∘Θ_𝑇(𝜔)𝛿𝜏_𝑇𝑢(𝜔𝐹(𝜔̆)𝛿𝜏)=𝐄_𝑇𝑢(𝜔). (26) Since this is valid for any cylindrical 𝐹, (24) holds for 𝑢∈ℒ². Now, for 𝑢 in the domain of divergence (see [37, 38]), 𝛿𝑢=_𝑖𝑢,ℎ_𝑖ℒ2𝛿ℎ_𝑖−∇𝑢,ℎ_𝑖⊗ℎ_{𝑖ℒ2⊗ℒ2}, (27) where (ℎ_𝑖,𝑖∈𝐍) is an orthonormal basis of ℒ²([0,𝑇];𝐑^𝑛). Thus, we have 𝛿̆(𝑢(𝜔̆))(𝜔̆)=_𝑖𝑢(𝜔̆),ℎ_𝑖ℒ2𝛿̆ℎ_𝑖(𝜔̆)−∇̆𝑢(𝜔̆),ℎ_𝑖⊗ℎ_{𝑖ℒ2⊗ℒ2}=_𝑖𝑢(𝜔̆),ℎ_𝑖ℒ2𝛿𝜏_𝑇ℎ_𝑖𝜔)−(𝜔̆)(∇𝑢,𝜏_𝑇ℎ_𝑖⊗ℎ_{𝑖ℒ2⊗ℒ2}=_𝑖𝜏_𝑇𝑢(𝜔̆),𝜏_𝑇ℎ_𝑖ℒ2𝛿𝜏_𝑇ℎ_𝑖𝜔)−(∇𝜏_𝑇𝑢(𝜔̆),𝜏_𝑇ℎ_𝑖⊗𝜏_𝑇ℎ_{𝑖ℒ2⊗ℒ2}, (28) where we have taken into account that 𝜏_𝑇 is in an involution. Since (ℎ_𝑖,𝑖∈𝐍) is an orthonormal basis of ℒ²([0,𝑇];𝐑^𝑛), identity (24) is satisfied for any 𝑢 in the domain of 𝛿.

3.1. Causality and Quasinilpotence

In anticipative calculus, the notion of trace of an operator plays a crucial role, We refer to [39] for more details on trace.

Definition 4. Let 𝑉 be a bounded map from ℒ²([0,𝑇];𝐑^𝑛) into itself. The map 𝑉 is said to be trace class, whenever for one CONB (ℎ_𝑛,𝑛≥1) of ℒ²([0,𝑇];𝐑^𝑛), _𝑛≥1𝑉ℎ_𝑛,ℎ_𝑛ℒ2isfinite. (29) Then, the trace of 𝑉 is defined by trace(𝑉)=_𝑛≥1𝑉ℎ_𝑛,ℎ_𝑛ℒ2. (30)
It is easily shown that the notion of trace does not depend on the choice of the CONB.

Definition 5. A family 𝐸 of projections (𝐸_𝜆, 𝜆∈[0,1]) in ℒ²([0,𝑇];𝐑^𝑛) is called a resolution of the identity if it satisfies the conditions:(1)𝐸₀=0 and 𝐸₁=Id(2)𝐸_𝜆𝐸_𝜇=𝐸_𝜆∧𝜇(3)lim_𝜇↓𝜆𝐸_𝜇=𝐸_𝜆 for any 𝜆∈[0,1) and lim_𝜇↑1𝐸_𝜇=Id.
For instance, the family 𝐸=(𝑒_𝜆𝑇,𝜆∈[0,1]) is a resolution of the identity in ℒ²([0,𝑇];𝐑^𝑛).

Definition 6. A partition 𝜋 of [0,𝑇] is a sequence {0=𝑡₀<𝑡₁<⋯<𝑡_𝑛=𝑇}. Its mesh is denoted by |𝜋| and defined by |𝜋|=sup_𝑖|𝑡_𝑖+1−𝑡_𝑖|.
The causality plays a crucial role in what follows. The next definition is just the formalization in terms of operator of the intuitive notion of causality.

Definition 7. A continuous map 𝑉 from ℒ²([0,𝑇];𝐑^𝑛) into itself is said to be 𝐸 causal if and only if the following condition holds: 𝐸_𝜆𝑉𝐸_𝜆=𝐸_𝜆𝑉forany𝜆∈[0,1]. (31)
For instance, an operator 𝑉 in integral form 𝑉𝑓(𝑡)=∫^𝑇₀𝑉(𝑡,𝑠)𝑓(𝑠)d𝑠 is causal if and only if 𝑉(𝑡,𝑠)=0 for 𝑠≥𝑡, that is, computing 𝑉𝑓(𝑡) needs only the knowledge of 𝑓 up to time 𝑡 and not after. Unfortunately, this notion of causality is insufficient for our purpose, and we are led to introduce the notion of strict causality as in [40].

Definition 8. Let 𝑉 be a causal operator. It is a strictly causal operator, whenever for any 𝜀>0, there exists a partition 𝜋 of [0,𝑇] such that, for any 𝜋^′={0=𝑡₀<𝑡₁<⋯<𝑡_𝑛=𝑇}⊂𝜋, (𝐸_𝑡𝑖+1−𝐸_𝑡𝑖)𝑉(𝐸_𝑡𝑖+1−𝐸_𝑡𝑖)_ℒ2<𝜀,for𝑖=0,…,𝑛−1. (32)
Note carefully that the identity map is causal but not strictly causal. Indeed, if 𝑉=Id, for any 𝑠<𝑡, (𝐸_𝑡−𝐸_𝑠)𝑉(𝐸_𝑡−𝐸_𝑠)_ℒ2=𝐸_𝑡−𝐸_𝑠ℒ2=1 (33) since 𝐸_𝑡−𝐸_𝑠 is a projection. However, if 𝑉 is hyper-contractive, we have the following result.

Lemma 9. Assume the resolution of the identity to be either 𝐸=(𝑒_𝜆𝑇,𝜆∈[0,1]) or 𝐸=(𝐼𝑑−𝑒_{(1−𝜆)𝑇},𝜆∈[0,1]). If 𝑉 is an 𝐸 causal map continuous from ℒ² into ℒ^𝑝 for some 𝑝>2 then 𝑉 is strictly 𝐸 causal.

Proof. Let 𝜋 be any partition of [0,𝑇]. Assume that 𝐸=(𝑒_𝜆𝑇,𝜆∈[0,1]), and the very same proof works for the other mentioned resolution of the identity. According to Hölder formula, we have for any 0≤𝑠<𝑡≤𝑇, 𝐸_𝑡−𝐸_𝑠𝑉𝐸_𝑡−𝐸_𝑠𝑓_ℒ2=^𝑡_𝑠𝑉(𝑓𝟏_(𝑠,𝑡])(𝑢)²≤()d𝑢𝑡−𝑠^1−2/𝑝𝑉(𝑓𝟏_(𝑠,𝑡])_ℒ𝑝/2≤𝑐(𝑡−𝑠)^1−2/𝑝𝑓_ℒ2. (34) Then, for any 𝜀>0, there exists 𝜂>0 such that |𝜋|<𝜂 implies ‖(𝐸_𝑡𝑖+1−𝐸_𝑡𝑖)𝑉(𝐸_𝑡𝑖+1−𝐸_𝑡𝑖)𝑓‖_ℒ²≤𝜀 for any {0=𝑡₀<𝑡₁<⋯<𝑡_𝑛=𝑇}⊂𝜋 and any 𝑖=0,…,𝑛−1.

The importance of strict causality lies in the next theorem we borrow from [40].

