Time Reversal of Volterra Processes Driven Stochastic Differential Equations
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 is the Gaussian process which admits the following representation: for any , where is a one-dimensional Brownian motion and is a triangular kernel, that is, for , 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 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 , Feyel and de la Pradelle , and Russo and Vallois [3, 4]. Then, came the notion of rough paths was introduced by Lyons , whose application to fBm relies on the work of Coutin and Qian . 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 , and Gubinelli . The integral with respect to fBm was shown to exist as the unique process satisfying some characterization (analytic in the case of , algebraic in ). 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 . Within the probabilistic framework, one knows how to define a stochastic integral for any value of but one cannot prove existence and uniqueness of SDEs whatever the value of . 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 where is defined by and is the adjoint of in . In particular, there exists such that for any so that even if is adapted (with respect to the Brownian filtration), the process is anticipative. However, the stochastic integral process 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  in a different context. It was shown there that the time reversal of the solution of an fBm-driven SDE of the form 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 .
In what follows, there is no restriction on the dimension, but we need to assume that any component of is an fBm of Hurst index greater than . Consider that we want to solve the following equation: where is a deterministic function whose properties will be fixed below. It turns out that it is essential to investigate the more general equations: The strategy is then as follows. We will first consider the reciprocal problem:
The first critical point is that when we consider , this process solves an adapted, past-dependent, and stochastic differential equation with respect to a standard Brownian motion. Moreover, because is lower-triangular and sufficiently regular, the trace term vanishes in the equation defining . 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  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 and . 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  for a related application. In Section 4, we show that solving reduces to solve a past-dependent stochastic differential equation with respect to a standard Brownian motion; see below. Then, we prove existence, uniqueness, and some properties of this equation. Technical lemmas are postponed to Section 5.
2. Besov-Liouville Spaces
Let be fix real number. For a measurable function , we define by For , will represent the restriction of to , that is, . For any linear map , we denote by , its adjoint in . For , the space of -Hölder continuous functions on is equipped with the norm:
Its topological dual is denoted by . For (denoted by for short), the left and right fractional integrals of are defined by where and . For any , , any and where , we have
The Besov-Liouville space is usually equipped with the norm:
Analogously, the Besov-Liouville space is usually equipped with the norm:
We then have the following continuity results (see [2, 36]):
Proposition 1. Consider the following.(i) If , then is a bounded operator from into with .(ii) For any and any is continuously embedded in provided that .(iii) For any is compactly embedded in .(iv) For , the spaces and are canonically isomorphic. We will thus use the notation to denote any of these spaces.
3. Malliavin Calculus and Time Reversal
Our reference probability space is , the space of -valued, continuous functions, null at time . The Cameron-Martin space is denoted by and is defined as . In what follows, the space is identified with its topological dual. We denote by the canonical embedding from into . The probability measure on is such that the canonical map defines a standard -dimensional Brownian motion. A mapping from into some separable Hilbert space is called cylindrical if it is of the form , where for each , and is a sequence of . For such a function we define as where is the image of by the map . From the quasi-invariance of the Wiener measure , it follows that is a closable operator on , , and we will denote its closure with the same notation. The powers of are defined by iterating this procedure. For , we denote by the completion of -valued cylindrical functions under the following norm:
We denote by the space . The divergence, denoted as , is the adjoint of : belongs to whenever, for any cylindrical , and, for such a process ,
We introduced the temporary notation 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 an -dimensional standard Brownian motion and its time reversal. Consider the following map: and the commutative diagram: (18)
Note that since . For a function , we define the following:
The operator (resp., ) is the Malliavin gradient associated with a standard Brownian motion (resp., its time reversal). Since we can consider as a cylindrical function with respect to the standard Brownian motion. As such its gradient is given by
We thus have, for any cylindrical function ,
Since and is continuous from into itself for any , it is then easily shown that the spaces and (with obvious notations) coincide for any and that (22) holds for any element of one of these spaces. Hence we have proved the following theorem.
Theorem 2. For any and any integer , the spaces and coincide. For any for some ,
By duality, an analog result follows for divergences.
Theorem 3. A process belongs to the domain of if and only if belongs to the domain of , and, then, the following equality holds:
Proof. For , for cylindrical , we have on the one hand: and on the other hand, Since this is valid for any cylindrical , (24) holds for . Now, for in the domain of divergence (see [37, 38]), where is an orthonormal basis of . Thus, we have where we have taken into account that is in an involution. Since is an orthonormal basis of , identity (24) is satisfied for any in the domain of .
3.1. Causality and Quasinilpotence
In anticipative calculus, the notion of trace of an operator plays a crucial role, We refer to  for more details on trace.
