Reflected Backward Stochastic Differential Equation with Rank-based Data

TL;DR

This paper investigates reflected backward stochastic differential equations with rank-based data, establishing their regularity and connection to viscosity solutions of obstacle problems in a Markovian setting.

Contribution

It introduces a novel analysis of reflected BSDEs with rank-dependent coefficients and links their solutions to viscosity solutions of associated PDEs.

Findings

Established regularity properties of solutions.

Proved the uniqueness of viscosity solutions.

Connected reflected BSDE solutions to obstacle problems.

Abstract

In this paper, we study reflected backward stochastic differential equation (reflected BSDE in abbreviation) with rank-based data in a Markovian framework; that is, the solution to the reflected BSDE is above a prescribed boundary process in a minimal fashion and the generator and terminal value of the reflected BSDE depend on the solution of another stochastic differential equation (SDE in abbreviation) with rank-based drift and diffusion coefficients. We derive regularity properties of the solution to such reflected BSDE, and show that the solution at the initial starting time $t$ and position $x$, which is a deterministic function, is the unique viscosity solution to some obstacle problem (or variational inequality) for the corresponding parabolic partial differential equation.