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 Itô 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.
Backward stochastic differential equationItô formulaMalliavin calculuspartial differential equationGaussian Volterra process35K1060G2260H0560H0760H10French Ministry of Foreign Affairs17G1505The authors acknowledge the financial support by the program Hubert Curien Utique No. 17G1505 of the French Ministry of Foreign Affairs and the Tunisian Ministry of Education and Research. Introduction
A backward stochastic differential equation (BSDE) with a generator f:[0,T]×R×Rn→R, a terminal value ξT and driven by a stochastic process X=(X1,…,Xn) is given by the equation
Yt=ξT−∫tTf(s,Ys,Zs)ds+∫tTZsdXs,0⩽t⩽T.
A solution is a pair of square integrable processes Y and Z=(Z1,…,Zn) that are adapted to the filtration generated by X. Such equations appear especially in the context of asset pricing and hedging theory in finance and in the context of stochastic control problems. BSDEs may be considered as an alternative to the more familiar partial differential equations (PDE) since the solutions of BSDEs are closely related to classical or viscosity solutions of associated PDEs (see e.g. [17]). As a consequence, BSDEs may be used for the numerical solution of nonlinear PDEs.
BSDEs driven by Brownian motion have been studied extensively after the first general existence and uniqueness result proved by E. Pardoux and S.G. Peng [16]. For a synthesis of this research work we may refer to the recent textbooks [8, 9, 17, 18, 20, 24]. More recently BSDEs driven by fractional Brownian motions have been investigated (see, e.g., [4, 11–14, 3, 22, 23]). Since fractional Brownian motions (with Hurst index H∈(0,1/2)∪(1/2,1)) are neither martingales nor Markov processes, new methods have been developed to show the wellposedness of BSDEs in certain function spaces. In particular the integral ∫tTZsdXs has been defined in different ways, e.g. pathwise in the context of fractional analysis or as a divergence integral, and the notion of quasi-conditional expectation has been introduced. In fact, the classical notion of conditional expectation does not seem to be convenient for a proof of the existence and uniqueness of solutions to BSDEs whose driving process is not a martingale. Very few articles are concerned with BSDEs for more general Gaussian processes ([5, 6]) or in the context of the theory of rough paths [10]. In [5] the stochastic integral ∫tTZsdXs is understood in the Wick-Itô sense, and the existence and uniqueness of the solution of (1) is proved for a class of Gaussian processes which includes fractional Brownian motion. The proof is based on a transfer theorem that aims to reduce the question of wellposedness to BSDEs driven by Brownian motion. In [6] it is shown that the wellposedness of linear BSDEs with general square integrable terminal condition ξ holds true if and only if X is a martingale.
This paper is concerned with linear BSDEs driven by Gaussian Volterra processes X. This class of processes contains multifractional Brownian motions and multifractional Ornstein-Uhlenbeck processes. Contrary to fractional Brownian motion where the Hurst parameter H is constant, it becomes for multifractional Brownian motion a function h which is assumed here to be differentiable and with values in (1/2,1). The aim is to obtain the solution of the linear BSDE with the associated linear PDE whose solution is given explicitely. This generalizes a result in [4] obtained for fractional Brownian motion. We define the stochastic integral ∫tTZsdXs as a divergence integral, and extend an Itô formula in [2] to the multidimensional case. The Itô formula is then applied to the solution of the associated PDE in order to get a solution of the BSDE. Special attention is given to the fact that the variance of Volterra processes is not necessarily an increasing function of time, but in general only of bounded variation. The explicit solution of the associated PDE contains this variance and is given by a Feynman-Kac type formula on time intervals where it is increasing. The application of this formula to the BSDEs is therefore restricted to time intervals where this variance is increasing.
In this section we define the class of Volterra processes X we have in mind and the linear BSDEs and the associated PDE. Section 2 is concerned with complements on the Skorohod integral with respect to Volterra processes. The Itô formula is proved in Section 3 and applied in Section 4 to the linear BSDE.
Gaussian Volterra processes
Let X={Xt,0⩽t⩽T} be a zero mean continuous Gaussian process given by
Xt=∫0TK(t,s)dWs
where W={Wt,0⩽t⩽T} is a standard Brownian motion and K:[0,T]2→R is a square integrable kernel, i.e. ∫[0,T]2K(t,s)2dtds<+∞. 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. Gaussian Volterra processes and their stochastic analysis have been studied e.g. in [2, 21] and [19]. In [2] K is called regular if it satisfies
(H) For all s∈(0,T], ∫0T∣K∣((s,T],s)2ds<∞, where ∣K∣((s,T],s) denotes the total variation of K(.,s) on (s,T].
We assume the following condition on K(t,s) which is more restrictive than (H) ([2, 21]):
(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<12, there is a finite constant c>00$]]> such that
|∂K∂t(t,s)|⩽c(t−s)α−1(ts)β,for all0<s<t<T.
The covariance function of X is given by
R(t,s):=EXtXs=∫0min(t,s)K(t,u)K(s,u)du.
We discuss shortly some examples of Gaussian Volterra processes that satisfy (H1) and (H2).
