General existence and uniqueness of viscosity solutions for impulse control of jump-diffusions

Roland C. Seydel

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Submission date: 25. Apr. 2008 (revised version: July 2009)
Pages: 48
published in: Stochastic processes and their applications, 119 (2009) 10, p. 3719-3748 
DOI number (of the published article): 10.1016/j.spa.2009.07.004
Bibtex
with the following different title: Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions
MSC-Numbers: 35B37, 35D05, 45K05, 49L25, 49N25, 60G5, 93E20
Keywords and phrases: impulse control, combined stochastic control, jump-diffusion processes, viscosity solutions, quasi-variational inequalities
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Abstract:
General theorems for existence and uniqueness of viscosity solutions for Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVI) with integral term are established. Such nonlinear partial integro-differential equations (PIDE) arise in the study of combined impulse and stochastic control for jump-diffusion processes. The HJBQVI consists of an HJB part (for stochastic control) combined with a nonlocal impulse intervention term. Existence results are proved via stochastic means, whereas our uniqueness (comparison) results adapt techniques from viscosity solution theory. This paper is to our knowledge the first treating rigorously impulse control for jump-diffusion processes in a general viscosity solution framework; the jump part may have infinite activity. In the proofs, no prior continuity of the value function is assumed, quadratic costs are allowed, and elliptic and parabolic results are presented for solutions possibly unbounded at infinity.