Theorem 10. The set of strictly causal operators coincides with the set of quasinilpotent operators, that is, trace-class operators such that trace(𝑉^𝑛)=0 for any integer 𝑛≥1.

Moreover, we have the following stability theorem.

Theorem 11. The set of strictly causal operators is a two-sided ideal in the set of causal operators.

Definition 12. Let 𝐸 be a resolution of the identity in ℒ²([0,𝑇];𝐑^𝑛). Consider the filtration ℱ^𝐸 defined as ℱ^𝐸_𝑡=𝜎𝛿^W𝐸_𝜆ℎ,𝜆≤𝑡,ℎ∈ℒ². (35) An ℒ²-valued random variable 𝑢 is said to be ℱ^𝐸 adapted if, for any ℎ∈ℒ², the real valued process ⟨𝐸_𝜆𝑢,ℎ⟩ is ℱ^𝐸-adapted. We denote by 𝔻^𝐸_𝑝,𝑘(ℌ) the set of ℱ^𝐸 adapted random variables belonging to 𝔻_𝑝,𝑘(ℌ).

If 𝐸=(𝑒_𝜆𝑇,𝜆∈[0,1]), the notion of ℱ^𝐸 adapted processes coincides with the usual one for the Brownian filtration, and it is well known that a process 𝑢 is adapted if and only if ∇^W_𝑟𝑢(𝑠)=0 for 𝑟>𝑠. This result can be generalized to any resolution of the identity.

Theorem 13 (Proposition 3.1 of [33]). Let 𝑢 belongs to 𝕃_𝑝,1. Then 𝑢 is ℱ^𝐸 adapted if and only if ∇^𝑊𝑢 is 𝐸 causal.

We then have the following key theorem.

Theorem 14. Assume the resolution of the identity to be 𝐸=(𝑒_𝜆𝑇,𝜆∈[0,1]) either 𝐸=(𝐼𝑑−𝑒_{(1−𝜆)𝑇},𝜆∈[0,1]) and that 𝑉 is an 𝐸-strictly causal continuous operator from ℒ² into ℒ^𝑝 for some 𝑝>2. Let 𝑢 be an element of 𝔻^𝐸_2,1(ℒ²). Then, 𝑉∇^𝑊𝑢 is of trace class and we have trace(𝑉∇^𝑊𝑢)=0.

Proof. Since 𝑢 is adapted, ∇^W𝑢 is 𝐸-causal. According to Theorem 11, 𝑉∇^W𝑢 is strictly causal and the result follows by Theorem 10.

In what follows, 𝐸⁰ is the resolution of the identity in the Hilbert space ℒ² defined by 𝑒_𝜆𝑇𝑓=𝑓𝟏_[0,𝜆𝑇] and 𝐸̆⁰ is the resolution of the identity defined by 𝑒̆_𝜆𝑇𝑓=𝑓𝟏_{[(1−𝜆)𝑇,𝑇]}. The filtrations ℱ^𝐸0 and ℱ^̆𝐸0 are defined accordingly. Next lemma is immediate when 𝑉 is given in the form of 𝑉𝑓(𝑡)=∫^𝑡₀𝑉(𝑡,𝑠)𝑓(𝑠)d𝑠. Unfortunately such a representation as an integral operator is not always available. We give here an algebraic proof to emphasize the importance of causality.

Lemma 15. Let 𝑉 be a map from ℒ²([0,𝑇];𝐑^𝑛) into itself such that 𝑉 is 𝐸⁰-causal. Let 𝑉^∗ be the adjoint of 𝑉 in ℒ²([0,𝑇];𝐑^𝑛). Then, the map 𝜏_𝑇𝑉^∗_𝑇𝜏_𝑇 is 𝐸̆⁰-causal.

Proof. This is a purely algebraic lemma once we have noticed that 𝜏_𝑇𝑒_𝑟=Id−𝑒_𝑇−𝑟𝜏_𝑇forany0≤𝑟≤𝑇. (36) For, it suffices to write 𝜏_𝑇𝑒_𝑟𝑓(𝑠)=𝑓(𝑇−𝑠)𝟏_[0,𝑟]()()𝟏𝑇−𝑠=𝑓𝑇−𝑠_{[𝑇−𝑟,𝑇]}(𝑠)=Id−𝑒_𝑇−𝑟𝜏_𝑇𝑓(𝑠),forany0≤𝑠≤𝑇. (37) We have to show that 𝑒_𝑟𝜏_𝑇𝑉^∗_𝑇𝜏_𝑇𝑒_𝑟=𝑒_𝑟𝜏_𝑇𝑉^∗_𝑇𝜏_𝑇orequivalently𝑒_𝑟𝜏_𝑇𝑉𝜏_𝑇𝑒_𝑟=𝜏_𝑇𝑉𝜏_𝑇𝑒_𝑟, (38)(39) since 𝑒^∗_𝑟=𝑒_𝑟 and 𝜏^∗_𝑇=𝜏_𝑇. Now, (37) yields 𝑒_𝑟𝜏_𝑇𝑉𝜏_𝑇𝑒_𝑟=𝜏_𝑇𝑉𝜏_𝑇𝑒_𝑟−𝑒_𝑇−𝑟𝑉𝜏_𝑇𝑒_𝑟. (40) Use (37) again to obtain 𝑒_𝑇−𝑟𝑉𝜏_𝑇𝑒_𝑟=𝑒_𝑇−𝑟𝑉Id−𝑒_𝑇−𝑟𝜏_𝑇=𝑒_𝑇−𝑟𝑉−𝑒_𝑇−𝑟𝑉𝑒_𝑇−𝑟𝜏_𝑇=0, (41) since 𝑉 is 𝐸-causal.

3.2. Stratonovitch Integrals

In what follows, 𝜂 belongs to (0,1] and 𝑉 is a linear operator. For any 𝑝≥2, we set the following.

Hypothesis 1 (𝑝,𝜂). The linear map 𝑉 is continuous from ℒ^𝑝([0,𝑇];𝐑^𝑛) into the Banach space Hol(𝜂).

Definition 16. Assume that Hypothesis 1 (𝑝,𝜂) holds. The Volterra process associated to 𝑉, denoted by 𝑊^𝑉, is defined by 𝑊^𝑉(𝑡)=𝛿^W𝑉𝟏_[0,𝑡],∀𝑡∈[0,𝑇]. (42)

For any subdivision 𝜋 of [0,𝑇], that is, 𝜋={0=𝑡₀<𝑡₁<⋯<𝑡_𝑛=𝑇}, of mesh |𝜋|, we consider the Stratonovitch sums: 𝑅^𝜋(𝑡,𝑢)=𝛿^W_{𝑡𝑖∈𝜋}1𝜃_𝑖^𝑡𝑖+1_𝑡^∧𝑡_𝑖∧𝑡𝑉𝑢(𝑟)d𝑟𝟏_{[𝑡𝑖,𝑡𝑖+1)}+_{𝑡𝑖∈𝜋}1𝜃_{𝑖[𝑡𝑖∧𝑡,𝑡𝑖+1∧𝑡]}²𝑉∇^W_𝑟𝑢(𝑠)d𝑠d𝑟. (43)

Definition 17. We say that 𝑢 is 𝑉-Stratonovitch integrable on [0,𝑡] whenever the family 𝑅^𝜋(𝑡,𝑢), defined in (43), converges in probability as |𝜋| goes to 0. In this case the limit will be denoted by ∫^𝑡₀𝑢(𝑠)∘d𝑊^𝑉(𝑠).