The multi-fractional Brownian motion (mBm) (Bth(t),0⩽t⩽T) with Hurst function h:[0,T]→[a,b]⊂(12,1). Its kernel is given by [7]
K(t,s)=s1/2−h(t)∫st(y−s)h(t)−3/2yh(t)−1/2dy,
where h is assumed to be continously differentiable with bounded derivative. We get
∂K∂t(t,s)=h′(t)s12−h(t)∫st(y−s)h(t)−32yh(t)−12ln((ys−1)y)dy+s12−h(t)(t−s)h(t)−32th(t)−12.
A straightforward calculation shows that (H2) is satisfied with α=a−12, β=b+ϵ−12 with ϵ small enough and c depends on a, b, T and ϵ. The mBm generalizes fractional Brownian motion (fBm) with Hurst index H>1/21/2$]]>. MBm is a more flexible model than fBm since the Hölder continuity of its trajectories varies with h. The trajectories of mBm B·h(·) are in fact locally Hölder continous of order h(t) at t ([7], Proposition 6).
The multi-fractional Ornstein-Uhlenbeck process U={Ut,0⩽t⩽T} given by Ut=∫0te−θ(t−s)dBsh(s), where θ>00$]]> is a parameter and Bh is the mbm of Example 1. The kernel of U is given by
K(t,r)=∫rte−θ(t−s)∂K∂s(s,r)ds=K(t,r)−θ∫rte−θ(t−s)K(s,r)ds.
In fact we have
∫0tK(t,r)dWr=∫0tK(t,r)dWr−θ∫0t∫rte−θ(t−s)K(s,r)dsdWr=Bth(t)−θ∫0te−θ(t−s)Bsh(s)ds.
An integration by parts gives the representation of U. We notice that in the framework of the divergence integral (Section 2) the integral with respect to mbm can be reduced to an integral with respect to Brownian motion. (H2) is satisfied with the same values of α and β as in Example 1.
The Liouville multi-fractional Brownian motion (BtL,h(t),t∈[0,T]) with Hurst function h as in Example 1. Its kernel is given by K˜(t,r)=(t−r)h(t)−121(0,t](r). We refer to [21] for the Liouville fractional Brownian motion.
Linear backward stochastic differential equations
Let W=(W1,…,Wn) a standard Brownian motion in Rn, defined on the probability space (Ω,F,P), and let F={Ft⊂F, t∈[0,T]} be the filtration generated by W and augmented by the P-null sets. We consider the Rn-valued Volterra processes X=(X1,…,Xn) given by
Xtj=∫0tKj(t,s)dWsj,j=1,…,n,
where Kj:[0,T]2→R satisfies the conditions (H1) and (H2). Let σj, j=1,…,n be bounded functions on [0,T], and let bj∈C1((0,T),R)∩C([0,T],R), j=1,…,n. The process N:=(N1,…,Nn) is defined by
Ntj=btj+∫0tσsjδXsj,t∈[0,T],j=1,…,n,
where the integral ∫tTZsδXs is defined as a divergence integral and will be studied in Sections 2 and 3. Let t0⩾0 be fixed, and denote by L2(F,Rn) the set of F-adapted Rn-valued processes Z such that E(∫t0T∣Zt∣2dt)<∞. We consider the linear BDSE for the processes Y=(Yt, t∈[t0,T])∈L2(F,R) and Z=((Zt1,…,Ztn), t∈[t0,T])∈L2(F,Rn) given by
Yt=g(NT)−∫tT[f(s)+A1(s)Ys−A2(s)Zs]ds+∫tTZsδXs,t∈[t0,T],
where the real-valued functions g, f, A1 and the Rn-valued function A2 are supposed to be known. Equation (7) is associated to the following second order linear PDE with terminal condition
∂u∂t(t,x)=−12∑j=1nddtVar(Ntj)∂2u∂xj2(t,x)+∑j=1n[σtjA2j(t)−ddtbtj]∂u∂xj(t,x)+A1(t)u(t,x)+f(t),u(T,x)=g(x),(t,x)∈[t0,T)×Rn.
By means of the Itô formula of Section 3 we show that
Yt:=u(t,Nt)andZtj:=−σtj∂u∂xj(t,Nt),j=1,…,n
is a solution of (7). Equation (8) is solved explicitely in Section 4.
On the divergence integral for Gaussian Volterra processes
The kernel K in (2) defines a linear operator in L2([0,T]) given by (Kσ)t=∫0tK(t,s)σsds, σ∈L2([0,T]). Let E be the set of step functions on [0,T], and let KT∗:E→L2([0,T]) be defined by
(KT∗σ)u:=∫uTσs∂K∂s(s,u)ds.
The operator KT∗ is the adjoint of K ([2], Lemma 1).
a) For s>tt$]]>, we have (KT∗σ1[0,t])s=0, and we will denote (KT∗σ1[0,t])s by (Kt∗σ)s where Kt∗ is the adjoint of the operator K in the interval [0,t].
b) If K(u,u)=0 for all u∈[0,T], (KT∗1[0,r])u=K(r,u) for u<r. Indeed, if u⩽r, we have
(KT∗1[0,r])u=∫uT1[0,r](s)∂K∂s(s,u)ds=∫ur∂K∂s(s,u)ds.
