A Note on Indefinite Stochastic Riccati Equations

TL;DR

This paper introduces a novel approach to solving indefinite stochastic Riccati equations by transforming them into a system of BSDEs, ensuring the algebraic constraints are inherently satisfied, advancing the understanding of these complex equations.

Contribution

The paper presents a new method to solve indefinite stochastic Riccati equations by converting them into a system of BSDEs, simplifying the solution process.

Findings

Existence of solutions for the transformed BSDE system

The approach enforces algebraic constraints automatically

Applicability to equations driven by one-dimensional Brownian motion

Abstract

An indefinite stochastic Riccati Equation is a matrix-valued, highly nonlinear backward stochastic differential equation together with an algebraic, matrix positive definiteness constraint. We introduce a new approach to solve a class of such equations (including the existence of solutions) driven by one-dimensional Brownian motion. The idea is to replace the original equation by a system of BSDEs (without involving any algebraic constraint) whose existence of solutions automatically enforces the original algebraic constraint to be satisfied.