Abstract
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 Ito 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.
Highlights
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 TYt = ξT − f (s, Ys, Zs )ds + ZsdXs, 0 t T . (1)t t www.vmsta.orgH
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
BSDEs may be used for the numerical solution of nonlinear PDEs
Summary
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 = 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 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. 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. We define the class of Volterra processes X we have in mind and the linear BSDEs and the associated PDE.
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