For σ,σ˜∈E this equality may be extended to X(σ):=∫0t(Kt∗σ)sdWs by
E[X(σ)X(σ˜)]=<KT∗σ,KT∗σ˜>L2([0,T])._{{L^{2}}([0,T])}}.\]]]>
Let H be the closure of the linear span of the indicator functions 1[0,t], t∈[0,T] with respect to the scalar product
<1[0,t],1[0,s]>H:=<KT∗1[0,t],KT∗1[0,s]>L2([0,T])._{\mathcal{H}}}:=<{K_{T}^{\ast }}{1_{[0,t]}},{K_{T}^{\ast }}{1_{[0,s]}}{>_{{L^{2}}([0,T])}}.\]]]>
The operator KT∗ is an isometry between H and a closed subspace of L2([0,T]), and ∥·∥H is a semi-norm on H. Furthermore, for φ,ψ∈H,
<KT∗φ,KT∗ψ>L2([0,T])=∫0T∫0T(∫0min(r,s)∂K∂r(r,t)∂K∂s(s,t)dt)φrψsdsdr._{{L^{2}}([0,T])}}={\int _{0}^{T}}{\int _{0}^{T}}\Big({\int _{0}^{min(r,s)}}\frac{\partial K}{\partial r}(r,t)\frac{\partial K}{\partial s}(s,t)dt\Big){\varphi _{r}}{\psi _{s}}dsdr.\]]]>
For further use let
ϕ(r,s):=∫0min(r,s)∂K∂r(r,t)∂K∂s(s,t)dt,r≠s,ϕ˜(r,s):=∫0min(r,s)|∂K∂r(r,t)||∂K∂s(s,t)|dt,r≠s.
Note that ϕ(r,s)=∂2R∂s∂r(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
∥φ∥∣H∣2=∫0T∫tT∣φr∣|∂K∂r(r,t)|dr2dt=2∫0Tdr∫0rdsϕ˜(r,s)∣φr∣∣φs∣.
We briefly recall some basic elements of the stochastic calculus of variations with respect to X. We refer to [15] for a more complete presentation. Let S be the set of random variables of the form F=f(X(φ1),…,X(φn)), where n⩾1, f∈Cb∞(Rn) (f and its derivatives are bounded) and φ1,…,φn∈H. The derivative of FDXF:=∑j=1n∂f∂xj(X(φ1),…,X(φn))φj,
is an H-valued random variable, and DX is a closable operator from Lp(Ω) to Lp(Ω;H) for all p⩾1. We denote by D1,pX the closure of S with respect to the semi-norm
‖F‖1,pp=E|F|p+E‖DXF‖Hp.
We denote by Dom(δX) the subset of L2(Ω,H) composed of those elements u for which there exists a positive constant c such that
|E[<DXF,u>H]|⩽cE[F2],for allF∈D1,2X._{\mathcal{H}}}\Big]\Big|\leqslant c\sqrt{\mathbb{E}[{F^{2}}]},\hspace{0.1667em}\text{for all}\hspace{2.5pt}F\in {\mathbb{D}_{1,2}^{X}}\text{.}\]]]>
For u∈L2(Ω;H) in Dom(δX), δX(u) is the element in L2(Ω) defined by the duality relationship
E[FδX(u)]=E[<D·XF,u·>H],F∈D1,2X._{\mathcal{H}}}\Big],F\in {\mathbb{D}_{1,2}^{X}}.\]]]>
We also use the notation ∫0TutδXt for δX(u). A class of processes that belong to the domain of δX is given as follows: let SH be the class of H-valued random variables u=∑j=1nFjhj (Fj∈S, hj∈H).
In the same way D1,pX(∣H∣) is defined as the completion of S∣H∣ under the semi-norm
∥u∥1,p,∣H∣p:=E∥u∥∣H∣p+E∥DXu∥∣H∣⊗∣H∣p,
where
∥DXu∥∣H∣⊗∣H∣2=∫[0,T]4∣DsXut∣∣Dt′Xus′∣ϕ˜(s,s′)ϕ˜(t,t′)dsdtds′dt′.
The space D1,2X(∣H∣) is included in the domain of δX, and we have, for u∈D1,2X(∣H∣),
E(δX(u)2)⩽E∥u∥∣H∣2+E∥DXu∥∣H∣⊗∣H∣2.
For F∈S, let
DsXF:=∫0Tϕ(s,t)DtXFdt,s∈[0,T].
Then (14) implies
∫[0,T]2DsXutDtXvsdsdt=∫[0,T]4DsXutDt′Xvs′ϕ(s,s′)ϕ(t,t′)dsdtds′dt′.
Let f,g∈D1,2X(∣H∣). Then the integralsδX(f)andδX(g)exist inL2(Ω)andE[δX(f)δX(g)]=E<f,g>H+∫0Tds∫0TdtE[DtXfsDsXgt]._{\mathcal{H}}}+{\int _{0}^{T}}ds{\int _{0}^{T}}dt\mathbb{E}\Big[{\mathbb{D}_{t}^{X}}{f_{s}}{\mathbb{D}_{s}^{X}}{g_{t}}\Big].\]]]>
With the choice f=g Proposition 1 implies D1,2X(∣H∣)⊂dom(δX). In fact, for f∈D1,2X(∣H∣),
∥f∥H⩽∥f∥∣H∣and∫[0,T]2DsXftDtXfsdsdt⩽∥DXf∥∣H∣⊗∣H∣2.
