Linear Backward Stochastic Differential Equations with Gaussian Volterra processes

Explicit solutions for a class of linear backward stochastic differential equations (BSDE) driven by Gaussian Volterra processes are given. These processes include the multifractional brownian motion and the multifractional Ornstein-Uhlenbeck process. By an It\^o formula, proven in the context of Malliavin calculus, the BSDE is associated to a linear second order partial differential equation with terminal condition whose solution is given by a Feynman-Kac type formula. An application to self-financing trading strategies is discussed.


Introduction
Backward stochastic differential equations (BSDE) driven by a brownian motion have been introduced by Bismut [1] in the linear case. Non linear BSDE have been studied first by Paradoux and Peng [9]. Since then BSDE have been of interest due to the connections with partial differential equations (PDE) and their applications, especially in mathematical finance, stochastic differential games and stochastic control.
In this paper, we study the BSDE where X = {X t , 0 t T } is a zero mean continuous Gaussian process given by where W = {W t , 0 t T } is a standard Brownian motion and K : [0, T ] 2 → R is a square integrable kernel, i.e. [0,T ] 2 K(t, s) 2 dtds < +∞. We assume that K is of Volterra type, i.e, K(t, s) = 0 whenever t < s. Usually, the representation (2) is called a Volterra representation of X. The kernel K in (2) defines a linear operator in L 2 ([0, T ]) given by (Kσ) t = t 0 K(t, s)σ s ds, σ ∈ L 2 ([0, T ]). The process (N t , 0 t T ) is given by with σ being a deterministic function and K * the adjoint operator of K ( [3], lemma 1) given by (11) (see also [6]). f is called the generator of the BSDE, g(N T ) the terminal condition. In [3] K is called regular if it satisfies We assume the following condition on K(t, s) which is more restrictive than (H) ( [3], [4]): (H1) K(t, s) is continuous for all 0 < s t < T and continuously differentiable in the variable t in 0 < s < t < T , (H2) For some 0 < α, β < 1 2 , there is a finite constant c > 0 such that ∂K ∂t (t, s) c(t − s) α−1 t s β , for all 0 < s < t < T.
The covariance function of X is given by The aim of this paper is to study the nonlinear BSDE (1) and we establish the relation to the associated partial differential equation (6) that opens the possibility to solve the BSDE by means of classical or viscosity solutions of the PDE. This generalizes a resultat in [5] obtained for fractional brownian motion. For the existence and uniqueness for the solution of the BSDE (1), essentially two methods were applied. The existence and uniqueness of the solution of (1) is addressed in Theorem 2.1 by means of the associated PDE. Another proof will be treated in a separate paper without making reference to this PDE, but with probabilistic and functional theoretic methods. In this paper, we prove in Theorem 2.3 that Z t = −σ t ∂ ∂x u(t, x) for BSDE's with generators f ∈ C 0,1 ([t 0 , T ] × R 3 ) of polynomial growth and under the injectivity hypothesis for the adjoint operator to K. We discuss this hypothesis in Remark 4.5. This hypothesis is satisfied for the mbf (with H > 1 2 ) and comes from the preliminary Lemma 4.4 that shows a kind of orthogonality between Lebesgue and divergence integrals. The proof of this Lemma generalizes the proof given by Y. Hu and S. Peng in 2009 for fractional brownian motion ( [5]). The proof of Lemma 4.4 depends itself on Proposition 4.2 where we show that, for any continuous function of exponential growth h, h(N t ) admits a representation as a divergence integral of the heat kernel operator evaluated for N. We generalize in Theorem 2.5 a comparison theorem, known for BSDE with respect to brownian motion (see for example [10]), to the solution of the BSDE (1) and study the continuity of the law of Y in Theorem 2.6.
Here is the organisation of the paper. In Section 2 we state the main results on the solution of (1). In Section 3 we give some definitions and complements on the Skorohod integral with respect to Volterra processes. Section 4 is devoted to the proofs of the results and contains other results of independent interest, like an It formula for N proven in the framework of Malliavin calculus (Theorem 4.1) and a transfer formula (Proposition 4.6).

