The paper presents the study on the existence and uniqueness (strong and in law) of a class of non-Markovian SDEs whose drift contains the derivative in the sense of distributions of a continuous function.

This paper discusses in detail a framework of one-dimensional stochastic differential equations (henceforth abbreviated by SDEs) with distributional drift and possible path-dependency. To our best knowledge, this is the first paper which approaches a class of non-Markovian SDEs with distributional drifts.

The main objective of this paper is to analyze the solution (existence and uniqueness) of the martingale problem associated with SDEs of the type

Path-dependent SDEs were investigated under several aspects. Under standard Lipschitz regularity conditions on the coefficients, it is known (see, e.g., Theorem 11.2 [

The Markovian case (

In [

Let us come back to the objective of the present paper in which the path-dependent drift contains the derivative in the sense of distributions of a continuous function

The strategy of this paper consists in eliminating the distributional drift of the so-called Zvonkin’s transform, see [

Moreover, Corollary

Several results of the present paper can be partially extended to the multidimensional case, by using the techniques developed in the Markovian case in [

Let

We recall some notions from [

An

Any

If

As in the case of Markovian SDEs, it is possible to formulate the notions of strong existence, pathwise uniqueness, existence and uniqueness in law for path-dependent SDEs of the type (

The previous equation will be denoted by

Let

Suppose

When

We say that a continuous stochastic process

We will also say that the couple

If a solution exists, we say that

We say that

We remark that in the classical literature of martingale problems, see [

In the sequel, when the measurable space

Below we introduce the analogous notion of strong existence and pathwise uniqueness for our martingale problem.

Let

We say that

We say that

In this section we recall some basic notations and results from [

Equation (

(1) We observe

(2) The proof follows by setting

We now formulate a standing assumption.

Σ

Under Assumption

It is easy to verify that Assumption

When

We assume that Assumption

By Remark

By Notation

Let us discuss the converse implication. Suppose that

By setting

In [

By Proposition 3.2 of [

Let

The first result explains how to reduce our path-dependent martingale problem to a path-dependent SDE.

(1) We start proving the direct implication. According to (

In particular, by Proposition

Next, we prove the converse implication. Suppose that

(2) The converse implication follows in the same way as for item (1). The proof of the direct implication follows directly by Itô’s formula. □

By Remark

If

An immediate consequence of Proposition

We fix here the same conventions as in Section

At this point, we introduce the following technical assumption, which is in particular verified if Γ is bounded and

Let

Proposition

Next, we need a slight adaptation of the Dambis–Dubins–Schwarz theorem to the case of a finite interval. For the sake of completeness, we give the details here.

Let us define

The proposition below is an adaptation of a well-known argument for Markov diffusions.

Let

Let

Let

Proposition

By Proposition

We use here again the notation

Let

By means of localization (similarly to Proposition 5.3.10 in [

By Girsanov’s theorem, under

Before exploring conditions for strong existence and uniqueness for the martingale problem, we state and prove Proposition

Let

One typical example of nonanticipative functional which satisfies (3) is given by

The proof of the Proposition

Let

We take the expectation, and applying Fubini’s theorem in (

We come back to the framework of the beginning of Section

It follows by using Assumptions

By Proposition

In order to prove existence, we will apply Theorem

By Theorem

As a consequence of Proposition

Let us suppose below that

Let

We say that

We say that pathwise uniqueness holds for equation

Let

We say that existence in law holds for

Let

We say that

The authors are very grateful to both referees for their very careful reading of the first submitted version and for their relevant comments which allowed to drastically improve the paper.