Since (16) is a standard property of the divergence integral (adapted to the actual framework), we omit here its proof.
Itô formula
Let F∈C1,2([0,T]×Rn) and suppose that
max(|F(t,x)|,|∂F∂t(t,x)|,|∂F∂xj(t,x)|,|∂2F∂xj2(t,x)|,j=1,…,n)⩽ceλ∣x∣2
for all t∈[0,T] and x∈Rn, where c, λ are positive constants such that λ<14mini(supt∈[0,T]Var(Nti))−1. This implies
E|F(t,Nt)|2⩽c2Eexp(2λ∣Nt∣2)=c2(2π)n/2∏j=1n1(Var(Ntj))1/2∫exp2λxj2−12(xj−btj)2Var(Ntj)dxj=c2(2π)n/2∏j=1n1(Var(Ntj))1/2×∫exp−xj2(1−4λVarNtj)2VarNtj+btjVarNtjxj−(btj)22VarNtjdxj=c2(2π)n/2∏j=1n1(Var(Ntj))1/22πVarNtj1−4λVarNtj×exp(btj)22VarNtj(1−4λVarNtj)−(btj)22VarNtj=c2(2π)(n−1)/2∏j=1n11−4λVarNtjexp2λ(btj)21−4λVarNtj<∞,
and the same inequalities holds for ∂F∂t(t,x), ∂F∂xj(t,x) and ∂2F∂xj2(t,x), j=1,…,n.
Let N be given by (6), and suppose that, forj=1,…,n, the kernelsKjofXjsatisfy (H1) and (H2),bj∈C1((0,T),R)∩C([0,T],R), andσ={σtj,t∈[0,T],j=1,…,n}is bounded. IfF∈C1,2([0,T]×Rn)satisfies (17),∂F∂xj(·,N·)∈D1,2Xj(∣Hj∣),j=1,…,nand, for allt∈[0,T],F(t,Nt)=F(0,0)+∫0t∂F∂s(s,Ns)ds+∑j=1n∫0t∂F∂xj(s,Ns)(ddsbsjds+σsjδXsj)+12∑j=1n∫0t∂2F∂xj2(s,Ns)ddsVar(Nsj)ds.
a) The growth assumption (17) on F may be unexpected for the proof an Itô formula. In fact, in [1] an Itô formula is shown for fractional Brownian motion for any F∈C1,2 by means of a method of localization. The reason for hypothesis (17) is that it implies the finiteness of the second moment of F(t,Nt), shown in (18). When applied to the solution u of the PDE (8), it implies, together with Theorem 2, the finiteness of the second moment of the solution (Y,Z) of the BSDE, and this seems to be an important ingredient for the proof of the uniqueness of the solution ([11, 14] for BSDE with fBm). Moreover, Lemma 1 shows that (17) is also reasonable from the point of view of the PDE (8).
b) A more general model for N than in Section 1.2 is
N˜ti=bti+∑j=1n∫0tσ˜si,jδXsj,i=1,…,n,
where σ˜=(σ˜ti,j, i,j=1,…,n) is a matrix of bounded functions σ˜i,j defined on [0,T]. Let N˜=(N˜1,…,N˜n). The components of N˜ are dependent since N˜i depends not only on a single random perturbation Xi, but on the others Xj (j≠i) as well. The model of Section 1.2 may be recovered by choosing the matrix σ˜ diagonal with functions σj:=σ˜j,j in the diagonal. An Itô formula can be shown for F(t,Nt˜) too, but, instead of the variances of N, the covariances of N˜ appear now in the second order term. It reads
F(t,N˜t)=F(0,0)+∫0t∂F∂s(s,N˜s)ds+∑i=1n∫0t∂F∂xi(s,N˜s)(ddsbsids+∑j=1nσ˜si,jδXsj)+12∑i=1n∑j=1n∫0t∂2F∂xi∂xj(s,N˜s)ddsCov(N˜si,N˜sj)ds.
The derivatives of the variances of N in the second order term of (8) have therefore to be replaced by the derivatives of the covariances of N˜, and one has to assume that the matrix (ddtCov(N˜ti,N˜tj), i,j=1,…,n) is positive definite. We notice that
Var(Ntj)=E∫0tKt∗,jσjsδWsj2=∫0t(Kt∗,jσj)s2ds,andddtVar(Ntj)=2σtjDtjNtj,
c) In the model where σ˜ is diagonal, Xi and Xj are defined with independent Brownian motions Wi and Wj if i≠j. This differs from the model where all the Volterra processes X‾j are defined with the same Brownian motion W‾ as follows:
X‾tj=∫0tKj(t,s)δW‾s,N‾tj=btj+∫0tσsjδX‾sj,t∈[0,T],j=1,…,n.
In this case the processes N‾j are again correlated, and the matrix (ddtCov(N‾ti,N‾tj), i,j=1,…,n) is not diagonal and not necessarily positive semidefinite.
Let us prove now Theorem 1.
1. First we show that ∂F∂xj(·,N·)∈Dom(δXj) for all j=1,…,n. For this we show that ∂F∂xj(·,N·)∈D1,2Xj(∣Hj∣), where ∣Hj∣ is the space defined in Section 2 with X replaced by Xj. The terms ϕj and ϕj˜ refer now to the kernel Kj of Xj. The constants in the inequalities below may vary from line to line.
