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Open AccessArticle

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

The Institute of Theoretical Electrical Engineering, Ruhr University of Bochum, Universitätsstrasse 150, D-44801 Bochum, Germany
Symmetry 2020, 12(3), 383; https://doi.org/10.3390/sym12030383
Received: 19 January 2020 / Revised: 12 February 2020 / Accepted: 19 February 2020 / Published: 3 March 2020
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. View Full-Text
Keywords: picard iteration; doléans-dade exponential; exponential splitting; stochastic differential equation; iterative splitting; splitting analysis picard iteration; doléans-dade exponential; exponential splitting; stochastic differential equation; iterative splitting; splitting analysis
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Geiser, J. Numerical Picard Iteration Methods for Simulation of Non-Lipschitz Stochastic Differential Equations. Symmetry 2020, 12, 383.

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