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 Teor. Veroyatnost. i Primenen., 2019, Volume 64, Issue 2, Pages 228–257 (Mi tvp5234)

Approximate Wiener–Hopf factorization and the Monte Carlo methods for Lévy processes

O. E. Kudryavtsev

Rostov Branch of Russian Customs Academy

Abstract: In the present paper, we justify the convergence formulas for approximate Wiener–Hopf factorization to exact formulas for factors from a broad class of Lévy processes. Another result obtained here is the analysis of the convergence of Monte Carlo methods that are based on time randomization and explicit Wiener–Hopf factorization formulas. The paper puts forward two generalized approaches to the construction of a Monte Carlo method in the case of Lévy models that do not admit explicit Wiener–Hopf factorization. Both methods depend on approximate formulas that do for Wiener–Hopf factors. In the first approach, the simulation of the supremum and infimum processes at exponentially distributed time moments is effected by inverting their approximate cumulative distribution functions. The second approach, which does not require a partition of the path, involves direct simulation of terminal values of the infimum (supremum) process, and can be used for the simulation of the joint distribution of a Lévy process and the corresponding extrema of the process.

Keywords: Lévy processes, Wiener–Hopf factorization, numerical methods, Monte Carlo methods, the Laplace transform.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00910 This work was supported by the Russian Foundation for Basic Research (grant 18-01-00910).

DOI: https://doi.org/10.4213/tvp5234

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English version:
Theory of Probability and its Applications, 2019, 64:2, 186–208

Bibliographic databases:

Revised: 22.11.2018
Accepted:25.10.2018

Citation: O. E. Kudryavtsev, “Approximate Wiener–Hopf factorization and the Monte Carlo methods for Lévy processes”, Teor. Veroyatnost. i Primenen., 64:2 (2019), 228–257; Theory Probab. Appl., 64:2 (2019), 186–208

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp5234
• https://doi.org/10.4213/tvp5234
• http://mi.mathnet.ru/eng/tvp/v64/i2/p228

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Beliaysky G., Danilova N., Ougolnitsky G., “Calculation of Probability of the Exit of a Stochastic Process From a Band By Monte-Carlo Method: a Wiener-Hopf Factorization”, Mathematics, 7:7 (2019), 581
2. “Abstracts of talks given at the 4th International Conference on Stochastic Methods”, Theory Probab. Appl., 65:1 (2020), 121–172
3. O. Kudryavtsev, P. Luzhetskaya, “The Wiener-Hopf factorization for pricing options made easy”, Eng. Lett., 28:4 (2020), 1310–1317
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