E‖∂F∂xj(.,N.)‖∣Hj∣2=E∫0T(∫sT|∂F∂xj(t,Nt)||∂Kj∂t(t,s)|dt)2ds⩽cE∫0T(∫sTexp(λ∣Nt∣2)(t−s)αj−1(ts)βjdt)2ds.
Applying Hölder’s inequality to 1<pj<11−αj<2 and qj its conjugate, we get
(∫sTexp(λ∣Nt∣2)(t−s)αj−1(ts)βjdt)2⩽(∫sTexp(qjλ∣Nt∣2)dt)2qj(∫sT(t−s)pj(αj−1)(ts)pjβjdt)2pj.
Then
E(∫0Texp(qjλ∣Nt∣2)dt)2qj⩽E(∫0Texp(qjλsupt∈[0,T]∣Nt∣2)dt)2qj⩽T2/qj∏i=1nEexp(2λsupt∈[0,T](Nti)2)dt.
The right side of the inequality above is finite for λ<14mini(supt∈[0,T]Var(Nti))−1, see [2]. Moreover,
∫sT(t−s)pj(αj−1)(ts)pjβjdt=s−pjβj∫sT(t−s)pj(αj−1)tpjβjdt⩽s−pjβjTpjβj(T−s)pj(αj−1)+1pj(αj−1)+1,
and
∫0T(∫sT(t−s)pj(αj−1)(ts)pjβjdt)2pjds⩽T2βj(pj(αj−1)+1)2pj∫0T(s−pjβj(T−s)pj(αj−1)+1)2pjds=T2βj(pj(αj−1)+1)2pj∫0Ts−2βjT2(αj−1)+2pj(1−sT)2(αj−1)+2pjds=T2(αj−1)+2pj+1(pj(αj−1)+1)2pjB(1−2βj,2(αj−1)+2pj+1),
where B is the beta function. It remains to show that E‖DXj∂F∂xj(.,N.)‖∣Hj∣⊗∣Hj∣<∞.
E‖DXj∂F∂xj(.,N.)‖∣Hj∣⊗∣Hj∣2=E∫[0,T]4|DsXj∂F∂xj(t,Nt)||Dt′Xj∂F∂xj(s′,Ns′)|ϕj˜(s,s′)ϕj˜(t,t′)dsdtds′dt′=E∫[0,T]4|∂2F∂xj2(t,Nt)σsj||∂2F∂xj2(s′,Ns′)σt′j|ϕj˜(s,s′)ϕj˜(t,t′)dsdtds′dt′⩽c∫0Tdt∫0Tds′E∂2F∂xj2(t,Nt)2+E∂2F∂xj2(s′,Ns′)2×∫0Tdt′ϕj˜(t,t′)∫0Tdsϕj˜(s′,s).
By (18) with F replaced by ∂2F∂xj2, E(∂2F∂xj2(t,Nt))2 stays bounded in t∈[0,T]. The finiteness of the remaining integrals follows from (H1), (H2) applied to Kj.
2. We proceed now to the outline of the proof of the Itô formula. Let
Ntj,ε=btj+∫0t(∫stσrj∂Kj∂r(r+ε,s)dr)δWsj=btj+∫0t(∫0rσrj∂Kj∂r(r+ε,s)δWsj)dr,
for t⩽T−ε. Then, F(t,Nt1,ε,…,Ntn,ε) has locally bounded variation, and we can write
dF(t,Nt1,ε,…,Ntn,ε)=∂F∂t(t,Nt1,ε,…,Ntn,ε)dt+∑i=1n∂F∂xi(t,Nt1,ε,…,Ntn,ε)dNti,ε=∂F∂t(t,Ntε)+∑j=1n∂F∂xj(t,Ntε)ddtbtjdt+∑j=1n∂F∂xj(t,Ntε)σtj∫0t∂Kj∂t(t+ε,s)δWsjdt
with the notation Nε=(N1,ε,…,Nn,ε). Furthermore,
∂F∂xj(t,Ntε)σtj∫0t∂Kj∂t(t+ε,s)δWsj=σtj[∫0t∂F∂xj(t,Ntε)∂Kj∂t(t+ε,s)δWsj+∫0tDsWj(∂F∂xj(t,Ntε))∂Kj∂t(t+ε,s)ds],
where
DsWj(∂F∂xj(t,Ntε))=∂2F∂xj2(t,Ntε)DsWjNtj,ε=∂2F∂xj2(t,Ntε)∫stσrj∂Kj∂r(r+ε,s)dr.
Therefore
F(t,Ntε)=F(0,0)+∫0t∂F∂s(s,Nsε)+∑j=1n∂F∂xj(s,Nsε)ddsbsjds+∑j=1n∫0t(∫stσrj∂F∂xj(r,Nrε)∂Kj∂r(r+ε,s)dr)δWsj+12∑j=1n∫0t∂2F∂xj2(r,Nrε)∂∂r(∫0r(Kr∗,ε,jσj)s2ds)dr.
