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Backward SDEs with constrained jumps and quasi-variational inequalities. (English) Zbl 1205.60114

The authors give solutions to backward stochastic differential equations (BSDEs) driven by Brownian motion and a Poisson random measure, and subject to constraints on the jump component. Some impulse control problems are the motivation for investigation of such BSDEs. A problem with constraints is considered in this paper, so we may not have the uniqueness of the solution. Therefore the authors look for the minimal solution using a penalization approach. They 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 is the dynamic programming equation associated to the impulse control problems. Moreover, the authors give a probabilistic representation for solutions to QVIs and a new stochastic formula for value functions of a class of impulse control problems. This also suggests a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs. As a direct consequence, this suggests a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs. Such BSDEs are also useful in problems of mathematical finance, namely in problems of hedging of payoffs.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
35K86 Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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