Numerical Schemes for Multivalued Backward Stochastic Differential Systems

We define some approximation schemes for different kinds of generalized backward stochastic differential systems, considered in the Markovian framework. We propose a mixed approximation scheme for a decoupled system of forward reflected SDE and backward stochastic variational inequality. We use an Euler scheme type, combined with Yosida approximation techniques.Comment: 13 page

Convergence rate and averaging of nonlinear two-time-scale stochastic approximation algorithms

The first aim of this paper is to establish the weak convergence rate of nonlinear two-time-scale stochastic approximation algorithms. Its second aim is to introduce the averaging principle in the context of two-time-scale stochastic approximation algorithms. We first define the notion of asymptotic efficiency in this framework, then introduce the averaged two-time-scale stochastic approximation algorithm, and finally establish its weak convergence rate. We show, in particular, that both components of the averaged two-time-scale stochastic approximation algorithm simultaneously converge at the optimal rate $\sqrt{n}$.Comment: Published at http://dx.doi.org/10.1214/105051606000000448 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Development and Application of the Fourier Method to the Mean-Square Approximation of Iterated Ito and Stratonovich Stochastic Integrals

The article is devoted to the mean-square approximation of iterated Ito and Stratonovich stochastic integrals in the context of the numerical integration of Ito stochastic differential equations. The expansion of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ and expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 5 have been obtained. Considerable attention is paid to expansions based on multiple Fourier-Legendre series. The exact and approximate expressions for the mean-square error of approximation of iterated Ito stochastic integrals are derived. The results of the article will be useful for numerical integration of Ito stochastic differential equations with non-commutative noise.Comment: 56 pages. Minor changes along the text in the whol