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A Numerical Realization of the Wiener–Hopf Method for the Kolmogorov Backward Equation

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Modern Methods in Operator Theory and Harmonic Analysis (OTHA 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 291))

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Abstract

We propose a new numerical method for a certain type of boundary value problems for 3-dimensional partial differential equations, which are related to first passage time distributions of Itô diffusions. We consider the Kolmogorov backward equation, which arises in various applications including mathematical finance. The technique presented is based on a probabilistic interpretation of the problem, which involves a Markov chain approximation, and a Wiener–Hopf factorization. First, we use Carr’s time randomization and approximate the second component of the related diffusion process with a Markov chain. As the result, we reduce the original problem to a sequence of 1-dimensional differential equations with suitable boundary conditions, associated with Gaussian processes, whose constant coefficients are defined by the Markov chain constructed. We also suggest an improvement for the approximation procedure, which lowers the number of nodes used. Then we express an analytic solution to each problem in terms of a probabilistic form of Wiener–Hopf factorization. We implement explicit and approximate factorization formulae numerically using the Fast Fourier Transform algorithm and provide the results of numerical experiments to illustrate the performance of the method developed.

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Acknowledgements

The reported study was funded by the Russian Foundation for Basic Research according to the research project No 18-01-00910.

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Authors and Affiliations

  1. Russian Customs Academy Rostov branch, Budennovskiy 20, Rostov-on-Don, 344002, Russia

    Oleg Kudryavtsev

  2. Institute for Mathematics, Mechanics, and Computer Science in the name of I.I. Vorovich (Southern Federal University), Milchakova str. 8a, Rostov-on-Don, 344090, Russia

    Vasily Rodochenko

Authors
  1. Oleg Kudryavtsev
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  2. Vasily Rodochenko
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Correspondence to Oleg Kudryavtsev .

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Editors and Affiliations

  1. Institute of Mathematics, Mechanic and Computer Sciences and Regional Mathematical Center, Southern Federal University, Rostov-on-Don, Russia

    Alexey Karapetyants

  2. Department of Mathematics, Cinvestav, Santiago de Querétaro, Querétaro, Mexico

    Vladislav Kravchenko

  3. Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel

    Elijah Liflyand

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Kudryavtsev, O., Rodochenko, V. (2019). A Numerical Realization of the Wiener–Hopf Method for the Kolmogorov Backward Equation. In: Karapetyants, A., Kravchenko, V., Liflyand, E. (eds) Modern Methods in Operator Theory and Harmonic Analysis. OTHA 2018. Springer Proceedings in Mathematics & Statistics, vol 291. Springer, Cham. https://doi.org/10.1007/978-3-030-26748-3_23

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