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Numerical Solutions of Stochastic Differential Equations with Jumps and Measurable Drifts

Author

Listed:
  • Maryam Siddiqui

    (Department of Mathematics, College of Science, King Saud University (KSU), P.O. Box 2455, Riyadh 11451, Saudi Arabia)

  • Mhamed Eddahbi

    (Department of Mathematics, College of Science, King Saud University (KSU), P.O. Box 2455, Riyadh 11451, Saudi Arabia)

  • Omar Kebiri

    (Department of Stochastics and Its Applications, Brandenburg University of Technology (BTU) Cottbus-Senftenberg, 01968 Senftenberg, Germany)

Abstract

This paper deals with numerical analysis of solutions to stochastic differential equations with jumps (SDEJs) with measurable drifts that may have quadratic growth. The main tool used is the Zvonkin space transformation to eliminate the singular part of the drift. More precisely, the idea is to transform the original SDEJs to standard SDEJs without singularity by using a deterministic real-valued function that satisfies a second-order differential equation. The Euler–Maruyama scheme is used to approximate the solution to the equations. It is shown that the rate of convergence is 1 2 . Numerically, two different methods are used to approximate solutions for this class of SDEJs. The first method is the direct approximation of the original equation using the Euler–Maruyama scheme with specific tests for the evaluation of the singular part at simulated values of the solution. The second method consists of taking the inverse of the Euler–Maruyama approximation for Zvonkin’s transformed SDEJ, which is free of singular terms. Comparative analysis of the two numerical methods is carried out. Theoretical results are illustrated and proved by means of an example.

Suggested Citation

  • Maryam Siddiqui & Mhamed Eddahbi & Omar Kebiri, 2023. "Numerical Solutions of Stochastic Differential Equations with Jumps and Measurable Drifts," Mathematics, MDPI, vol. 11(17), pages 1-14, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:17:p:3755-:d:1230339
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    References listed on IDEAS

    as
    1. Ioannis Lestas & Glenn Vinnicombe & Johan Paulsson, 2010. "Fundamental limits on the suppression of molecular fluctuations," Nature, Nature, vol. 467(7312), pages 174-178, September.
    2. Pellegrini, Clément, 2010. "Existence, uniqueness and approximation of the jump-type stochastic Schrodinger equation for two-level systems," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1722-1747, August.
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