Example 18. The first example is the so-called Lévy fractional Brownian motion of Hurst index 𝐻>1/2 defined as 1Γ(𝐻+1/2)^𝑡₀(𝑡−𝑠)^𝐻−1/2d𝐵_𝑠=𝛿𝐼_𝑇^𝐻−1/2₋𝟏_[0,𝑡]. (44) This amounts to say that 𝑉=𝐼_𝑇^𝐻−1/2₋. Thus Hypothesis 1  (𝑝,𝐻−1/2−1/𝑝) holds provided that 𝑝(𝐻−1/2)>1.

Example 19. The other classical example is the fractional Brownian motion with stationary increments of Hurst index 𝐻>1/2, which can be written as ^𝑡₀𝐾_𝐻(𝑡,𝑠)d𝐵(𝑠), (45) where 𝐾_𝐻()=()𝑡,𝑟𝑡−𝑟^𝐻−(1/2)Γ((𝐹1𝐻+1/2))21−𝐻,𝐻−21,𝐻+2𝑡,1−𝑟×1_[0,𝑡)(𝑟). (46) The Gauss hypergeometric function 𝐹(𝛼,𝛽,𝛾,𝑧) (see [41]) is the analytic continuation on ℂ×ℂ×ℂ⧵{−1,−2,…}×{𝑧∈ℂ,Arg|1−𝑧|<𝜋} of the power series: ^+∞_𝑘=0(𝛼)_𝑘𝛽_𝑘𝛾_𝑘𝑧𝑘!^𝑘,(𝑎)₀(𝑎)=1,_𝑘=Γ(𝑎+𝑘)Γ(𝑎)=𝑎(𝑎+1)⋯(𝑎+𝑘−1). (47) We know from [36] that 𝐾_𝐻 is an isomorphism from ℒ^𝑝([0,1]) onto ℐ⁺_{𝐻+1/2,𝑝} and 𝐾_𝐻𝑓=𝐼¹₀⁺𝑥^𝐻−1/2𝐼₀^𝐻−1/2₊𝑥^1/2−𝐻𝑓. (48) Consider that 𝒦_𝐻=𝐼₀⁻¹⁺∘𝐾_𝐻. Then it is clear that ^𝑡₀𝐾_𝐻(𝑡,𝑠)d𝐵(𝑠)=^𝑡₀𝒦_𝐻^∗_𝑇𝟏_[0,𝑡](𝑠)d𝐵(𝑠); (49) hence we are in the framework of Definition 17 provided that we take 𝑉=(𝒦_𝐻)^∗_𝑇. Hypothesis 1  (𝑝,𝐻−1/2−1/𝑝) is satisfied provided that 𝑝(𝐻−1/2)>1.

The next theorem then follows from [26].

Theorem 20. Assume that Hypothesis 1 (𝑝,𝜂) holds. Assume that 𝑢 belongs to 𝕃_𝑝,1. Then 𝑢 is 𝑉-Stratonovitch integrable, and there exists a process which we denote by 𝐷^𝑊𝑢 such that 𝐷^𝑊𝑢 belongs to 𝐿^𝑝(𝐏⊗𝑑𝑠) and ^𝑇₀𝑢(𝑠)∘d𝑊^𝑉(𝑠)=𝛿^𝑊(𝑉𝑢)+^𝑇₀𝐷^𝑊𝑢(𝑠)d𝑠. (50) The so-called “trace-term” satisfies the following estimate: 𝐄^𝑇₀𝐷^𝑊𝑢(𝑟)^𝑝d𝑟≤𝑐𝑇^𝑝𝜂‖𝑢‖^𝑝_𝕃𝑝,1, (51) for some universal constant 𝑐. Moreover, for any 𝑟≤𝑇, 𝑒_𝑟𝑢 is 𝑉-Stratonovitch integrable and ^𝑟₀𝑢(𝑠)∘d𝑊^𝑉(𝑠)=^𝑇₀𝑒_𝑟𝑢(𝑠)∘d𝑊^𝑉(𝑠)=𝛿^𝑊𝑉𝑒_𝑟𝑢+^𝑟₀𝐷^𝑊𝑢(𝑠)d𝑠, (52) and we have the maximal inequality: 𝐄^.₀𝑢(𝑠)∘d𝑊^𝑉(𝑠)^𝑝_{𝐻𝑜𝑙(𝜂)}≤𝑐‖𝑢‖^𝑝_𝕃𝑝,1, (53) where 𝑐 does not depend on 𝑢.

The main result of this Section is the following theorem which states that the time reversal of a Stratonovitch integral is an adapted integral with respect to the time-reversed Brownian motion. Due to its length, its proof is postponed to Section 5.1.

Theorem 21. Assume that Hypothesis 1 (𝑝,𝜂) holds. Let 𝑢 belong to 𝕃_𝑝,1 and let 𝑉̆_𝑇=𝜏_𝑡𝑉𝜏_𝑇. Assume furthermore that 𝑉 is 𝐸̆₀-causal and that 𝑢̆=𝑢∘Θ_𝑇⁻¹ is ℱ^̆𝐸0-adapted. Then, ^𝑇−𝑟_𝑇−𝑡𝜏_𝑇𝑢(𝑠)∘d𝑊^𝑉(𝑠)=^𝑡_𝑟𝑉̆_𝑇𝟏_[𝑟,𝑡]𝑢̆(𝑠)d𝐵̆^𝑇(𝑠),0≤𝑟≤𝑡≤𝑇, (54) where the last integral is an Itô integral with respect to the time reversed Brownian motion 𝐵̆^𝑇(𝑠)=𝐵(𝑇)−𝐵(𝑇−𝑠)=Θ_𝑇(𝐵)(𝑠).

Remark 22. Note that, at a formal level, we could have an easy proof of this theorem. For instance, consider the Lévy fBm, and a simple computation shows that 𝑉̆_𝑇=𝐼₀^𝐻−1/2₊ for any 𝑇. Thus, we are led to compute trace(𝐼₀^𝐻−1/2₊∇𝑢). If we had sufficient regularity, we could write trace𝐼₀^𝐻−1/2₊∇𝑢=^𝑇₀^𝑠₀(𝑠−𝑟)^𝐻−3/2∇_𝑠𝑢(𝑟)d𝑟d𝑠=0, (55) since ∇_𝑠𝑢(𝑟)=0 for 𝑠>𝑟 for 𝑢 adapted. Obviously, there are many flaws in these lines of proof. The operator 𝐼₀^𝐻−1/2₊∇𝑢 is not regular enough for such an expression of the trace to be true. Even more, there is absolutely no reason for 𝑉̆_𝑇∇𝑢 to be a kernel operator so we cannot hope such a formula. These are the reasons that we need to work with operators and not with kernels.

4. Volterra-Driven SDEs

Let 𝔊 be the group of homeomorphisms of 𝐑^𝑛 equipped with the distance. We introduce a distance 𝑑 on 𝔊 by 𝑑𝜑,𝜙=𝜌𝜑,𝜙+𝜌𝜑⁻¹,𝜙⁻¹, (56) where 𝜌𝜑,𝜙=^∞_𝑁=12^−𝑁sup_{|𝑥|≤𝑁}𝜑(𝑥)−𝜙(𝑥)1+sup_{|𝑥|≤𝑁}𝜑(𝑥)(𝑥⋅−𝜙) (57) Then, 𝔊 is a complete topological group. Consider the equations: 𝑋_𝑟,𝑡=𝑥+^𝑡_𝑟𝜎𝑋_𝑟,𝑠∘d𝑊^𝑉(𝑠),0≤𝑟≤𝑡≤𝑇,𝑌_𝑟,𝑡=𝑥−^𝑡_𝑟𝜎𝑌_𝑠,𝑡∘d𝑊^𝑉(𝑠),0≤𝑟≤𝑡≤𝑇. (A)(B) As a solution of (A) is to be constructed by “inverting” a solution of (B), we need to add to the definition of a solution of (A) or (B) the requirement of being a flow of homeomorphisms. This is the meaning of the following definition.

