Pathwise Iteration for Backward SDEs

TL;DR

This paper presents a new numerical method for solving backward stochastic differential equations and related PDEs, enabling iterative improvement of bounds with applications in high-dimensional finance.

Contribution

It introduces a pathwise iteration technique that efficiently approximates solutions and bounds for complex stochastic dynamic programs.

Findings

Effective in high-dimensional financial models

Enables iterative refinement of solution bounds

Improves computational efficiency for nested expectations

Abstract

We introduce a novel numerical approach for a class of stochastic dynamic programs which arise as discretizations of backward stochastic differential equations or semi-linear partial differential equations. Solving such dynamic programs numerically requires the approximation of nested conditional expectations, i.e., iterated integrals of previous approximations. Our approach allows us to compute and iteratively improve upper and lower bounds on the true solution starting from an arbitrary and possibly crude input approximation. We demonstrate the benefits of our approach in a high dimensional financial application.