Itô–Taylor expansions for systems of stochastic differential equations with applications to stochastic partial differential equations
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Date
2017
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Publisher
Springer New York LLC
Abstract
Stochastic differential equations (SDEs) are playing a growing role in financial mathematics, actuarial sciences, physics, biology and engineering. For example, in financial mathematics, fluctuating stock prices and option prices can be modeled by SDEs. In this chapter, we focus on a numerical simulation of systems of SDEs based on the stochastic Taylor series expansions. At first, we apply the vector-valued Itô formula to the systems of SDEs, then, the stochastic Taylor formula is used to get the numerical schemes. In the case of higher dimensional stochastic processes and equations, the numerical schemes may be expensive and take more time to compute. We deal with systems with standard n-dimensional systems of SDEs having correlated Brownian motions. One the main issue is to transform the systems of SDEs with correlated Brownian motions to the ones having standard Brownian motion, and then, to apply the Itô formula to the transformed systems. As an application, we consider stochastic partial differential equations (SPDEs). We first use finite difference method to approximate the space variable. Then, by using the stochastic Taylor series expansions we obtain the discrete problem. Numerical examples are presented to show the efficiency of the approach. The chapter ends with a conclusion and an outlook to future studies. © 2017, Springer International Publishing AG.
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Keywords
Correlated Brownian motions, Itô–Taylor expansions, Stochastic partial differential equations, Systems of SDEs, Vector-valued Itô formula
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Citation
0
WoS Q
Scopus Q
Q4
Source
Springer Proceedings in Mathematics and Statistics -- 3rd International Conference on Dynamics, Games and Science, DGS 2014 -- 17 February 2014 through 21 February 2014 -- Porto -- 199879
Volume
195
Issue
Start Page
513
End Page
532