Definition 23. By a solution of (A), we mean a measurable map: []×[]()↦Ω×0,𝑇0,𝑇→𝔊𝜔,𝑟,𝑡𝑥↦𝑋_𝑟,𝑡(𝜔,𝑥) (58) such that the following properties are satisfied.(1)For any 0≤𝑟≤𝑡≤𝑇, for any 𝑥∈𝐑^𝑛, 𝑋_𝑟,𝑡(𝜔,𝑥) is 𝜎{𝑊^𝑉(𝑠),𝑟≤𝑠≤𝑡}-measurable.(2)For any 0≤𝑟≤𝑇, for any 𝑥∈𝐑^𝑛, the processes (𝜔,𝑡)↦𝑋_𝑟,𝑡(𝜔,𝑥) and (𝜔,𝑡)↦𝑋⁻¹_𝑟,𝑡(𝜔,𝑥) belong to 𝕃_𝑝,1 for some 𝑝≥2. (3)For any 0≤𝑟≤𝑠≤𝑡, for any 𝑥∈𝐑^𝑛, the following identity is satisfied: 𝑋_𝑟,𝑡(𝜔,𝑥)=𝑋_𝑠,𝑡𝜔,𝑋_𝑟,𝑠(𝜔,𝑥). (59)(4)Equation (A) is satisfied for any 0≤𝑟≤𝑡≤𝑇𝐏-a.s.

Definition 24. By a solution of (B), we mean a measurable map: []×[]()↦Ω×0,𝑇0,𝑇→𝔊𝜔,𝑟,𝑡𝑥↦𝑌_𝑟,𝑡(𝜔,𝑥) (60) such that the following properties are satisfied. (1)For any 0≤𝑟≤𝑡≤𝑇, for any 𝑥∈𝐑^𝑛, 𝑌_𝑟,𝑡(𝜔,𝑥) is 𝜎{𝑊^𝑉(𝑠),𝑟≤𝑠≤𝑡} measurable. (2)For any 0≤𝑟≤𝑇, for any 𝑥∈𝐑^𝑛, the processes (𝜔,𝑟)↦𝑌_𝑟,𝑡(𝜔,𝑥) and (𝜔,𝑟)↦𝑌⁻¹_𝑟,𝑡(𝜔,𝑥) belong to 𝕃_𝑝,1 for some 𝑝≥2. (3)Equation (B) is satisfied for any 0≤𝑟≤𝑡≤𝑇𝐏-a.s.. (4)For any 0≤𝑟≤𝑠≤𝑡, for any 𝑥∈𝐑^𝑛, the following identity is satisfied: 𝑌_𝑟,𝑡(𝜔,𝑥)=𝑌_𝑟,𝑠𝜔,𝑌_𝑠,𝑡(𝜔,𝑥). (61)

At last consider the equation, for any 0≤𝑟≤𝑡≤𝑇, 𝑍_𝑟,𝑡=𝑥−^𝑡_𝑟𝑉̆_𝑇𝜎∘𝑍_.,𝑡𝟏_[𝑟,𝑡](𝑠)d𝐵̆^𝑇(𝑠), (C) where 𝐵 is a standard 𝑛-dimensional Brownian motion.

Definition 25. By a solution of (C), we mean a measurable map: []×[]()↦Ω×0,𝑇0,𝑇→𝔊𝜔,𝑟,𝑡𝑥↦𝑍_𝑟,𝑡(𝜔,𝑥) (62) such that the following properties are satisfied.(1)For any 0≤𝑟≤𝑡≤𝑇, for any 𝑥∈𝐑^𝑛, 𝑍_𝑟,𝑡(𝜔,𝑥) is 𝜎{𝐵̆^𝑇(𝑠),𝑠≤𝑟≤𝑡} measurable. (2)For any 0≤𝑟≤𝑡≤𝑇, for any 𝑥∈𝐑^𝑛, the processes (𝜔,𝑟)↦𝑍_𝑟,𝑡(𝜔,𝑥) and (𝜔,𝑟)↦𝑍⁻¹_𝑟,𝑡(𝜔,𝑥) belong to 𝕃_𝑝,1 for some 𝑝≥2. (3)Equation (C) is satisfied for any 0≤𝑟≤𝑡≤𝑇 𝐏-a.s..

Theorem 26. Assume that 𝑉̆_𝑇 is an 𝐸⁰ causal map continuous from ℒ^𝑝 into ℐ_𝛼,𝑝 for 𝛼>0 and 𝑝≥4 such that 𝛼𝑝>1. Assume that 𝜎 is Lipschitz continuous and sublinear; see (96) for the definition. Then, there exists a unique solution to (C). Let 𝑍 denote this solution. For any (𝑟,𝑟^′), 𝐄𝑍_𝑟,𝑇−𝑍_𝑟′,𝑇^𝑝≤𝑐𝑟−𝑟^′𝑝𝜂. (63) Moreover, (𝜔,𝑟)↦𝑍_𝑟,𝑠𝜔,𝑍_𝑠,𝑡(𝜔,𝑥)∈𝕃_𝑝,1,𝑓𝑜𝑟𝑎𝑛𝑦𝑟≤𝑠≤𝑡≤𝑇. (64)

Since this proof needs several lemmas, we defer it to Section 5.2.

Theorem 27. Assume that 𝑉̆_𝑇 is an 𝐸⁰-causal map continuous from ℒ^𝑝 into ℐ_𝛼,𝑝 for 𝛼>0 and 𝑝≥2 such that 𝛼𝑝>1. For fixed 𝑇, there exists a bijection between the space of solutions of (B) on [0,𝑇] and the set of solutions of (C).

Proof. Set 𝑍_𝑟,𝑇(𝜔̆,𝑥)=𝑌_{𝑇−𝑟,𝑇}Θ_𝑇⁻¹(𝜔̆),𝑥 (65) or equivalently 𝑌_𝑟,𝑇(𝜔,𝑥)=𝑍_{𝑇−𝑟,𝑇}Θ_𝑇(𝜔),𝑥. (66) According to Theorem 21, 𝑌 satisfies (B) if and only if 𝑍 satisfies (C). The regularity properties are immediate since ℒ^𝑝 is stable by 𝜏_𝑇.

The first part of the next result is then immediate.