Statement of the main results
We consider X defined on the probability space (Ω, F , P ) and given by (2). Let F = {F t ⊂ F , t ∈ [0, T ]} the filtration generated by X and augmented by the P -null sets. Let N be given by (3), where σ is a bounded function on [0, T ] and suppose that d dt V ar(N t ) > 0 for all t ∈ (0, T ). Let t 0 0 be fixed, and denote by L 2 (F, R) the set of F-adapted R-valued processes Z such that E( T t0 | Z t | 2 dt) < ∞. We consider the non linear BDSE for the processes Y = (Y t , t ∈ [t 0 , T ]) ∈ L 2 (F, R) and Z = (Z t , t ∈ [t 0 , T ]) ∈ L 2 (F, R) given by We show that (5) is associated to the following second order PDE with terminal condition The association of this PDE to the BSDE (5) is proven by means of the Itô formula for the class of functions F ∈ C 1,2 ([0, T ]×R) that satisfy, together with their partial derivatives, the growth condition for all t ∈ [0, T ] and x ∈ R, where c, λ are positive constants such that λ < 1 4 ( sup This implies and the same property holds for ∂/∂tF (t, x), ∂/∂xF (t, x) and ∂ 2 /∂x 2 F (t, x).
The main results of this paper is stated below: Theorem 2.1. If (6) has a classical solution u(t, x) that satisfies (7) with F replaced by u, then (Y t , Z t ) := (u(t, N t ), −σ t ∂u ∂x (t, N t )), t ∈ (t 0 , T ) satisfies (5) and Y, Z ∈ L 2 (F, R). Remark 2.2. The mild (or evolution) solution of (6) is given by .
In fact, theorem 4.1 ([6]) applied to n = 1 gives u(t, x) = G(T, t, x − y)g(y)dy in the linear case and (9) follows by classical arguments for the mild form of a nonlinear PDE.

Remark 2.4. If the PDE (6) has a unique classical solution, then under the hypotheses of Theorem 2.3 the BSDE (5) has a unique solution with
In fact, let us suppose that (5) By uniqueness of solutions of (6), where Suppose that both BSDE's satisfy the hypotheses of Theorem 2.1 and Theorem 2.3. If f 1 (·, ·, y, z) f 2 (·, ·, y, z) for all y, z, and Theorem 2.6. Suppose that Y is a solution of the BSDE (5) and u is a classical solution of the PDE (6).
, the law of Y is absolutely continuous.

Preliminaries
In this section, we recall important definitions and results concerning the Malliavin calculus for Volterra process. These results will be used to study the BSDE (5).
Let E be the set of step functions of [0, T ], and let K * Therefore For σ, σ ∈ E this may be extended to The operator K * T is an isometry between H and a closed subspace of L 2 ([0, T ]), and · H is a semi-norm on H. Furthermore, for ϕ, ψ ∈ H, For further use let Note that φ(r, s) = ∂ 2 /∂s∂rR(r, s) (r = s) (φ may be infinite on the diagonal r = s). Let | H | be the closure of the linear span of indicator functions with respect to the semi-norm given by We briefly recall some basic elements of the stochastic calculus of variations with respect to X given by (2). We refer to [2] and [7] for a more complete presentation. Let S be the set of random variables of the form F = f (X(ϕ 1 ), ...., X(ϕ n )), where n 1, f : R n → R is a C ∞ -function such that f and its partial derivatives have at most polynomial growth, and ϕ 1 , ..., ϕ n ∈ H. The derivative of F is an H-valued random variable, and D X is a closable operator from L p (Ω) to L p (Ω; H) for all p 1. We denote by D X 1,p the closure of S with respect to the norm We denote by Dom(δ X ) the subset of L 2 (Ω, H) composed of those elements u for which there exists a positive constant c such that For u ∈ L 2 (Ω; H) in Dom(δ X ), δ X (u) is the element in L 2 (Ω) defined by the duality relationship We also use the notation T 0 u t δX t for δ X (u). A class of processes that belong to the domain of δ X is given as follows: let S |H| be the class of H-valued random variables u = n j=1 F j h j (F j ∈ S, h j ∈| H |). In the same way D X 1,p (| H |) is defined as the completion of S |H| under the semi-norm The space D X 1,2 (| H |) is included in the domain of δ X , and we have, for u ∈ D X 1,2 (| H |), Remark 3.3. Let N t be given by (3), then we have: 4 Proofs of the main results

Proof of Theorem 2.1
Theorem 2.1 is proven by means of an It formula given in [6]. For the convenience of the reader we state it here. We have the following theorem that is proved in [6].
Now, we are in position to prove Theorem 2.1.
By plugging (6) into the first term on the right side of the equation above we get Therefore the pair (Y, Z) given by Y t = u(t, N t ) and Z t = −σ t ∂u ∂x (t, N t ) is a solution of (5). From (7) we conclude E T t0 Z 2 s ds < ∞ and E T t0 Y 2 s ds < ∞. This proves the theorem.

Proof of Theorem 2.3
The proof of this theorem needs some auxiliary results. Let Let k be a continuous function, such that the following is well defined: A straightforward calculation shows The following proposition will be needed in the proof of Lemma 4.4 below.

