Conditional Expectation Backward Stochastic Differential Equations and Related Backward Stochastic Differential Equations with Conditional Reflection

TL;DR

This paper introduces conditional expectation BSDEs, a new class of backward stochastic differential equations depending on conditional expectations, with applications to decision-making under partial information.

Contribution

It establishes well-posedness for these equations, compares them to classical and mean-field BSDEs, and constructs solutions for reflected variants without left-continuity assumptions.

Findings

Conditional expectation BSDEs generalize classical and mean-field BSDEs.

Well-posedness of conditional expectation BSDEs is proven under mild conditions.

Solutions to conditional reflected BSDEs are constructed as limits of penalized equations.

Abstract

In this paper, we introduce a new type of backward stochastic differential equations (BSDEs), called conditional expectation BSDEs, whose drivers depend not only on the value of the solutions but also on their conditional expectations with respect to a certain sub-{\sigma}-algebra. The collection of these sub-{\sigma}-algebra forms a subfiltration, which stands for partial information that is common for decision making applications. The classical BSDEs and the mean-field BSDEs can be regarded as two special and extreme cases of conditional expectation BSDEs. We establish the well-posedness for conditional expectation BSDEs under mild conditions and discuss the comparison results. Then, we provide an alternative construction for the solutions to conditional reflected BSDEs without the left-continuity assumption for the subfiltration, which can be seen as the limit of a sequence of penalized conditional expectation BSDEs.