Corollary 28. Assume that 𝑉̆_𝑇 is an 𝐸⁰-causal map continuous from ℒ^𝑝 into ℐ_𝛼,𝑝 for 𝛼>0 and 𝑝≥2 such that 𝛼𝑝>1. Then (B) has one and only one solution and for any 0≤𝑟≤𝑠≤𝑡, for any 𝑥∈𝐑^𝑛, the following identity is satisfied: 𝑌_𝑟,𝑡(𝜔,𝑥)=𝑌_𝑟,𝑠𝜔,𝑌_𝑠,𝑡(𝜔,𝑥). (67)

Proof. According to Theorems 27 and 26,  (B) has at most one solution since (C) has a unique solution. As to the existence, points from (1) to (3) are immediately deduced from the corresponding properties of 𝑍 and (66).
According to Theorem 26, (𝜔,𝑟)↦𝑌_𝑟,𝑠(𝜔,𝑌_𝑠,𝑡(𝜔,𝑥)) belongs to 𝕃_𝑝,1; hence, we can apply the substitution formula and 𝑌_𝑟,𝑠𝜔,𝑌_𝑠,𝑡(𝜔,𝑥)=𝑌_𝑠,𝑡(𝜔,𝑥)−^𝑠_𝑟𝜎𝑌_𝜏,𝑠(𝜔,𝑥)∘d𝑊^𝑉(𝜏)_{𝑥=𝑌𝑠,𝑡(𝜔,𝑥)}=𝑥−^𝑡_𝑠𝜎𝑌_𝜏,𝑡(𝜔,𝑥)∘d𝑊^𝑉(𝜏)−^𝑠_𝑟𝜎𝑌_𝜏,𝑠𝜔,𝑌_𝑠,𝑡(𝜔,𝑥)∘d𝑊^𝑉(𝜏). (68) Set 𝑅_𝜏,𝑡=𝑌_𝜏,𝑡()𝑌𝜔,𝑥for𝑠≤𝜏≤𝑡_𝜏,𝑠𝜔,𝑌_𝑠,𝑡(𝜔,𝑥)for𝑟≤𝜏≤𝑠. (69) Then, in view of (68), 𝑅 appears to be the unique solution (B) and thus 𝑅_𝑠,𝑡(𝜔,𝑥)=𝑌_𝑠,𝑡(𝜔,𝑥). Point (4) is thus proved.

Corollary 29. For 𝑥 fixed, the random field (𝑌_𝑟,𝑡(𝑥),0≤𝑟≤𝑡≤𝑇) admits a continuous version. Moreover, 𝐄𝑌_𝑟,𝑠(𝑥)−𝑌_{𝑟′,𝑠′}(𝑥)^𝑝≤𝑐1+|𝑥|^𝑝𝑠^′−𝑠^𝑝𝜂+𝑟−𝑟^′𝑝𝜂. (70) We still denote by 𝑌 this continuous version.

Proof. Without loss of generality, assume that 𝑠≤𝑠^′ and remark that 𝑌_{𝑠,𝑠′(𝑥)} thus belongs to 𝜎{𝐵̆^𝑇_𝑢,𝑢≥𝑠}: 𝐄𝑌_𝑟,𝑠(𝑥)−𝑌_{𝑟′,𝑠′}(𝑥)^𝑝≤𝑐𝐄𝑌_𝑟,𝑠(𝑥)−𝑌_𝑟′,𝑠(𝑥)^𝑝+𝐄𝑌_𝑟′,𝑠(𝑥)−𝑌_{𝑟′,𝑠′}(𝑥)^𝑝=𝑐𝐄𝑌_𝑟,𝑠(𝑥)−𝑌_𝑟′,𝑠(𝑥)^𝑝+𝐄𝑌_𝑟′,𝑠(𝑥)−𝑌_𝑟′,𝑠𝑌_𝑠,𝑠′(𝑥)^𝑝=𝑐𝐄𝑍_{𝑠−𝑟,𝑠}(𝑥)−𝑍_{𝑠−𝑟′,𝑠}(𝑥)^𝑝+𝐄𝑍_{𝑠−𝑟′,𝑠}(𝑥)−𝑍_{𝑠−𝑟′,𝑠}𝑌_𝑠,𝑠′(𝑥)^𝑝. (71) According to Theorem 37, 𝐄𝑍_{𝑠−𝑟,𝑠}(𝑥)−𝑍_{𝑠−𝑟′,𝑠}(𝑥)^𝑝≤𝑐𝑟−𝑟^′𝑝𝜂1+|𝑥|^𝑝. (72) In view of Theorem 21, the stochastic integral which appears in (C) is also a Stratonovitch integral; hence, we can apply the substitution formula and say 𝑍_{𝑠−𝑟′,𝑠}𝑌_𝑠,𝑠′(𝑥)=𝑍_{𝑠−𝑟′,𝑠}(𝑦)_{𝑦=𝑌′𝑠,𝑠(𝑥)}. (73) Thus we can apply Theorem 37 and obtain that 𝐄𝑍_{𝑠−𝑟′,𝑠}(𝑥)−𝑍_{𝑠−𝑟′,𝑠}(𝑌_𝑠,𝑠′(𝑥))^𝑝≤𝑐𝐄𝑥−𝑌_𝑠,𝑠′(𝑥)^𝑝. (74) The right hand side of this equation is in turn equal to 𝐄[|𝑍_0,𝑠′−𝑍_{𝑠′−𝑠,𝑠′}(𝑥)|^𝑝] thus, we get 𝐄𝑍_{𝑠−𝑟′,𝑠}(𝑥)−𝑍_{𝑠−𝑟′,𝑠}𝑌_𝑠,𝑠′(𝑥)^𝑝≤𝑐1+|𝑥|^𝑝𝑠^′−𝑠^𝑝𝜂. (75) Combining (72) and (75) gives 𝐄𝑌_𝑟,𝑠(𝑥)−𝑌_{𝑟′,𝑠′}(𝑥)^𝑝≤𝑐1+|𝑥|^𝑝𝑠^′−𝑠^𝑝𝜂+𝑟−𝑟^′𝑝𝜂, (76) hence the result comes

Thus, we have the main result of this paper.

Theorem 30. Assume that 𝑉̆_𝑇 is an 𝐸⁰-causal map continuous from ℒ^𝑝 into ℐ_𝛼,𝑝 for 𝛼>0 and 𝑝≥4 such that 𝛼𝑝>1. Then (A) has one and only one solution.

Proof. Under the hypothesis, we know that (B) has a unique solution which satisfies (67). By definition a solution of (B), the process 𝑌⁻¹∶(𝜔,𝑠)↦𝑌⁻¹_𝑠𝑡(𝜔,𝑥) belongs to 𝕃_𝑝,1; hence, we can apply the substitution formula. Following the lines of proof of the previous theorem, we see that 𝑌⁻¹ is a solution of (A).
In the reverse direction, two distinct solutions of (A) would give rise to two solutions of (B) by the same principles. Since this is definitely impossible in view of Corollary 28  (A) has at most one solution.

5. Technical Proofs

5.1. Substitution Formula

The proof of Theorem 21 relies on several lemmas including one known in anticipative calculus as the substitution formula, compare [38].

Theorem 31. Assume that Hypothesis 1 (𝑝,𝜂) holds. Let 𝑢 belong to 𝕃_𝑝,1. If 𝑉∇^𝑊𝑢 is of trace class, then ^𝑇₀𝐷^𝑊𝑢(𝑠)d𝑠=trace𝑉∇^𝑊𝑢. (77) Moreover, 𝐄trace(𝑉∇^𝑊𝑢)^𝑝≤𝑐‖𝑢‖^𝑝_𝕃𝑝,1. (78)

Proof. For each 𝑘, let (𝜙_𝑘,𝑚,𝑚=1,…,2^𝑘) be the functions 𝜙_𝑘,𝑚=2^𝑘/2𝟏_[(𝑚−1)2^−𝑘_,𝑚2^−𝑘₎. Let 𝑃_𝑘 be the projection onto the span of the 𝜙_𝑘,𝑚; since ∇^W𝑉𝑢 is of trace class, we have (see [42]) trace𝑉∇^W𝑝_𝑡𝑢=lim_𝑘→+∞trace𝑃_𝑘𝑉∇^W𝑝_𝑡𝑢𝑃_𝑘. (79) Now, trace𝑃_𝑘𝑉∇^W𝑢𝑃_𝑘=^𝑘_𝑚=1𝑉∇^W𝑝_𝑡𝑢,𝜙_𝑘,𝑚⊗𝜙_{𝑘,𝑚ℒ2⊗ℒ2}=^𝑘_𝑚=12^{𝑘𝑚2−𝑘}₍₎₂^∧𝑡_{𝑚−1−𝑘∧𝑡}^𝑚2−𝑘₍₎₂^∧𝑡_{𝑚−1−𝑘∧𝑡}𝑉∇^W_𝑟𝑢(𝑠)d𝑠d𝑟. (80) According to the proof of Theorem 20, the first part of the theorem follows. The second part is then a rewriting of (51).