Proposition 4.2. Let V ar(N t ) be increasing. Assume that h is a continuous function, and suppose that there exist positive constants c and λ
Proof. We want to apply the Itô formula (Theorem 4.1) to F (t, N t ) = P V ar(Ns)−V ar(Nt) h (N t ), t < s. We begin by verifying the hypotheses of the Itô formula: where the last inequality follows by choosing M = c √ 1−4λV arNT and λ = 2λ ′ . The proofs for the upper bounds of ∂ ∂x F (t, x) and ∂ 2 ∂x 2 F (t, x) are similar.
Moreover, for any ǫ > 0, there is a constant K ǫ such that Then, we get with a suitable constant M and 2λ ′ = λ.
M ′ e λx 2 , (for fixed t, t < s) with a suitable constant M ′ and 2λ ′ = λ. Moreover, for fixed t, t < s we have ∂ ∂t Furthermore, we have Therefore since h is a continuous function and ∂ ∂x P V ar(Ns)−V ar(Nt) h (N t ) ∈ D X 1,2 (| H |) ⊂ Dom(δ X ).
In this case we must add the following hypothesis in the previous proposition V ar(Ns) |V ar(Ns)−V ar(Nt)| 1, for all 0 t < s < T. We need this hypothesis to verify (7) and to apply the Itô formula.
The following lemma will play an important role in this paper.
Proof. First we show that In fact, on the one hand, for all u ∈ (0, s), we have

Moreover, we have
2V ar(Ns ) 2πV ar(N s ) dx, since N s ∼ N (0, V ar(N s )). Therefore, On the other hand, we apply Proposition 4.2 to h(x) = b(s, x) (for fixed s) and In fact, for F ∈ L 2 (Ω, H) Therefore for all z ∈ R ( [5]). Now by differentiating with respect to t, we get for all t > u and z ∈ R. An integration by parts formula yields R p V ar(Nt)−V ar(Nu) (z − y) ∂ ∂y b(t, y)dy = 0.
Let u → t then we see that ∂ ∂y b(t, y) = 0 for all t ∈ (t 0 , T − ǫ) and y ∈ R. This means that there is a b 1 (t) such that b(t, y) = b 1 (t). Now from (28) we have This implies that b 1 (t) = 0 for all t ∈ (t 0 , T ) and accordingly b(t, y) = 0 for all t ∈ (t 0 , T ) and y ∈ R. Thus, a(t, y) = 0. which implies that K * t Z(·, N · ) u = An evident hypothesis is the injectivity of K * t Z(·, N · ) u as a function of u ∈ [t 0 , t]. Let us look for a sufficient condition for injectivity: , we obtain a contradiction. Therefore, a sufficient condition for K * T to be injective is K t0 (s) = 0 on [t 0 , T ]. This last hypothesis is satisfied in particular if ∂ ∂s K(s, u) > 0 (or < 0) for u ∈ (t 0 , s) for all s ∈ (t 0 , T ).
We are now ready to prove Theorem 2.3. By Theorem 4.1 we get Moreover, we can write We evaluate for t = t 0 and we make the subtracting with the above equation. We obtain for all Using ( Since u, v satisfy (7) and f ∈ C 0,1 ([0, T ] × R 3 ) is of polynomial growth, we apply now Lemma 4.4 and obtain v(t, x) = −σ t ∂ ∂x u(t, x), ∀t ∈ (t 0 , T ), x ∈ R.

Proof of Theorem 2.5 and Theorem 2.6
We start with proving this preliminary result: Proposition 4.6. (Transfer formula) Let X be given by (2). Let D W be the Malliavin derivative with respect to the Brownian motion. Then, K * T D X . = D W . on D X 1,2 .
We extend the equality (K * T D X . F ) t = D W t F to the closure of the linear combinations of S by means of the norm F 2 1,2 = E|F | 2 + E D X F 2 H . We have F 1 , F 2 ∈ S, a 1 , a 2 ∈ R : (K * T D X . (a 1 F 1 + a 2 F 2 )) t = D W t (a 1 F 1 + a 2 F 2 ) because K * , D X and D W are linear operators. Let (F n ) n∈N ⊂ span(S) such that (F n ) n∈N converges in norm . 1,2 to F. Then, F ∈ D X 1,2 . Thus, D X F − D X F n H → 0, with n −→ ∞. Thus, K * T D X F − K * T D X F n L 2 (Ω,L 2 (0,T )) → 0, with n −→ ∞ by isometry. Since K * T D X F n = D W F n , (D W F n ) n∈N is a convergent sequence in L 2 (Ω, L 2 (0, T )). By proposition 1.2.1 ([7]) the limit in L 2 (Ω, L 2 (0, T )) is D W F. Therefore, K * T D X . = D W . .

Proof of Theorem 2.5
Let u i (t, x) be the solution to (6) with f replaced by f i , g replaced by g i . Then the solution to (10) is given by Y i t = u i (t, N t ). It suffices to prove u 1 (t, x) u 2 (t, x). Denote ρ t = d dt V ar(N t ), and ζ t = t 0 ρ s δW s , where W is standard Brownian motion. Applying Itô's formula with respect to W, we have du i (t, ζ t ) = ρ t ∂u i ∂x (t, ζ t )dW t + ∂u i ∂t (t, ζ t )dt + 1 2