A Dual Method For Backward Stochastic Differential Equations with Application to Risk Valuation

TL;DR

This paper introduces a dual numerical method for solving backward stochastic differential equations to evaluate risk, combining stochastic control, maximum principle, and dynamic programming, with applications in financial risk management.

Contribution

It develops a novel dual control approach using maximum principle and dynamic programming for risk valuation via backward stochastic differential equations.

Findings

Effective approximation of risk measures on short intervals

Extension to finite horizon via dynamic programming

Successful application to financial risk management and portfolio valuation

Abstract

We propose a numerical recipe for risk evaluation defined by a backward stochastic differential equation. Using dual representation of the risk measure, we convert the risk valuation to a stochastic control problem where the control is a certain Radon-Nikodym derivative process. By exploring the maximum principle, we show that a piecewise-constant dual control provides a good approximation on a short interval. A dynamic programming algorithm extends the approximation to a finite time horizon. Finally, we illustrate the application of the procedure to financial risk management in conjunction with nested simulation and on an multidimensional portfolio valuation problem.