For 𝑝≥1, let Γ_𝑝 be the set of random fields: 𝑢∶𝐑^𝑚→𝕃_𝑝,1𝑥↦((𝜔,𝑠)↦𝑢(𝜔,𝑠,𝑥)) (81) equipped with the seminorms, 𝑝_𝐾(𝑢)=sup_𝑥∈𝐾‖𝑢(𝑥)‖_𝕃𝑝,1 (82) for any compact 𝐾 of 𝐑^𝑚.

Corollary 32 (substitution formula). Assume that Hypothesis 1 (𝑝,𝜂) holds. Let {𝑢(𝑥),𝑥∈𝐑^𝑚} belong to Γ_𝑝. Let 𝐹 be a random variable such that ((𝜔,𝑠)↦𝑢(𝜔,𝑠,𝐹)) belongs to 𝕃_𝑝,1. Then, ^𝑇₀𝑢(𝑠,𝐹)∘d𝑊^𝑉(𝑠)=^𝑇₀𝑢(𝑠,𝑥)∘d𝑊^𝑉_{𝑠𝑥=𝐹}. (83)

Proof. Simple random fields of the form: 𝑢(𝜔,𝑠,𝑥)=^𝐾_𝑙=1𝐻_𝑙(𝑥)𝑢_𝑙(𝜔,𝑠) (84) with 𝐻_𝑙 smooth and 𝑢_𝑙 in 𝕃_𝑝,1 are dense in Γ_𝑝. In view of (53), it is sufficient to prove the result for such random fields. By linearity, we can reduce the proof to random fields of the form 𝐻(𝑥)𝑢(𝜔,𝑠). Now for any partition 𝜋, 𝛿^W_{𝑡𝑖∈𝜋}1𝜃_𝑖^𝑡𝑖+1_𝑡^∧𝑡_𝑖∧𝑡𝐻(𝐹)𝑉(𝑢(𝜔,.))(𝑟)d𝑟𝟏_{[𝑡𝑖,𝑡𝑖+1)}=𝐻(𝐹)𝛿^W_{𝑡𝑖∈𝜋}1𝜃_𝑖^𝑡𝑖+1_𝑡^∧𝑡_𝑖∧𝑡𝑉(𝑢(𝜔,.))(𝑟)d𝑟𝟏_{[𝑡𝑖,𝑡𝑖+1)}−_{𝑡𝑖∈𝜋}^𝑡𝑖+1_𝑡^∧𝑡_𝑖∧𝑡^𝑡𝑖+1_𝑡^∧𝑡_𝑖∧𝑡𝐻^′(𝐹)∇^W_𝑠𝐹𝑉𝑢(𝑟)d𝑠d𝑟. (85) On the other hand, ∇^W_𝑠(𝐻(𝐹)𝑢(𝜔,𝑟))=𝐻^′(𝐹)∇^W_𝑠𝐹𝑢(𝑟), (86) Hence, _{𝑡𝑖∈𝜋}1𝜃_{𝑖[𝑡𝑖∧𝑡,𝑡𝑖+1∧𝑡]}²𝑉∇^W_𝑟𝐻(𝐹)𝑢(𝑠)d𝑠d𝑟=_{𝑡𝑖∈𝜋}1𝜃_{𝑖[𝑡𝑖∧𝑡,𝑡𝑖+1∧𝑡]}²𝐻^′(𝐹)∇^W_𝑠𝐹𝑉𝑢(𝑟)d𝑠d𝑟. (87) According to Theorem 20, (83) is satisfied for simple random fields.

Definition 33. For any 0≤𝑟≤𝑡≤𝑇, for 𝑢 in 𝕃_𝑝,1, we define ∫^𝑡_𝑟𝑢(𝑠)∘d𝑊^𝑉(𝑠) as ^𝑡_𝑟𝑢(𝑠)∘𝑑𝑊^𝑉(𝑠)=^𝑡₀𝑢(𝑠)∘𝑑𝑊^𝑉(𝑠)−^𝑟₀𝑢(𝑠)∘𝑑𝑊^𝑉(𝑠)=^𝑇₀𝑒_𝑡𝑢(𝑠)𝑑𝑊^𝑉(𝑠)−^𝑇₀𝑒_𝑟𝑢(𝑠)∘𝑑𝑊^𝑉(𝑠)=𝛿^W𝑉𝑒_𝑡−𝑒_𝑟𝑢+^𝑡_𝑟𝐷^W𝑢(𝑠)𝑑𝑠. (88)

By the very definition of trace class operators, the next lemma is straightforward.

Lemma 34. Let 𝐴 and 𝐵 be two continuous maps from ℒ²([0,𝑇];𝐑^𝑛) into itself. Then, the map 𝜏_𝑇𝐴⊗𝐵 (resp. 𝐴𝜏_𝑇⊗𝐵) is of trace class if and only if the map 𝐴⊗𝜏_𝑇𝐵 (resp. 𝐴⊗𝐵𝜏_𝑇) is of trace class. Moreover, in such a situation, trace𝜏_𝑇𝐴⊗𝐵=trace𝐴⊗𝜏_𝑇𝐵,𝑟𝑒𝑠𝑝.trace𝐴𝜏_𝑇⊗𝐵=trace𝐴⊗𝐵𝜏_𝑇. (89)

The next corollary follows by a classical density argument.

Corollary 35. Let 𝑢∈𝕃_2,1 such that ∇^𝑊⊗𝜏_𝑇𝑉𝑢 and ∇^𝑊⊗𝑉𝜏_𝑇𝑢 are of trace class. Then, 𝜏_𝑇∇^𝑊⊗𝑉𝑢 and ∇^𝑊𝜏_𝑇⊗𝑉𝑢 are of trace class. Moreover, we have trace∇^𝑊⊗𝜏_𝑇𝑉𝑢=trace𝜏_𝑇∇^𝑊,∇⊗𝑉𝑢trace^𝑊⊗𝑉𝜏_𝑇𝑢=trace∇^𝑊𝜏_𝑇⊗𝑉𝑢. (90)

Proof of Theorem 21. We first study the divergence term. In view of Theorem 3 we have 𝛿^𝐵𝑉𝑒_𝑇−𝑟−𝑒_𝑇−𝑡𝜏_𝑇𝑢̆∘Θ_𝑇=𝛿^𝐵𝑉𝜏_𝑇𝑒_𝑡−𝑒_𝑟𝑢̆∘Θ_𝑇=𝛿^𝐵𝜏_𝑇𝑉̆_𝑇𝑒_𝑡−𝑒_𝑟𝑢̆∘Θ_𝑇=𝛿̆𝑉̆_𝑇𝑒_𝑡−𝑒_𝑟𝑢̆(𝜔̆)=^𝑡_𝑟𝑉̆_𝑇𝟏_[𝑟,𝑡]𝑢̆(𝑠)d𝐵^𝑇(𝑠). (91) According to Lemma 15  (𝑉̆_𝑇)^∗ is 𝐸̆₀ causal, and, according to Lemma 9, it is strictly 𝐸̆₀ causal. Thus, Theorem 14 implies that ∇̆𝑉(𝑒_𝑡−𝑒_𝑟)𝑢̆ is of trace class and quasinilpotent. Hence Corollary 35 induces that 𝜏_𝑇𝑉̆_𝑇𝜏_𝑇⊗𝜏_𝑇∇̆𝜏_𝑇𝑒_𝑡−𝑒_𝑟𝑢̆ (92) is trace class and quasinilpotent. Now, according to Theorem 2, we have 𝜏_𝑇𝑉̆_𝑇𝜏_𝑇⊗𝜏_𝑇∇̆𝜏_𝑇𝑒_𝑡−𝑒_𝑟𝑢̆=𝑉∇𝜏_𝑇𝑒_𝑇−𝑟−𝑒_𝑇−𝑡𝑢̆∘Θ_𝑇. (93) According to Theorem 20, we have proved (54).