Definition 4. Let be a bounded map from into itself. The map is said to be trace class, whenever for one CONB of ,
Then, the trace of is defined by
It is easily shown that the notion of trace does not depend on the choice of the CONB.
Definition 5. A family of projections , in is called a resolution of the identity if it satisfies the conditions:(1) and (2)(3) for any and .
For instance, the family is a resolution of the identity in .
Definition 6. A partition of is a sequence . Its mesh is denoted by and defined by .
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 from into itself is said to be causal if and only if the following condition holds:
For instance, an operator in integral form is causal if and only if for , that is, computing needs only the knowledge of up to time 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 .
Definition 8. Let be a causal operator. It is a strictly causal operator, whenever for any , there exists a partition of such that, for any ,
Note carefully that the identity map is causal but not strictly causal. Indeed, if , for any , since is a projection. However, if is hyper-contractive, we have the following result.
Lemma 9. Assume the resolution of the identity to be either or . If is an causal map continuous from into for some then is strictly causal.
Proof. Let be any partition of . Assume that , and the very same proof works for the other mentioned resolution of the identity. According to Hölder formula, we have for any , Then, for any , there exists such that implies for any and any .
The importance of strict causality lies in the next theorem we borrow from .
Theorem 10. The set of strictly causal operators coincides with the set of quasinilpotent operators, that is, trace-class operators such that for any integer .
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 be a resolution of the identity in . Consider the filtration defined as An -valued random variable is said to be adapted if, for any , the real valued process is -adapted. We denote by the set of adapted random variables belonging to .
If , the notion of adapted processes coincides with the usual one for the Brownian filtration, and it is well known that a process is adapted if and only if for . This result can be generalized to any resolution of the identity.
Theorem 13 (Proposition 3.1 of ). Let belongs to . Then is adapted if and only if is causal.
We then have the following key theorem.
Theorem 14. Assume the resolution of the identity to be either and that is an -strictly causal continuous operator from into for some . Let be an element of . Then, is of trace class and we have .
Proof. Since is adapted, is -causal. According to Theorem 11, is strictly causal and the result follows by Theorem 10.
In what follows, is the resolution of the identity in the Hilbert space defined by and is the resolution of the identity defined by . The filtrations and are defined accordingly. Next lemma is immediate when is given in the form of . 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 be a map from into itself such that is -causal. Let be the adjoint of in . Then, the map is -causal.
Proof. This is a purely algebraic lemma once we have noticed that For, it suffices to write We have to show that since and . Now, (37) yields Use (37) again to obtain since is -causal.
3.2. Stratonovitch Integrals
In what follows, belongs to and is a linear operator. For any , we set the following.
Hypothesis 1 (). The linear map is continuous from into the Banach space .
Definition 16. Assume that Hypothesis 1 () holds. The Volterra process associated to , denoted by , is defined by
For any subdivision of , that is, , of mesh , we consider the Stratonovitch sums:
Definition 17. We say that is -Stratonovitch integrable on whenever the family , defined in (43), converges in probability as goes to . In this case the limit will be denoted by .
Example 18. The first example is the so-called Lévy fractional Brownian motion of Hurst index defined as This amounts to say that . Thus Hypothesis 1 holds provided that .
Example 19. The other classical example is the fractional Brownian motion with stationary increments of Hurst index , which can be written as where The Gauss hypergeometric function (see ) is the analytic continuation on of the power series: We know from  that is an isomorphism from onto and Consider that . Then it is clear that hence we are in the framework of Definition 17 provided that we take . Hypothesis 1 is satisfied provided that .
The next theorem then follows from .
Theorem 20. Assume that Hypothesis 1 () holds. Assume that belongs to . Then is -Stratonovitch integrable, and there exists a process which we denote by such that belongs to and The so-called “trace-term” satisfies the following estimate: for some universal constant . Moreover, for any , is -Stratonovitch integrable and and we have the maximal inequality: where does not depend on .
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 () holds. Let belong to and let . Assume furthermore that is -causal and that is -adapted. Then, where the last integral is an Itô integral with respect to the time reversed Brownian motion .
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 for any . Thus, we are led to compute . If we had sufficient regularity, we could write since for for adapted. Obviously, there are many flaws in these lines of proof. The operator is not regular enough for such an expression of the trace to be true. Even more, there is absolutely no reason for 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 be the group of homeomorphisms of equipped with the distance. We introduce a distance on by where Then, is a complete topological group. Consider the equations: As a solution of is to be constructed by “inverting” a solution of , we need to add to the definition of a solution of or the requirement of being a flow of homeomorphisms. This is the meaning of the following definition.
