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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.

Fractional Brownian motion (fBm for short) of Hurst index

In the probabilistic approach [

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

In [

In what follows, there is no restriction on the dimension, but we need to assume that any component of

The first critical point is that when we consider

This paper is organized as follows. After some preliminaries on fractional Sobolev spaces, often called Besov-Liouville space, we address, in Section

Let

Its topological dual is denoted by

The Besov-Liouville space

Analogously, the Besov-Liouville space

We then have the following continuity results (see [

Consider the following.

If

For any

For any

For

Our reference probability space is

We denote by

We introduced the temporary notation

Note that

The operator

We thus have, for any cylindrical function

Since

For any

By duality, an analog result follows for divergences.

A process

For

In anticipative calculus, the notion of trace of an operator plays a crucial role, We refer to [

Let

It is easily shown that the notion of trace does not depend on the choice of the CONB.

A family

For instance, the family

A partition

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.

A continuous map

For instance, an operator

Let

Note carefully that the identity map is causal but not strictly causal. Indeed, if

Assume the resolution of the identity to be either

Let

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

The set of strictly causal operators coincides with the set of quasinilpotent operators, that is, trace-class operators such that

Moreover, we have the following stability theorem.

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

Let

If

Let

We then have the following key theorem.

Assume the resolution of the identity to be

Since

In what follows,

Let

This is a purely algebraic lemma once we have noticed that

In what follows,

The linear map

Assume that Hypothesis

For any subdivision

We say that

The first example is the so-called Lévy fractional Brownian motion of Hurst index

The other classical example is the fractional Brownian motion with stationary increments of Hurst index

The next theorem then follows from [

Assume that Hypothesis

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

Assume that Hypothesis

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

Let

By a solution of

For any

For any

For any

Equation

By a solution of

For any

For any

Equation

For any

At last consider the equation, for any

By a solution of

For any

For any

Equation

Assume that

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

Assume that

Set

The first part of the next result is then immediate.

Assume that

According to Theorems

According to Theorem

For

Without loss of generality, assume that

Thus, we have the main result of this paper.

Assume that

Under the hypothesis, we know that

In the reverse direction, two distinct solutions of

The proof of Theorem

Assume that Hypothesis

For each

For

Assume that Hypothesis

Simple random fields of the form:

For any

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

Let

The next corollary follows by a classical density argument.

Let

We first study the divergence term. In view of Theorem

Assume that Hypothesis

Let

Let

Since

Following [

Assume that Hypothesis

Existence, uniqueness, and homeomorphy of a solution of

Assume that Hypothesis

According to ([

Now, since

According to [

for every

there exists

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.