BSDEs and SDEs with time-advanced and -delayed coefficients

ABSTRACT

This paper introduces a class of backward stochastic differential equations (BSDEs), whose coefficients not only depend on the value of its solutions of the present but also the past and the future. For a sufficiently small time delay or a sufficiently small Lipschitz constant, the existence and uniqueness of such BSDEs is obtained. As an adjoint process, a class of stochastic differential equations (SDEs) is introduced, whose coefficients also depend on the present, the past and the future of its solutions. The existence and uniqueness of such SDEs is proved for a sufficiently small time advance or a sufficiently small Lipschitz constant. A duality between such BSDEs and SDEs is established.

KEYWORDS:

• Backward stochastic differential equation
• stochastic differential equation
• time-delayed coefficients
• time-advanced coefficients
• comparison theorem

AMS SUBJECT CLASSIFICATION:

• 60H10

Acknowledgements

The authors would like to thank the anonymous referee for valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is supported by National Natural Science Foundation of China [grant number 11571024] and a program of Hebei province [grant number QN2017116]. The second author is supported by the National Natural Science Foundation of China [grant number 11501325 and 11671229] and the China Postdoctoral Science Foundation [grant number 2018T110706 and 2018M642641].