Definition 23. By a solution of , we mean a measurable map: such that the following properties are satisfied.(1)For any , for any , is -measurable.(2)For any , for any , the processes and belong to for some . (3)For any , for any , the following identity is satisfied: (4)Equation is satisfied for any -a.s.
Definition 24. By a solution of , we mean a measurable map: such that the following properties are satisfied. (1)For any , for any , is measurable. (2)For any , for any , the processes and belong to for some . (3)Equation is satisfied for any -a.s.. (4)For any , for any , the following identity is satisfied:
At last consider the equation, for any , where is a standard -dimensional Brownian motion.
Definition 25. By a solution of , we mean a measurable map: such that the following properties are satisfied.(1)For any , for any , is measurable. (2)For any , for any , the processes and belong to for some . (3)Equation is satisfied for any -a.s..
Theorem 26. Assume that is an causal map continuous from into for and such that . Assume that is Lipschitz continuous and sublinear; see (96) for the definition. Then, there exists a unique solution to . Let denote this solution. For any , Moreover,
Since this proof needs several lemmas, we defer it to Section 5.2.
Theorem 27. Assume that is an -causal map continuous from into for and such that . For fixed , there exists a bijection between the space of solutions of on and the set of solutions of .
Proof. Set or equivalently According to Theorem 21, satisfies if and only if satisfies . The regularity properties are immediate since is stable by .
The first part of the next result is then immediate.
Corollary 28. Assume that is an -causal map continuous from into for and such that . Then has one and only one solution and for any , for any , the following identity is satisfied:
Proof. According to Theorems 27 and 26, has at most one solution since has a unique solution. As to the existence, points from to are immediately deduced from the corresponding properties of and (66).
According to Theorem 26, belongs to ; hence, we can apply the substitution formula and Set Then, in view of (68), appears to be the unique solution and thus . Point is thus proved.
Corollary 29. For fixed, the random field admits a continuous version. Moreover, We still denote by this continuous version.
Proof. Without loss of generality, assume that and remark that thus belongs to : According to Theorem 37, In view of Theorem 21, the stochastic integral which appears in is also a Stratonovitch integral; hence, we can apply the substitution formula and say Thus we can apply Theorem 37 and obtain that The right hand side of this equation is in turn equal to thus, we get Combining (72) and (75) gives hence the result comes
Thus, we have the main result of this paper.
Theorem 30. Assume that is an -causal map continuous from into for and such that . Then has one and only one solution.
Proof. Under the hypothesis, we know that has a unique solution which satisfies (67). By definition a solution of , the process belongs to ; hence, we can apply the substitution formula. Following the lines of proof of the previous theorem, we see that is a solution of .
In the reverse direction, two distinct solutions of would give rise to two solutions of by the same principles. Since this is definitely impossible in view of Corollary 28 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 .
Theorem 31. Assume that Hypothesis 1 () holds. Let belong to . If is of trace class, then Moreover,
Proof. For each , let be the functions . Let be the projection onto the span of the ; since is of trace class, we have (see ) Now, According to the proof of Theorem 20, the first part of the theorem follows. The second part is then a rewriting of (51).
For , let be the set of random fields: equipped with the seminorms, for any compact of .
Corollary 32 (substitution formula). Assume that Hypothesis 1 () holds. Let belong to . Let be a random variable such that belongs to . Then,
Proof. Simple random fields of the form: with smooth and in are dense in . 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 . Now for any partition , On the other hand, Hence, According to Theorem 20, (83) is satisfied for simple random fields.
Definition 33. For any , for in , we define as
By the very definition of trace class operators, the next lemma is straightforward.
Lemma 34. Let and be two continuous maps from into itself. Then, the map (resp. ) is of trace class if and only if the map (resp. ) is of trace class. Moreover, in such a situation,
The next corollary follows by a classical density argument.
Corollary 35. Let such that and are of trace class. Then, and are of trace class. Moreover, we have
Proof of Theorem 21. We first study the divergence term. In view of Theorem 3 we have According to Lemma 15 is causal, and, according to Lemma 9, it is strictly causal. Thus, Theorem 14 implies that is of trace class and quasinilpotent. Hence Corollary 35 induces that is trace class and quasinilpotent. Now, according to Theorem 2, we have According to Theorem 20, we have proved (54).
5.2. The Forward Equation
Lemma 36. Assume that Hypothesis 1 holds and that is Lipschitz continuous. Then, for any , the map is Lipschitz continuous and Gâteaux differentiable. Its differential is given by Assume furthermore that is sublinear; that is, Then, for any , for any ,
Proof. Let and be two continuous functions, since is continuously embedded in and belongs to . Moreover,
since is Lipschitz continuous.
Let and be two continuous functions on . Since is Lipschitz continuous, we have