Dynamic Utility and related nonlinear SPDE driven by Lévy Noise

Anis Matoussi () and Mohamed Mrad
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Anis Matoussi: LMM - Laboratoire Manceau de Mathématiques - UM - Le Mans Université
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: This work concerns the study of consistent dynamic utilities in a financial market with jumps. We extend the results established in the paper [EKM13] to this framework. The ideas are similar but the difficulties are different due to the presence of the Lévy process. An additional complexity is clearly the interpretation of the terms of jumps in the different problems primal and dual one and relate them to each other. To do, we need an extension of the Itô-Ventzel's formula to jump's frame. By verification, we show that the dynamic utility is solution of a non-linear second order stochastic partial integro-differential equation (SPIDE). The main difficulty is that this SPIDE is forward in time, so there are no results in the literature that ensure the existence of a solution or simply allow us to deduce important properties, in our study, such as concavity or monotonicity. Our approach is based on a complete study of the primal and the dual problems. This allows us, firstly, to establish a connection between the utility-SPIDE and two SDEs satisfied by the optimal processes. Based on this connection and the SDE's theory, stochastic flow technics and characteristic method allow us, secondly, to completely solve the equation; existence, uniqueness, monotony and concavity. * This research benefited from the support of the "Chair Risques Émergents ou atypiques en Assurance", under the aegis of Fondation du Risque, a joint initiative by Le Mans Université, École polytechnique and l'Entreprise MMA and the support of the "Labex MME-DII".

Keywords: forward utility; stochastic PDE with jumps; Stochastic flows; stochastic characteristics method; performance criteria; horizon-unbiased utility; consistent utility; progressive utility; portfolio optimization; duality (search for similar items in EconPapers)
Date: 2022-02-28
New Economics Papers: this item is included in nep-upt
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Published in International Journal of Theoretical and Applied Finance, 2022

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