Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression

Emmanuel Gobet (1) , Plamen Turkedjiev (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We design a numerical scheme for solving a Dynamic Programming equation with Malliavin weights arising from the time-discretization of backward stochastic differential equations with the integration by parts-representation of the Z-component by [Ma-Zhang 2002]. When the sequence of conditional expectations is computed using empirical least-squares regressions, we establish, under general conditions, tight error bounds as the time-average of local regression errors only (up to logarithmic factors). We compute the algorithm complexity by a suitable optimization of the parameters, depending on the dimension and the smoothness of value functions, in the limit as the number of grid times goes to infinity. The estimates take into account the regularity of the terminal function.

Mots clés: Backward stochastic differential equations; Malliavin calculus; dynamic programming equation; empirical regressions; non-asymptotic error estimates

Domaines: Probabilités [math.PR]

Emmanuel Gobet, Plamen Turkedjiev. Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression. Bernoulli, 2016, 22 (1), ⟨10.3150/14-BEJ667⟩. ⟨hal-00855760v2⟩