The divergence integral coincides, up to ε, with the integral that appears in the statement of the theorem. The last term coincides, up to ε, with the term at the end of Remark b) after the theorem. It remains to show that the terms above converge in L2(Ω) towards the terms in the statement of the theorem as ε→0. This can be done for each integral similarly as in the proof of Theorem 4 in [2]. □
Solvability of linear BSDEs
As mentioned in the introduction the aim is to apply the Itô formula (19) for F replaced by the solution u of the PDE (8) and to show that Y and Z defined by (9) satisfy the BSDE (7). We will show later in this section that u in fact satisfies the growth condition (17) under a suitable growth condition on the final condition in (7). This implies by (18) that Y and Z are square integrable.
The Itô formula (19), with F replaced by u reads
u(t,Nt)=u(T,NT)−∫tT∂u∂s(s,Ns)ds−∑j=1n∫tT∂u∂xj(s,Ns)(ddsbsjds+σsjδXsj)−12∑j=1n∫tT∂2u∂xj2(s,Ns)ddsVar(Nsj)ds.
An application of (8) to the second term on the right hand side of (22) yields
u(t,Nt)=u(T,NT)−∫tT(f(s)+A1(s)u(s,Ns)+∑j=1nA2j(s)σsj∂u∂xj(s,Ns))ds−∑j=1n∫tTσsj∂u∂xj(s,Ns)δXsj.
We get (7) by setting Yt:=u(t,Nt) and Ztj:=−σtj∂u∂xj(t,Nt), i.e. (Y,(Z1,…,Zn)) solves (7) and is adapted to F. As in Section 1.2 we consider this equation for t∈[t0,T], for some fixed t0⩾0.
In order to solve (8) explicitely, we have to assume, in addition to (H1) and (H2) for the kernels Kj, some regularity and integrability conditions. (H5) will be discussed later.
(H3) There exist constants c,C>00$]]> such that c<σj<C, j=1,…,n, and A1, f, A2:=(A21,…,A2n) are bounded.
(H4) g is continuous, and there exist positive constants c′ and λ′<minj=1,…,n(16supt∈[0,T]Var(Ntj))−1 such that ∣g(x)∣⩽c′eλ′∣x∣2 for all x∈Rn.
(H5)ddtVar(Ntj)>00$]]> for all t∈[t0,T] and ∫t0T(Var(NTj)−Var(Ntj))−1/2dt<∞, j=1,…,n.
Assume that (H1)–(H5) hold, and letv(t,z)=(2πt)−1/2exp(−z2/2t). Then(Y,(Z1,…,Zn))given by (9) solves (7), whereu(t,x)=−∫tTexp(∫stA1(r)dr)f(s)ds+exp(−∫tTA1(s)ds)×∫Rng(y)∏j=1nv(Var(NTj)−Var(Ntj),xj−∫tT(σsjA2j(s)−ddsbsj)ds−yj)dysolves (8).
An explicit calculation of the partial derivatives of u shows that u is in fact a classical solution of (8). Sufficient conditions for the uniqueness of the solution of (8) (even in the general nonlinear case) can be found in [12], Theorem 2.4. The question of uniqueness of the solution (Y,(Z1,…,Zn)) of (7) is more delicate for equations with Volterra processes than for equations with fractional Brownian motion and will be adressed in a separate paper. Here we notice that (7) has a unique solution of the form (9) if the solution of (8) is unique. We show now that u verifies the growth condition (17).
Let u be given by (24). Then there are positive constants M andλ<minj=1,…n(4supt∈[0,T]Var(Ntj))−1such that
We prove 2), the proofs of 1) and 3) are simpler or similar. Let us write rsi for σsiA2i(s)−ddsbsi and Dti for Var(NTi)−Var(Nti).
∂u∂xi(t,x)=e−∫tTA1(s)ds∫Rng(y)∂∂ξiv(Dti,ξi)|ξi=xi−∫tTrsids−yi×∏j≠inv(Dtj,xj−∫tTrsjds−yj)dy=e−∫tTA1(s)ds∫Rng(y)12πDti(−xi−∫tTrsids−yiDti)×exp(−(xi−∫tTrsids−yi)22Dti)×∏j≠in12πDtjexp(−(xj−∫tTrsjds−yj)22Dtj)dy.
By (H4) and the change of variables zj=xj−∫tTrsjds−yj, j=1,…,n we get
|∂u∂xi(t,x)|⩽C∫Rn12πDti|ziDti|exp(−zi22Dti)exp(λ′|x−∫tTrsds−z|2)×∏j≠in12πDtjexp(−zj22Dtj)dz.
Let us prove that
exp(λ′|x−∫tTrsds−z|2)∏j≠iexp(−zj22Dtj)2πDtj⩽exp(2λ′|x−∫tTrsds|2)exp(2λ′zi2)×∏j≠iexp[−zj2Dtj(12−2λ′VarNTj)]2πDtj.
In fact,
exp(λ′|x−∫tTrsds−z|2)⩽exp(2λ′|x−∫tTrsds|2)exp(2λ′∣z∣2)=exp(2λ′|x−∫tTrsds|2)exp(2λ′zi2)∏j≠iexp(2λ′zj2)⩽exp(2λ′|x−∫tTrsds|2)exp(2λ′zi2)∏j≠iexp(2λ′VarNTjDtjzj2).
