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Existence, uniqueness, and approximation of solutions of jump-diffusion SDEs with discontinuous drift

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  • Przybyłowicz, Paweł
  • Szölgyenyi, Michaela

Abstract

In this paper we study jump-diffusion stochastic differential equations (SDEs) with a discontinuous drift coefficient and a possibly degenerate diffusion coefficient. Such SDEs appear in applications such as optimal control problems in energy markets. We prove existence and uniqueness of strong solutions. In addition we study the strong convergence order of the Euler–Maruyama scheme and recover the optimal rate 1/2.

Suggested Citation

  • Przybyłowicz, Paweł & Szölgyenyi, Michaela, 2021. "Existence, uniqueness, and approximation of solutions of jump-diffusion SDEs with discontinuous drift," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    • Handle: RePEc:eee:apmaco:v:403:y:2021:i:c:s0096300321002812
    DOI: 10.1016/j.amc.2021.126191
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    References listed on IDEAS

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    1. Anton A. Shardin & Michaela Szölgyenyi, 2016. "Optimal Control Of An Energy Storage Facility Under A Changing Economic Environment And Partial Information," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(04), pages 1-27, June.
    2. Deng, Shounian & Fei, Chen & Fei, Weiyin & Mao, Xuerong, 2019. "Generalized Ait-Sahalia-type interest rate model with Poisson jumps and convergence of the numerical approximation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 533(C).
    3. Menoukeu Pamen, Olivier & Taguchi, Dai, 2017. "Strong rate of convergence for the Euler–Maruyama approximation of SDEs with Hölder continuous drift coefficient," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2542-2559.
    4. Ngo, Hoang-Long & Taguchi, Dai, 2017. "Strong convergence for the Euler–Maruyama approximation of stochastic differential equations with discontinuous coefficients," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 55-63.
    5. Dereich, Steffen & Heidenreich, Felix, 2011. "A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1565-1587, July.
    6. Anton A. Shardin & Michaela Szolgyenyi, 2016. "Optimal Control of an Energy Storage Facility Under a Changing Economic Environment and Partial Information," Papers 1602.04662, arXiv.org, revised Apr 2016.
    7. Benth, Fred Espen & Klüppelberg, Claudia & Müller, Gernot & Vos, Linda, 2014. "Futures pricing in electricity markets based on stable CARMA spot models," Energy Economics, Elsevier, vol. 44(C), pages 392-406.
    8. Nikolaos Halidias & P. E. Kloeden, 2006. "A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient," International Journal of Stochastic Analysis, Hindawi, vol. 2006, pages 1-6, May.
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