Backward Stochastic Differential Equations with Nonlinear Expectation Reflection

TL;DR

This paper investigates a class of constrained backward stochastic differential equations with nonlinear expectation reflection, establishing existence, uniqueness, and comparison properties, and applies these results to risk management in finance.

Contribution

It introduces a new type of BSDE with nonlinear expectation reflection and proves key theoretical properties, extending the framework for risk management applications.

Findings

Existence and uniqueness of solutions are proven.

Comparison properties for solutions are established.

Application to superhedging under risk constraints.

Abstract

In this paper, we study a kind of constrained backward stochastic differential equations (BSDEs) such that the nonlinear expectation of the composition of a loss function and the solution remains above zero. The existence and uniqueness result is established with the help of the Skorokhod problem and the method of contraction mapping. We provide the comparison properties for the pointwise value of the solutions and the expectation of the solutions, respectively. In addition, a similar BSDE with risk measure reflection is proposed, which can be applied to the superhedging for contingent claims under risk management constraints.