Proposition de stage M2 Master Orleans-Ho Chi Minh

Constrained Backward SDEs with jumps and applications

Encadrant : Huyen PHAM
Laboratoire de Probabilit´es et Mod`eles Aleatoires
Universit´es Paris 6-Paris 7, UMR 7599

The theory of Backward Stochastic Differential Equations (BSDEs) initiated by Peng and Pardoux (1990), see the survey by Pardoux (1998), generated over these last years a very active research field. This interest for BSDEs is due in particular to its nice connections with mathematical finance, stochastic control, partial differential equations and applications for probabilistic numerical methods (see e.g. Gobet et Lemor (2005)). Recently, Kharroubi et al. (2008) introduced a new class of constrained BSDEs with jumps, and related these BSDEs with a system of quasi variational inequalities (QVIs), arising in particular in impulse control problems. These stochastic control problems attracted an increasing renewed interest in economy and finance as they provide a suitable framework in real options literature and for optimal investment in presence of liquidity risk. The constrained BSDEs with jumps lead to original approximation for QVIs by simulation of these BSDEs. The purpose of this internship is twofold: in the first part, the candidate will synthetize the relation between constrained BSDEs with jumps, QVIs and impulse controls. In the second part, the student will numerically implement this BSDE methodology in some examples of interest, and will compare it with existing approaches.

• I. Kharroubi, J. Ma, H. Pham and J. Zhang (2008): “Backward SDEs with constrained jumps and quasi-variational inequalities”, Preprint, PMA, paper available also on: http://felix.proba.jussieu.fr/pageperso/pham/BSDE-consjump.pdf

• E. Pardoux (1998): “Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order”, in Stochastic Analysis and Related Topics : The Geilo Workshop, 1996, L. Decreusefond, J. Gjerde, B. Oksendal,

A.S. Ustunel eds., Birkhuser, 79-127, 1998, http://www.cmi.univ-mrs.fr/ pardoux/geilo.ps

_• E. Gobet and J.P. Lemor (2005): “Numerical simulation of BSDEs using empirical regression methods: theory and practice”, Proceedings of the Fifth Colloquium on BSDEs (29th May -1st June 2005, Shangai). arXiv:0806.4447v1

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