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Fractional Anticipating Semilinear Equations
In this subsection we discuss the existence and uniqueness of solutions to anticipating semilinear equations driven by a fractional Brownian motion B with Hurst parameter H ∈ (0, 1/2). This type of equation was studied by Jien and Ma [57], and since it motivates the approach in our work, we give it in details. We consider the fractional anticipating equation ∫ t ∫ t Xt = ξ + b(s, Xs)ds + γsXsdBs, t ∈ [0, T ]. (3.1) 0 0 Here ξ ∈ Lp(Ω), p > 2, and b : Ω × [0, T ] × R → R is a measurable function such that: (H2) There exist ν ∈ L1([0, T ]), ν ≥ 0, and a positive constant L such that |b(ω, t, x) − b(ω, t, y)| ≤ νt|x − y|, ∫0T νtdt ≤ L, |b(ω, t, 0)| ≤ L, for all x, y ∈ R and almost all ω ∈ Ω. We observe that the above assumption guarantees that the pathwise equation ∫ t ζ (x) = x + ε−1 (T )b(T , s, ε (T )ζ (x))ds, t ∈ [0, T ], t 0 s s s s s s has a unique solution. Henceforth, we denote it by ζ.
Now we can state the existence of a unique solution equation (3.1): Theorem 3.1. Under Hypotheses (H1) and (H2), the process Xt = εtζt(At, ξ(At)) (3.2) is the unique solution in L2(Ω × [0, T ]) of the equation (3.1), such that γX1[0,t] ∈ Dom δ, for all t ∈ [0, T ].
Proof: We first show that the process X given in (3.2) is a solution of equation (3.1). For this we first observe that X belongs to L2(Ω × [0, T ]) and we let F ∈ SK. Then, by the integration by parts formula and the Girsanov theorem, together with the fact that dF (Tt) = γt(K∗KDF )(Tt, t), we have dt.
Fractional Backward Doubly Stochastic Differential Equations
In this section we state some of the main results of this chapter. Namely, the existence and uniqueness of backward doubly stochastic differential equations driven by both a fractional Brownian motion B and a standard Brownian motion W .
Let {Bt : 0 ≤ t ≤ T } be a one-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1/2), defined on the classical Wiener space (Ω′, FB , P B ) with Ω′ = C0([0, T ]; R), and {Wt = (Wt1, Wt2, · · · , Wtd) : 0 ≤ t ≤ T } a d-dimensional canonical Brow-nian motion defined on the classical Wiener space (Ω′′, F W , P W ) with Ω′′ = C0([0, T ]; Rd). We put (Ω, F0, P ) = (Ω ′, FB , P B ) ⊗ (Ω′′, FW , P W ) and let F = F0 ∨ N , where N is the class of the P -null sets. We denote again by B and W their canonical extension from Ω′ and Ω′′, respectively, to Ω. We let Ft,TW = σ{WT − Ws, t ≤ s ≤ T } ∨ N , FtB = σ{Bs, 0 ≤ s ≤ t} ∨ N , and Gt = Ft,TW ∨ FtB , t ∈ [0, T ]. Let us point out that Ft,TW is decreasing and FtB is increasing in t, but Gt is neither decreasing nor increasing. We denote the family of σ-fields {Gt}0≤t≤T by G. Moreover, we shall also introduce the backward filtrations H = {Ht = Ft,TW ∨ FTB }t∈[0,T ] and FW = {Ft,TW }t∈[0,T ].
Let SK′ denote the class of smooth random variables of the form F = f (B(ϕ1), . . . , B(ϕn), W (ψ1), . . . , W (ψm)), (3.3) where ϕ1, . . . , ϕn are elements of the domain of the operator K∗K, ψ1, . . . , ψm ∈ C([0, T ], Rd), f ∈ Cp∞(Rn+m) and n, m ≥ 1. Here, Cp∞(Rn+m) is the set of all C∞ functions f : Rn+m → R such that f and all its partial derivatives have polynomial growth. The Malliavin derivative of the smooth random variable F w.r.t. B is the Λ1T/2−H – valued random variable DB F defined by DB F = ∑n ∂f (B(ϕ1), . . . , B(ϕn), W (ψ1), . . . , W (ψm))ϕi.
The Associated Stochastic Partial Differential Equations
In this section we will use the following standard assumptions: (H4) i). The functions σ : Rd → Rd×d, b : Rd → Rd and Φ : Rd → R are Lipschitz. (H4) ii). The function f : [0, T ] × Rd × R × Rd R is continuous, f (t, ·, ·, ·) is Lipschitz, uniformly with respect to t and f ( ,0,0,0) ∈ L2(Ω × [0, T ]). t,x · We denote by (Xs )0≤s≤t the unique solution of the following stochastic differential equation: { Xt = x ∈ ( . ) ( ) dXst,x = −b Xst,x ds − σ Xst,x ↓ dWs, s ∈ [0, t] , (4.1) t,x Rd We note that this equation looks like a backward stochastic differential equation, but due to the backward Itˆo integral, the SDE (4.1) is indeed a classical forward SDE. Under our stan-dard assumptions on σ and b, it has a unique strong solution which is {Fs,tW }s∈[0,t]-adapted. The following result provides some standard estimates for the solution of equation (4.1) (cf. [85] Lemma 2.7).
Table of contents :
Acknowledgement
R´esum´e (en fran¸cais)
Introduction
1 Semilinear Backward Doubly Stochastic Differential Equations and SPDEs Driven by Fractional Brownian Motion with Hurst Parameter in (0,1/2)
1 Introduction
2 Preliminaries
2.1 Fractional Calculus
2.2 Fractional Brownian Motion
2.3 Girsanov Transformations
3 Semilinear Fractional SDEs and Fractional Backward Doubly SDEs
3.1 Fractional Anticipating Semilinear Equations
3.2 Fractional Backward Doubly Stochastic Differential Equations
4 The Associated Stochastic Partial Differential Equations
2 Fractional Backward Doubly Stochastic Differential Equations with Hurst Parameter in (1/2,1)
1 Introduction
2 Preliminaries
2.1 Fractional Brownian Motion and General Setting
2.2 Russo-Vallois Integral
3 A Generalized Itˆo Formula
4 Doss-Sussmann Transformation of Fractional BDSDE
5 Associated Stochastic Partial Differential Equations
3 Regularity Properties of Viscosity Solutions of Integro-Partial Differential Equations of Hamilton-Jacobi-Bellman Type
1 Introduction
2 Lipschitz Continuity
3 Semiconcavity
4 Appendix
Bibliography
