[Paper Review] Numerical Algorithms for 1-d Backward Stochastic Differential Equations: Convergence and Simulations

Shigē Péng, Mingyu Xu|arXiv (Cornell University)|Nov 28, 2006

Stochastic processes and financial applications22 references3 citations

TL;DR

This paper proposes and analyzes implicit and explicit numerical schemes for solving 1D backward stochastic differential equations (BSDEs) and reflected BSDEs using a random walk approximation of Brownian motion. It establishes convergence of these schemes under Lipschitz conditions and presents simulations demonstrating stability and accuracy, particularly for penalized and constrained BSDEs with one or two barriers.

ABSTRACT

In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are introduced. Then we prove the convergence of different algorithms and present simulation results for different types of BSDEs.

Motivation & Objective

• To develop and analyze numerical algorithms for solving 1D backward stochastic differential equations (BSDEs) with nonlinear generators.
• To extend these algorithms to reflected BSDEs with one or two barriers, including penalized and explicit-implicit schemes.
• To establish theoretical convergence results for the proposed numerical schemes under Lipschitz and integrability conditions.
• To implement and validate the algorithms via numerical simulations, demonstrating stability and accuracy in practical settings.
• To provide a user-friendly software package for computing solutions to BSDEs and constrained BSDEs with intuitive input-output interface.

Proposed method

• Discretizes the time interval using a random walk approximation of 1D Brownian motion with step size δ = T/n.
• Applies implicit and explicit schemes to solve the discrete BSDE system, with Z computed via conditional expectation on the Bernoulli increments.
• Uses a penalization method to enforce constraints on the solution Y or the control Z, introducing a penalty term proportional to pδ times the positive part of the constraint violation.
• Employs an implicit scheme by solving a fixed-point equation involving the inverse of a function derived from the generator and penalty term.
• Introduces an explicit-implicit scheme that uses conditional expectations of future values to approximate the current Y and Z iteratively.
• Validates the schemes through dynamic simulation of solution surfaces and trajectories, visualizing the effect of constraints and penalization.

Experimental results

Research questions

• RQ1 Can implicit and explicit numerical schemes for 1D BSDEs converge under Lipschitz conditions on the generator and integrability of the terminal condition?
• RQ2 How do penalized schemes perform in approximating reflected BSDEs with one or two barriers, and do they converge as the penalty parameter p increases and the time step δ decreases?
• RQ3 What is the impact of constraint enforcement on the stability and accuracy of numerical solutions, particularly when p√δ > 1?
• RQ4 How do the proposed schemes handle nonlinear generators such as g(t,y,z) = -2|y+z| - 1, and what is the behavior of the solution surface and process trajectories?
• RQ5 Can the numerical algorithms be effectively implemented in a user-friendly software package with a convenient interface for non-expert users?

Key findings

• The implicit and explicit schemes for standard BSDEs converge in L² and Lβ norms as n → ∞ and p → ∞, with β < 2, under Lipschitz and integrability conditions.
• For reflected BSDEs with one barrier, both implicit and explicit reflected schemes converge to the true solution as the time step δ → 0 and the penalty parameter p → ∞.
• The penalized explicit-implicit scheme for BSDEs with constraint on Y or Z converges in L² and Lβ norms, with β < 2, under the same conditions.
• Simulations show that when p√δ > 1, the numerical solution becomes unstable and may explode, indicating a critical threshold for stability.
• The solution surface Y^{p,n}(t, B_t^n) and the process A_t^{p,n} (penalization term) are clearly visualized, showing effective enforcement of constraints when Y or Z deviates from the allowed region.
• The software package successfully computes solutions with a convenient interface, enabling users to simulate complex BSDEs and reflected BSDEs with minimal expertise.