Solving some Stochastic Partial Differential Equations driven by Lévy Noise using two SDEs. *

Mohamed Mrad
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Mohamed Mrad: LAGA - Laboratoire Analyse, Géométrie et Applications - UP8 - Université Paris 8 Vincennes-Saint-Denis - UP13 - Université Paris 13 - Institut Galilée - CNRS - Centre National de la Recherche Scientifique

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Abstract: The method of characteristics is a powerful tool to solve some nonlinear second order stochastic PDEs like those satisfied by a consistent dynamic utilities, see [EM13, MM20]. In this situation the solution V (t, z) is theoretically of the formX t V (0,ξ t (z)) whereX and Y are solutions of a system of two SDEs,ξ is the inverse flow ofȲ and V (0, .) is the initial condition. Unfortunately this representation is not explicit except in simple cases whereX andȲ are solutions of linear equations. The objective of this work is to take advantage of this representation to establish a numerical scheme approximating the solution V using Euler approximations X N and ξ N of X and ξ. This allows us to avoid a complicated discretizations in time and space of the SPDE for which it seems really difficult to obtain error estimates. We place ourselves in the framework of SDEs driven by Lévy noise and we establish at first a strong convergence result, in L p-norms, of the compound approximation X N t (Y N t (z)) to the compound variable X t (Y t (z)), in terms of the approximations of X and Y which are solutions of two SDEs with jumps. We then apply this result to Utility-SPDEs of HJB type after inverting monotonic stochastic flows.

Keywords: Euler scheme; stochastic flow; method of stochastic characteristics; SPDE driven by Lévy noise; Utility-SPDE; Garsia-Rodemich-Rumsey lemma; strong approximation (search for similar items in EconPapers)
Date: 2022-02-01
New Economics Papers: this item is included in nep-upt
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Published in Stochastics: An International Journal of Probability and Stochastic Processes, 2022

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