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Multiple stochastic integrals with Mathematica

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  • Tocino, A.

Abstract

In the construction of numerical methods for solving stochastic differential equations it becomes necessary to calculate the expectation of products of multiple stochastic integrals. Well-known recursive relationships between these multiple integrals make it possible to express any product of them as a linear combination of integrals of the same type. This article describes how, exploiting the symbolic character of Mathematica, main recursive properties and rules of Itô and Stratonovich multiple integrals can be implemented. From here, a routine that calculates the expectation of any polynomial in multiple stochastic integrals is obtained. In addition, some new relations between integrals, found with the aid of the program, are shown and proved.

Suggested Citation

  • Tocino, A., 2009. "Multiple stochastic integrals with Mathematica," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(5), pages 1658-1667.
    • Handle: RePEc:eee:matcom:v:79:y:2009:i:5:p:1658-1667
    DOI: 10.1016/j.matcom.2008.08.005
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    References listed on IDEAS

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    1. P. E. Kloeden & Eckhard Platen, 1991. "Relations between multiple ito and stratonovich integrals," Published Paper Series 1991-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    2. P. E. Kloeden & Eckhard Platen & I. W. Wright, 1992. "The approximation of multiple stochastic integrals," Published Paper Series 1992-2, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    3. Gaines, J.G., 1995. "A basis for iterated stochastic integrals," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 38(1), pages 7-11.
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    Cited by:

    1. Chenxu Li, 2014. "Closed-Form Expansion, Conditional Expectation, and Option Valuation," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 487-516, May.
    2. Soheili, Ali R. & Amini, Mohammad & Soleymani, Fazlollah, 2019. "A family of Chaplygin-type solvers for Itô stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 296-304.
    3. Chenxu Li & Yu An & Dachuan Chen & Qi Lin & Nian Si, 2016. "Efficient computation of the likelihood expansions for diffusion models," IISE Transactions, Taylor & Francis Journals, vol. 48(12), pages 1156-1171, December.

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