Backward SDEs with constrained jumps and quasi-variational inequalities

Details Backward SDEs with constrained jumps and quasi-variational inequalities Backward SDEs with constrained jumps and quasi-variational inequalities

2008-05-30

arxiv.org/abs/0805.4676v2

Annals of Probability 2010, Vol. 38, No. 2, 794-840

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AOP496 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/09-AOP496

We consider a class of backward stochastic differential equations (BSDEs) driven by Brownian motion and Poisson random measure, and subject to constraints on the jump component. We prove the existence and uniqueness of the minimal solution for the BSDEs by using a penalization approach. Moreover, we show that under mild conditions the minimal solutions to these constrained BSDEs can be characterized as the unique viscosity solution of quasi-variational inequalities (QVIs), which leads to a probabilistic representation for solutions to QVIs. Such a representation in particular gives a new stochastic formula for value functions of a class of impulse control problems. As a direct consequence, this suggests a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs.

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