Backward Stochastic Differential Equations with Nonlinear Expectation Reflection
Hanwu Li
TL;DR
This work develops backward stochastic differential equations with nonlinear expectation reflection, constraining the solution through $\mathcal{E}[l(t,Y_t)]\ge0$ and enforcing Skorokhod-type minimality. The authors prove existence and uniqueness via contraction mappings, aided by a domination of $\mathcal{E}$ by a $g$-expectation $G^{\kappa}_{0,T}$, and analyze structural properties through the reflection operator $L_t$. They establish a Lipschitz bound for $L_t$ with respect to $G^{\kappa}_{0,T}$ and discuss comparison results under specific parameter structures. Finally, they apply the theory to superhedging under risk constraints, showing the reflected BSDE yields the superhedging price in constrained settings and framing risk measures (coherent or convex) within this nonlinear-expectation BSDE context.
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.