Consequently,
|∂u∂xi(t,x)|⩽Cexp(2λ′|x−∫tTrsds|2)∫Rn12πDti1Dti|zi|Dtiexp(2λ′zi2)exp(−zi22Dti)×∏j≠iexp[−zj2Dtj(12−2λ′VarNTj)]2πDtjdz.
Moreover, for any ϵ>00$]]>, there is a constant Kϵ such that |zi|Dti⩽Kϵexp(ϵDtizi2). Therefore
|∂u∂xi(t,x)|⩽CKϵexp(2λ′|x−∫tTrsds|2)∫Rn12πDti1Dtiexp(ϵDtizi2)×exp(2λ′zi2)exp(−zi22Dti)∏j≠iexp[−zj2Dtj(12−2λ′VarNTj)]2πDtjdz⩽CKϵexp(2λ′|x−∫tTrsds|2)×∫Rn−1[∫R12πDtiexp(−(12−ϵ−2λ′VarNTi)Dtizi2)dzi]×∏j≠iexp[−zj2Dtj(12−2λ′VarNTj)]2πDtjdz′,z′=(z1,…,zi−1,zi+1,…,zn)⩽CKϵexp(2λ′|x−∫tTrsds|2)∫Rn−1[12πDti2πDti1−2ϵ−4λ′VarNTi]×∏j≠iexp[−zj2Dtj(12−2λ′VarNtj)]2πDtjdz′=CKϵ11−2ϵ−4λ′VarNTi1Dtiexp(2λ′|x−∫tTrsds|2)×∏j≠i12πDtj2πDtj1−4λ′VarNTj⩽CKϵ11−2ϵ−4λ′VarNTi1Dtiexp(4λ′∣x∣2)exp(4λ′|∫tTrsds|2)×1[1−4λ′maxjVarNTj]n−1⩽MKε′Dtiexp(λ∣x∣2)
with a M=exp(4λ′max{|∫tTrsds|2,t∈[t0,T]}), λ=4λ′ and a suitable constant Kε′. □
Let us prove now Theorem 2.
It remains to show that (Y,Z) satisfies the BSDE (7). By the preceding lemma u satisfies (23) with T replaced by T−ε. Since Eexp(2λ∣Nt∣2) is bounded on [0,T] for λ<minj=1,…n(4supt∈[0,T]Var(Ntj))−1, we obtain
E∫tT|f(s)+A1(s)u(s,Ns)+∑j=1nA2j(s)σsj∂u∂xj(s,Ns)|ds⩽∑j=1n∫tT(Var(NTj)−Var(Ntj))−1/2dt<∞
for all t<T by (H5). By continuity of u and N, the terms in the first line of (23) with T replaced by T−ε converge to the terms in (23) as ε→0. The divergence integrals converge too in the sense
E∫tT−εσsj∂u∂xj(s,Ns)δXsjF→ε→0E∫tTσsj∂u∂xj(s,Ns)δXsjF
for all j=1,…,n and F∈S. □
We discuss now the hypothesis (H5). The positivity of ddtVar(Ntj) means that Var(Ntj) is (strictly) increasing on [t0,T]. We note that
ddtVar(Ntj)=ddt∫0t(Kt∗,jσj)s2ds=2∫0tσtjσuj∫0u∂Kj∂t(t,s)∂Kj∂u(u,s)dsdu=2σtj∫0tσujϕj(t,u)du.
Since σj>00$]]> by (H3), a sufficient condition for ddtVar(Ntj)>00$]]> for all t∈[t0,T] is ϕj>00$]]> on [0,T]2∖[0,t0]2. This is the case if ∂Kj∂u(u,s)>00$]]> for all (u,s)∈[0,T]2, u>ss$]]>, but the explicit calculation of Dtj:=Var(NTj)−Var(Ntj) below shows that this is a sufficient but not a necessary condition. We note that for fractional Brownian motion BH with Hurst index H>1/21/2$]]>ϕH(t,u)=∂2∂t∂uEBtHBuH=CH∣t−u∣2H−2>0,0,\]]]>
where CH>00$]]> is a constant depending on H.
Let us comment now on the hypothesis of integrability of (Dtj)−1/2 near T. We have
Dtj=E[(∫0TδWsj∫sTσrj∂Kj∂r(r,s)dr)2−(∫0tδWsj∫stσrj∂Kj∂r(r,s)dr)2]=E[(∫0TδWsj∫sTσrj∂Kj∂r(r,s)dr+∫0tδWsj∫stσrj∂Kj∂r(r,s)dr)×(∫0TδWsj∫sTσrj∂Kj∂r(r,s)dr−∫0tδWsj∫stσrj∂Kj∂r(r,s)dr)].
An explicit calculation shows
Dtj=∫0tdr∫tTdr′σrjσr′jϕj(r,r′)+∫0tds∫tTdrσrj∂Kj∂r(r,s)2+∫tTds∫sTdrσrj∂Kj∂r(r,s)2=:At1+At2+At3.
