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Itô–Taylor Expansions for Systems of Stochastic Differential Equations with Applications to Stochastic Partial Differential Equations

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Modeling, Dynamics, Optimization and Bioeconomics II (DGS 2014)

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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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Gazi University, Ankara, Turkey

    Fikriye Yılmaz

  2. Department of Mathematics, Atilim University, Ankara, Turkey

    Hacer Öz Bakan

  3. Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

    Gerhard-Wilhelm Weber

Authors
  1. Fikriye Yılmaz
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  2. Hacer Öz Bakan
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  3. Gerhard-Wilhelm Weber
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Corresponding author

Correspondence to Fikriye Yılmaz .

Editor information

Editors and Affiliations

  1. Department of Mathematics and LIAAD-INESC TEC, University of Porto, Porto, Portugal

    Alberto A. Pinto

  2. Department of Agricultural and Resource Economics, University of California, Berkeley, California, USA

    David Zilberman

Appendix

Appendix

Polar Marsaglia Method:

figure a

Approximation of \(I_{ij}\):

figure b

Approximation of \(I_{j0}\) and \(I_{0j}\):

figure c

The main file can be run as:

figure d

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Yılmaz, F., Öz Bakan, H., Weber, GW. (2017). Itô–Taylor Expansions for Systems of Stochastic Differential Equations with Applications to Stochastic Partial Differential Equations . In: Pinto, A., Zilberman, D. (eds) Modeling, Dynamics, Optimization and Bioeconomics II. DGS 2014. Springer Proceedings in Mathematics & Statistics, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-55236-1_25

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