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MiS Preprint
37/2008

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

Roland C. Seydel

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.

Received:
Apr 25, 2008
Published:
Apr 25, 2008
MSC Codes:
35B37, 35D05, 45K05, 49L25, 49N25, 60G5, 93E20
Keywords:
impulse control, combined stochastic control, jump-diffusion processes, viscosity solutions, quasi-variational inequalities

Related publications

inJournal
2009 Repository Open Access
Roland C. Seydel

Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions

In: Stochastic processes and their applications, 119 (2009) 10, pp. 3719-3748