Numerical methods for backward stochastic differential equations: A survey

Abstract

Backwards Stochastic Differential Equations (BSDEs) have been widely employed in various areas of applied and financial mathematics. In particular, BSDEs appear extensively in the pricing and hedging of financial derivatives, stochastic optimal control problems and optimal stopping problems. Most BSDEs cannot be solved analytically and thus numerical methods must be applied in order to approximate their solutions. There have been many numerical methods proposed over the past few decades, for the most part, in a complex and scattered manner, with each requiring a variety of different and similar assumptions and conditions. The aim of the present paper is thus to systematically survey various numerical methods for BSDEs, and in particular, compare and categorise them. To this end, we focus on the core features of each method: the main assumptions, the numerical algorithm itself, key convergence properties and advantages and disadvantages, in order to provide an exhaustive up-to-date coverage of numerical methods for BSDEs, with insightful summaries of each and useful comparison and categorization.

Convergence of a robust deep FBSDE method for stochastic control

“…In all examples, we use the neural network structure in Section 3.2. We use M train = 2 20 training data points and batch size M batch = 2 9 with K epoch = 15 epochs. This gives K batch = 2 10 = 1024 updates per epoch.…”

“…When this method converges, it is suitable for the approximation of high-dimensional FBSDE, see e.g., [11,12,13]. See [20] for a summary on forward and backward numerical methods for FBSDE.…”

Two Approaches for a Dividend Maximization Problem under an Ornstein-Uhlenbeck Interest Rate

“…The concept of BSDEs has proved itself very useful, especially in the context of finance and stochastic optimal control, see for example [15,20,18,24,19]. Moreover for high-dimensional problems, there exist numerical algorithms for BSDEs which do not suffer from a curse of dimensionality (see for example [16] or the survey on BSDE numerics [12]), whereas classical HJB-related methods like finite differences to solve the PDE associated to the stochastic control problem may be very inefficient.…”