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

Christel Geiss (1) , Céline Labart (2) , Antti Luoto (1)

1 JYU - Department of Mathematics and Statistics [Jyväskylä Univ]
2 LAMA - Laboratoire de Mathématiques

Let (Y, Z) denote the solution to a forward-backward SDE. If one constructs a random walk B n from the underlying Brownian motion B by Skorohod embedding, one can show L 2 convergence of the corresponding solutions (Y n , Z n) to (Y, Z). We estimate the rate of convergence in dependence of smoothness properties, especially for a terminal condition function in C 2,α. 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 PDE associated to the FBSDE as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by stochastic methods.

Mots clés: random walk approximation; 60H35; random walk approximation 2010 Mathematics Subject Classification: Primary 60H10; approximation scheme; Backward stochastic differential equations; convergence rate; 60G50 Secondary 60H30; finite difference equation

Domaines: Probabilités [math.PR]

Christel Geiss, Céline Labart, Antti Luoto. Mean square rate of convergence for random walk approximation of forward-backward SDEs. 2020. ⟨hal-01838449v2⟩