Mean square rate of convergence for random walk approximation of forward-backward SDEs

• Published in: Advances in applied probability Vol. 52; no. 3; pp. 735 - 771
• Main Authors: Geiss, Christel, Labart, Céline, Luoto, Antti
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.09.2020; Applied Probability Trust
• Subjects: Approximation; Brownian motion; Convergence; Difference equations; Differential calculus; Finite difference method; Mathematical analysis; Mathematics; Original Article; Original Articles; Partial differential equations; Probabilistic methods; Probability; Properties (attributes); Random variables; Random walk; Smoothness; Backward stochastic differential equations; 60H10; 60H30; 60G50; finite difference equation; random walk approximation; approximation scheme; convergence rate; 60H35; random walk approximation 2010 Mathematics Subject Classification: Primary 60H10; 60G50 Secondary 60H30
• ISSN: 0001-8678; 1475-6064
• DOI: 10.1017/apr.2020.17

Let (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk $B^n$ from the underlying Brownian motion B by Skorokhod embedding, one can show $L_2$-convergence of the corresponding solutions $(Y^n,Z^n)$ to $(Y, Z).$ We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in $C^{2,\alpha}$. The proof relies on an approximative representation of $Z^n$ and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.