Under the hypothesis (H3) for σj and if ϕj>00$]]>, a sufficient condition for ∫t0T(Dtj)−1/2dt<∞ is
At3=∫tT(KT∗,jσj)s2ds⩾c(T−t)a
for some constant c>00$]]> and a∈(0,2) as t↗T. For fractional Brownian motion BH with H>1/21/2$]]> this condition is satisfied with a=H+1/2. For the Volterra processes in Examples 1–3 this condition is satisfied with a=2h(T) if h is such that ∂K∂u(u,s)>00$]]>, for (u,s)∈(t0,T)2, u>ss$]]>.
Acknowledgments
The authors are grateful to the referees for the careful reading of the paper, their suggestions improved its presentation substantially.
ReferencesAlòs, E., Nualart, D.: Stochastic integration with respect to the fractional brownian motion. Stochastics75, 129–152 (2003). MR1978896. https://doi.org/10.1080/1045112031000078917Alòs, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gausssian processes. Ann. Probab.29, 766–801 (2001). MR1849177. https://doi.org/10.1214/aop/1008956692Bamba Sow, A., Kor Diouf, B.: Fractional anticipated BSDEs with stochastic Lipschitz coefficients. Random Oper. Stoch. Equ.26(3), 143–161 (2018). MR3849701. https://doi.org/10.1515/rose-2018-0012Bender, C.: Explicit solutions of a class of linear fractional BSDEs. Syst. Control Lett.54, 671–680 (2005). MR2142362. https://doi.org/10.1016/j.sysconle.2004.11.006Bender, C.: Backward SDE’s driven by gaussian processes. Stoch. Process. Appl.124, 2892–2916 (2014). MR3217428. https://doi.org/10.1016/j.spa.2014.03.013Bender, C., Viitasaari, L.: A general non-existence result for linear BSDEs driven by a Gaussian process. Stoch. Process. Appl.127, 1204–1233 (2017). MR3619268. https://doi.org/10.1016/j.spa.2016.07.012Boufoussi, B., Dozzi, M., Marty, R.: Local time and Tanaka formula for a Volterra-type multifractional Gaussian process. Bernoulli16, 1294–1311 (2010). MR2759180. https://doi.org/10.3150/10-BEJ261Carmona, R.: Lectures on BSDEs, Stochastic Control and Stochastic Differential Games with Financial Applications. SIAM (2016). MR3629171. https://doi.org/10.1137/1.9781611974249Crépey, S.: Financial Modeling, a Backward Stochastic Differential Equations Perspective. Springer (2013). MR3154654. https://doi.org/10.1007/978-3-642-37113-4Diehl, J., Friz, P.: Backward stochastic differential equations with rough drivers. Ann. Probab.40, 1715–1758 (2012). MR2978136. https://doi.org/10.1214/11-AOP660Hu, Y., Peng, S.: Backward stochastic differential equation driven by fractional brownian motion. SIAM J. Control Optim.48, 1675–1700 (2009). MR2516183. https://doi.org/10.1137/070709451Hu, Y., Ocone, D., Song, J.: Some results on backward stochastic differential equations driven by fractional brownian motion. Stochastic Analysis and Applications to Finance, Intediscip. Math. Sci13, 225–242 (2012). MR2986849. https://doi.org/10.1142/9789814383585_0012Jańczak-Borkowska, K.: Generalized BSDEs driven by fractional brownian motion. Stat. Probab. Lett.83, 805–811 (2013). MR3040307. https://doi.org/10.1016/j.spl.2012.11.029Maticiuc, L., Nie, T.: Fractional backward stochastic differential equations and fractional backward variational inequalities. J. Theor. Probab.28, 337–395 (2015). MR3320972. https://doi.org/10.1007/s10959-013-0509-9Nualart, D.: The Malliavin Calculus and Related Topics. 2nd edn.Springer (2006). MR2200233Pardoux, E., Peng, S.G.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett.14, 55–61 (1990). MR1037747. https://doi.org/10.1016/0167-6911(90)90082-6Pardoux, E., Răşcanu, A.: Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Springer (2014). MR3308895. https://doi.org/10.1007/978-3-319-05714-9Pham, H.: Continous-time Stochastic Control and Optimization with Financial Applications. Springer (2009). MR2533355. https://doi.org/10.1007/978-3-540-89500-8Sottinen, T., Viitasaari, L.: Stochastic analysis of gaussian processes via fredholm representation. Int. J. Stoch. Anal. (2016). MR3536393. https://doi.org/10.1155/2016/8694365Touzi, N.: Optimal Stochastic Control, Stochastic Target Problems and Backward SDE. Springer (2012). MR2976505. https://doi.org/10.1007/978-1-4614-4286-8Čoupek, P., Maslowski, B.: Stochastic evolution equations with Volterra noise. Stoch. Process. Appl.127, 877–900 (2017). MR3605714. https://doi.org/10.1016/j.spa.2016.07.003Wen, J., Shi, Y.: Anticipative backward stochastic differential equations driven by fractional brownian motion. Stat. Probab. Lett.122, 118–127 (2017). MR3584147. https://doi.org/10.1016/j.spl.2016.11.011Zhang, H.: Properties of solution of fractional backward stochastic differential equation. Appl. Math. Comput.228, 446–453 (2014). MR3151931. https://doi.org/10.1016/j.amc.2013.11.081Zhang, J.: Backward Stochastic Differential Equations, Probability Theory and Stochastic Modelling. Springer (2017). MR3699487. https://doi.org/10.1007/978-1-4939-7256-2