BSDEs driven by fractional Brownian motion with time-delayed generators

ABSTRACT

This paper deals with a class of backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1/2) with time-delayed generators. In this type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time-delay function, for instance, of the moving average type. We establish an existence and uniqueness result of solutions for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. The stochastic integral used throughout the paper is the divergence operator-type integral.

KEYWORDS:

• Fractional Brownian motion
• backward stochastic differential equation
• time-delayed generators
• Fubini's theorem

AMS SUBJECT CLASSIFICATIONS:

• 60H05
• 60H10
• 60G22
• 60G44

Acknowledgments

The authors would like to thank the referees for their relevant suggestions, in particular, that of giving an example to illustrate our main result.

Disclosure statement

No potential conflict of interest was reported by the author(s).