5.2. The Forward Equation

Lemma 36. Assume that Hypothesis 1(𝑝,𝜂) holds and that 𝜎 is Lipschitz continuous. Then, for any 0≤𝑎≤𝑏≤𝑇, the map 𝑉̆_𝑇∘𝜎∶𝐶[0,𝑇],𝐑^𝑛→𝐶[0,𝑇],𝐑^𝑛𝜙↦𝑉̆_𝑇𝜎∘𝜓𝟏_[𝑎,𝑏] (94) is Lipschitz continuous and Gâteaux differentiable. Its differential is given by 𝑑𝑉̆_𝑇∘𝜎𝜙𝜓=𝑉̆_𝑇𝜎^′∘𝜙𝜓. (95) Assume furthermore that 𝜎 is sublinear; that is, |𝜎(𝑥)|≤𝑐(1+|𝑥|),𝑓𝑜𝑟𝑎𝑛𝑦𝑥∈𝐑^𝑛. (96) Then, for any 𝜓∈𝐶([0,𝑇],𝐑^𝑛), for any 𝑡∈[0,𝑇], 𝑉̆_𝑇𝜎∘𝜓(𝑡)≤𝑐𝑇^𝜂+1/𝑝1+^𝑡₀𝜓(𝑠)^𝑝d𝑠≤𝑐𝑇^𝜂+1/𝑝1+𝜓_∞. (97)

Proof. Let 𝜓 and 𝜙 be two continuous functions, since 𝐶([0,𝑇],𝐑^𝑛) is continuously embedded in ℒ^𝑝 and 𝑉̆_𝑇(𝜎∘𝜓−𝜎∘𝜙) belongs to Hol(𝜂). Moreover, sup_𝑡≤𝑇𝑉̆_𝑇𝜎∘𝜓𝟏_[𝑎,𝑏](𝑡)−𝑉̆_𝑇𝜎∘𝜙𝟏_[𝑎,𝑏](𝑡𝑉̆)≤𝑐_𝑇((𝜎∘𝜓−𝜎∘𝜙)𝟏_[𝑎,𝑏])_Hol(𝜂)≤𝑐(𝜎∘𝜓−𝜎∘𝜙)𝟏_{[𝑎,𝑏]ℒ𝑝}≤𝑐𝜙−𝜓_{ℒ𝑝([𝑎,𝑏])}≤𝑐sup_𝑡≤𝑇𝜓(𝑡)−𝜙(𝑡), (98) since 𝜎 is Lipschitz continuous.
Let 𝜙 and 𝜓 be two continuous functions on [0,𝑇]. Since 𝜎 is Lipschitz continuous, we have 𝜎𝜓(𝑡)(𝑡)𝜓(𝑡)+𝜀𝜙=𝜎+𝜀¹₀𝜎^′𝑢𝜓(𝑡)+(1−𝑢)𝜙(𝑡)d𝑢. (99) Moreover, since 𝜎 is Lipschitz, 𝜎^′ is bounded and ^𝑇₀¹₀𝜎^′(𝑢𝜓(𝑡)+(1−𝑢)𝜙(𝑡))d𝑢^𝑝d𝑡≤𝑐𝑇. (100) This means that (𝑡↦∫¹₀𝜎^′(𝑢𝜓(𝑡)+(1−𝑢)𝜙(𝑡))d𝑢) belongs to ℒ^𝑝. Hence, according to Hypothesis 1, 𝑉̆_𝑇¹₀𝜎^′𝑢𝜓(.)+(1−𝑢)𝜙(.)d𝑢_𝐶≤𝑐𝑇. (101) Thus, lim_𝜀→0𝜀⁻¹𝑉̆_𝑇𝜎∘𝜓+𝜀𝜙−𝑉̆_𝑇𝜎∘𝜓exists, (102) and 𝑉̆_𝑇∘𝜎 is Gâteaux differentiable and its differential is given by (95).
Since 𝜎∘𝜓 belongs to 𝐶([0,𝑇],𝐑^𝑛), according to Hypothesis 1, we have 𝑉̆_𝑇𝜎∘𝜓(𝑡)≤𝑐^𝑡₀𝑠^𝜂𝑝𝜎(𝜓(𝑠))^𝑝d𝑠^1/𝑝≤𝑐𝑇^𝜂𝑡₀(1+𝜓(𝑠)^𝑝)d𝑠^1/𝑝≤𝑐𝑇^𝜂+1/𝑝1+𝜓^𝑝_∞^1/𝑝≤𝑐𝑇^𝜂+1/𝑝1+𝜓_∞. (103) The proof is thus complete.

Following [43], we then have the following nontrivial result.

Theorem 37. Assume that Hypothesis 1  (𝑝,𝜂) holds and that 𝜎 is Lipschitz continuous. Then, there exists one and only one measurable map from Ω×[0,𝑇]×[0,𝑇] into 𝔊 which satisfies the first two points of Definition (C). Moreover, 𝐄𝑍_𝑟,𝑡(𝑥)−𝑍_𝑟′,𝑡(𝑥^′)^𝑝≤𝑐1+|𝑥|^𝑝∨𝑥^′𝑝×𝑟−𝑟^′𝑝𝜂+𝑥−𝑥^′𝑝, (104) and, for any 𝑥∈𝐑^𝑛, for any 0≤𝑟≤𝑡≤𝑇, we have 𝐄𝑍_𝑟,𝑡(𝑥)^𝑝≤𝑐1+|𝑥|^𝑝𝑒^{𝑐𝑇𝜂𝑝+1}. (105) Note even if 𝑥 and 𝑥^′ are replaced by 𝜎{𝐵̆^𝑇(𝑢),𝑡≤𝑢} measurable random variables, the last estimates still hold.

Proof. Existence, uniqueness, and homeomorphy of a solution of (C) follow from [43]. The regularity with respect to 𝑟 and 𝑥 is obtained as usual by BDG inequality and Gronwall Lemma. For 𝑥 or 𝑥^′ random, use the independence of 𝜎{𝐵̆^𝑇(𝑢),𝑡≤𝑢} and 𝜎{𝐵̆^𝑇(𝑢),𝑟∧𝑟^′≤𝑢≤𝑡}.

Theorem 38. Assume that Hypothesis 1  (𝑝,𝜂) holds and that 𝜎 is Lipschitz continuous and sublinear. Then, for any 𝑥∈𝐑^𝑛, for any 0≤𝑟≤𝑠≤𝑡≤𝑇, (𝜔,𝑟)↦𝑍_𝑟,𝑠(𝜔,𝑍_𝑠,𝑡(𝑥)) and (𝜔,𝑟)↦𝑍⁻¹_𝑟,𝑡(𝜔,𝑥) belong to 𝕃_𝑝,1.

