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Pré-Publication, Document De Travail Année : 2023

Approximate solution for a class of stochastic differential equation driven by stable processes

Résumé

In this paper we consider a general class of stochastic differential equations driven by $\alpha$-stable processes with Lipschitz drift coefficients and non-Lipschitz diffusion coefficients. An Euler-Maruyama approximate solution is proved whenever the diffusion coefficient is H\"{o}lder continuous with exponent $\beta < \frac{\alpha}{2}$ with $\alpha \in (1,2)$. We obtain a strong rate of convergence of order $ n^{-\beta^{2}}$. Our proposed method is new in this context and is based on a truncation method by separating the large and small jumps of the $\alpha$ stable process in the L\'evy-It\^{o} decomposition. Along the paper we give some numerical simulations of stochastic models that match our results, namely some stable driven Ornstein-Uhlenbeck, Cox-Ingersoll-Ross and Lotka-Volterra type processes.
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Dates et versions

hal-04217398 , version 1 (25-09-2023)
hal-04217398 , version 2 (17-01-2024)
hal-04217398 , version 3 (03-02-2024)

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  • HAL Id : hal-04217398 , version 1

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Solym Manou-Abi. Approximate solution for a class of stochastic differential equation driven by stable processes. 2023. ⟨hal-04217398v1⟩
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