Backward stochastic differential equations with time delayed generators - results and counterexamples

TL;DR

This paper studies backward stochastic differential equations with generators depending on past solution values, establishing existence, uniqueness, and counterexamples, and analyzing properties like boundedness and measure solutions.

Contribution

It introduces a new class of BSDEs with time delayed generators, proving key properties and providing counterexamples to highlight limitations.

Findings

Existence and uniqueness for small time horizons or Lipschitz constants.

Counterexamples with multiple or no solutions.

Linear delayed generators can have solutions for any horizon and Lipschitz constant.

Abstract

We deal with backward stochastic differential equations with time delayed generators. In this new type of equations, 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 prove existence and uniqueness of a solution for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. We give examples of BSDE with time delayed generators that have multiple solutions or that have no solutions. We show for some special class of generators that existence and uniqueness may still hold for an arbitrary time horizon and for arbitrary Lipschitz constant. This class includes linear time delayed generators, which we study in more detail. We are concerned with different properties of a solution of a BSDE with time delayed generator, including the inheritance of boundedness from the terminal condition, the comparison principle, the existence of a measure solution and the BMO martingale property. We give examples in which they may fail.