The Annals of Probability

Backward SDEs with constrained jumps and quasi-variational inequalities

Idris Kharroubi, Jin Ma, Huyên Pham, and Jianfeng Zhang

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Abstract

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.

Article information

Source
Ann. Probab., Volume 38, Number 2 (2010), 794-840.

Dates
First available in Project Euclid: 9 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1268143533

Digital Object Identifier
doi:10.1214/09-AOP496

Mathematical Reviews number (MathSciNet)
MR2642892

Zentralblatt MATH identifier
1205.60114

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.) 35K85: Linear parabolic unilateral problems and linear parabolic variational inequalities [See also 35R35, 49J40]

Keywords
Backward stochastic differential equation jump-diffusion process jump constraints penalization quasi-variational inequalities impulse control problems viscosity solutions

Citation

Kharroubi, Idris; Ma, Jin; Pham, Huyên; Zhang, Jianfeng. Backward SDEs with constrained jumps and quasi-variational inequalities. Ann. Probab. 38 (2010), no. 2, 794--840. doi:10.1214/09-AOP496. https://projecteuclid.org/euclid.aop/1268143533


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