Numerical Picard Iteration Methods for Simulation of Non-Lipschitz Stochastic Differential Equations

Abstract

In this paper, we present splitting approaches for stochastic/deterministic coupled differential equations, which play an important role in many applications for modelling stochastic phenomena, e.g., finance, dynamics in physical applications, population dynamics, biology and mechanics. We are motivated to deal with non-Lipschitz stochastic differential equations, which have functions of growth at infinity and satisfy the one-sided Lipschitz condition. Such problems studied for example in stochastic lubrication equations, while we deal with rational or polynomial functions. Numerically, we propose an approximation, which is based on Picard iterations and applies the Doléans-Dade exponential formula. Such a method allows us to approximate the non-Lipschitzian SDEs with iterative exponential methods. Further, we could apply symmetries with respect to decomposition of the related matrix-operators to reduce the computational time. We discuss the different operator splitting approaches for a nonlinear SDE with multiplicative noise and compare this to standard numerical methods.

Iterative and Noniterative Splitting Methods of the Stochastic Burgers’ Equation: Theory and Application

“…-Picard's fixpoint scheme; see [8,35], this splitting approach is based on fixpoint-iterations; see [36], which are used to approximate ordinary or semi-discretised partial differential equations. The convergence theory is based on the local version of the theorem of Picard-Lindelöf; see [36].…”

“…These iterative schemes can be extended to stochastic partial differential equations. They are also effective and robust in the implementation; see [8,37]. -Iterative operator splitting; see [19], here we use an extension of the Picard's fixpoint scheme; see [16].…”

“…Our motivation is to explain the benefits of the splitting approaches, which allows to concentrate on each part of the equation; that is, the deterministic/nonlinear part and the stochastic/reaction part; see [1,7,8].…”

“…We discuss the most significant types, which are the noniterative and iterative approaches; see [16,17]. Their convergence and accuracy behaviour is studied for both deterministic and stochastic approaches; see [8,16].…”

“…In particular, we aim to discuss the benefits (e.g., they are fast to implement and simpler to control) and drawbacks (e.g., their accuracy and the lack of decomposing to different physical structures) of noniterative splitting approaches; see [15,18]. We will also discuss the alternative iterative splitting approaches, whose accuracy is of a higher order and which are physically more adequate but are more delicate to control and more complex to implement; see [8,16].…”