Proof. According to ([44], Theorem 3.1), the differentiability of 𝜔↦𝑍_𝑟,𝑡(𝜔,𝑥) is ensured. Furthermore, ∇_𝑢𝑍_𝑟,𝑡=−𝑉̆_𝑇𝜎∘𝑍_.,𝑡𝟏_[𝑟,𝑡](𝑢)−^𝑡_𝑟𝑉̆_𝑇𝜎^′𝑍_.,𝑡.∇_𝑢𝑍_.,𝑡𝟏_[𝑟,𝑡](𝑠)d𝐵̆(𝑠), (106) where 𝜎^′ is the differential of 𝜎. For 𝑀>0, let 𝜉_𝑀=inf𝜏,∇_𝑢𝑍_𝜏,𝑡^𝑝≥𝑀,𝑍^𝑀_𝜏,𝑡=𝑍_{𝜏∨𝜉𝑀,𝑡}. (107) Since 𝑉̆_𝑇 is continuous from ℒ^𝑝 into itself and 𝜎 is Lipschitz, according to BDG inequality, for 𝑟≤𝑢, 𝐄∇_𝑢𝑍^𝑀_𝑟,𝑡^𝑝≤𝑐𝐄𝑉̆_𝑇𝜎∘𝑍^𝑀_.,𝑡𝟏_[𝑟,𝑡](𝑢)^𝑝+𝑐𝐄^𝑡_𝑟𝑉̆_𝑇𝜎^′𝑍^𝑀_.,𝑡∇_𝑢𝑍^𝑀_.,𝑡𝟏_[𝑟,𝑡](𝑠)^𝑝d𝑠≤𝑐1+𝐄^𝑡_𝑟𝑢^𝑝𝜂𝑢_𝑟𝑍_𝜏,𝑡^𝑝d𝜏d𝑢+𝐄^𝑡_𝑟𝑠^𝑝𝜂𝑠_𝑟∇_𝑢𝑍^𝑀_𝜏,𝑡^𝑝d𝜏d𝑠≤𝑐1+𝐄^𝑡_𝑟𝑍_𝜏,𝑡^𝑝𝑡^𝑝𝜂+1−𝜏^𝑝𝜂+1d𝜏+𝐄^𝑡_𝑟∇_𝑢𝑍^𝑀_𝜏,𝑡^𝑝𝑡^𝑝𝜂+1−𝜏^𝑝𝜂+1d𝜏≤𝑐𝑡^𝑝𝜂+11+𝐄^𝑡_𝑟𝑍_𝜏,𝑡^𝑝d𝜏+𝐄^𝑡_𝑟∇_𝑢𝑍^𝑀_𝜏,𝑡^𝑝d𝜏. (108) Then, Gronwall Lemma entails that 𝐄∇_𝑢𝑍^𝑀_𝑟,𝑡^𝑝≤𝑐1+𝐄^𝑡_𝑟𝑍_𝜏,𝑡^𝑝d𝜏, (109) hence by Fatou lemma, 𝐄∇_𝑢𝑍_𝑟,𝑡^𝑝≤𝑐1+𝐄^𝑡_𝑟𝑍_𝜏,𝑡^𝑝d𝜏. (110) The integrability of 𝐄[|∇_𝑢𝑍_𝑟,𝑡|^𝑝] with respect to 𝑢 follows.
Now, since 0≤𝑟≤𝑠≤𝑡≤𝑇, 𝑍_𝑠,𝑡(𝑥) is independent of 𝑍_𝑟,𝑠(𝑥), thus the previous computations still hold and (𝜔,𝑟)↦𝑍_𝑟,𝑠(𝜔,𝑍_𝑠,𝑡(𝑥)) belongs to 𝕃_𝑝,1.
According to [45], to prove that 𝑍⁻¹_𝑟,𝑡(𝑥) belongs to 𝔻_𝑝,1, we need to prove that (1)for every ℎ∈ℒ², there exists an absolutely continuous version of the process (𝑡↦𝑍⁻¹_𝑟,𝑡(𝜔+𝑡ℎ,𝑥)), (2)there exists 𝐷𝑍⁻¹_𝑟,𝑡, an ℒ²-valued random variable such that, for every ℎ∈ℒ², 1𝑡𝑍⁻¹_𝑟,𝑡(𝜔+𝑡ℎ,𝑥)−𝑍⁻¹_𝑟,𝑡(𝜔,𝑥)−−−−→^𝑡→0𝑇₀𝐷𝑍⁻¹_𝑟,𝑡(𝑠)ℎ(𝑠)d𝑠, (111) where the convergence holds in probability, (3)𝐷𝑍⁻¹_𝑟,𝑡 belongs to ℒ²(Ω,ℒ²). We first show that 𝐄𝜕𝑍_𝑟,𝑡𝜕𝑥(𝜔,𝑍⁻¹_𝑟,𝑡(𝑥))^−𝑝isfinite. (112) Since 𝜕𝑍_𝑟,𝑡()𝜕𝑥𝜔,𝑥=Id+^𝑡_𝑟𝑉̆_𝑇𝜎^′𝑍_.,𝑡(𝑥)𝜕𝑍_.,𝑡(𝜔,𝑥)(𝑠)d𝐵̆(𝑠),𝜕𝑥 (113) let Θ_𝑣=sup_𝑢≤𝑣|𝜕_𝑥𝑍_𝑢,𝑡(𝑥)|. The same kind of computations as above entails that (for the sake of brevity, we do not detail the localisation procedure as it is similar to the previous one) 𝐄Θ_𝑣^2𝑞≤𝑐+𝑐𝐄^𝑡_𝑢Θ_𝑠^{2(𝑞−1)𝑠}_𝑢𝜕_𝑥𝑍_𝜏,𝑡(𝑥)^𝑝∣d𝜏^2/𝑝d𝑠+𝑐𝐄^𝑡_𝑢Θ_𝑠^𝑞−2𝑠_𝑢𝜕_𝑥𝑍_𝜏,𝑡(𝑥)^𝑝∣d𝜏^2/𝑝2d𝑠. (114) Hence, 𝐄Θ_𝑣^2𝑞≤𝑐1+^𝑡_𝑣𝐄Θ_𝑠^2𝑞d𝑠, (115) and (112) follows by Fatou and Gronwall lemmas. Since 𝑍_𝑟,𝑡(𝜔,𝑍⁻¹_𝑟,𝑡(𝜔,𝑥))=𝑥, the implicit function theorem implies that 𝑍⁻¹_𝑟,𝑡(𝑥) satisfies the first two properties and that ∇𝑍_𝑟,𝑡𝜔,𝑍⁻¹_𝑟,𝑡(𝑥)+𝜕𝑍_𝑟,𝑡𝜕𝑥𝜔,𝑍⁻¹_𝑟,𝑡(𝑥)∇𝑍⁻¹_𝑟,𝑡(𝜔,𝑥). (116) It follows by Hölder inequality and (112) that 𝐷𝑍⁻¹_𝑟,𝑡(𝑥))_𝑝,1≤𝑐𝑍_𝑟,𝑡(𝑥))_2𝑝,1(𝜕_𝑥𝑍_𝑟,𝑡(𝑥))⁻¹_2𝑝; (117) hence 𝑍⁻¹_𝑟,𝑡 belongs to 𝕃_𝑝,1.

Acknowledgment

The author would like to thank the anonymous referees for their thorough reviews, comments, and suggestions significantly contributed to improve the quality of this contribution.