Year: 2025
Type: article
Abstract: The composition of the solutions of two SDEs can be used to represent the solution of some SPDEs (method of stochastic characteristics). This representation holds, for example, in the context of consistent dynamic utilities (see Karoui and Mrad [An exact connection between two solvable SDEs and a nonlinear utility stochastic PDE, SIAM J. Financial Math., 2013;4(1):697–736] and Matoussi and Mrad [Dynamic utility and related nonlinear SPDEs driven by Lévy noise. Int. J. Theor. Appl. Finance, 2022;25(1):Paper No. 2250004, 45], where the SPDE can be used to establish a numerical scheme, approximating the SPDE's solution. This allows us to avoid a complicated discretization in time and space of a nonlinear SPDE. The purpose of this paper is to study the compound Euler scheme when the SDEs are driven by a Brownian motion and a Poisson random measure with infinite Lévy measure. The continuous case and the case of a finite Lévy measure have been treated in Gobet and Mrad [Convergence rate of strong approximations of compound random maps, application to SPDEs. Discrete Contin. Dyn. Syst. Ser. B, 2018;23(10):4455–4476] and Mrad [Solving some stochastic partial differential equations driven by Lévy noise using two SDEs. Stochastics, 2022;94(8):1265–1298]. Additional terms specific to the truncation method of the small jumps will appear in our error estimates. In many cases, an optimal choice of parameters allows us to find a standard convergence rate of order 1/n. However, we provide some examples of Lévy measures with much slower convergence rates.
Source: Stochastics
Authors: Mohamed Mrad, Alexandre Popier
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: green
Funder: Agence Nationale de la Recherche
OpenAlex ID: W4360800847