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Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions. (English) Zbl 1180.35164

Summary: 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, to our knowledge is 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.

MSC:

35D40 Viscosity solutions to PDEs
45K05 Integro-partial differential equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49N25 Impulsive optimal control problems
60G51 Processes with independent increments; Lévy processes
93E20 Optimal stochastic control
35R60 PDEs with randomness, stochastic partial differential equations
35R09 Integro-partial differential equations
35F21 Hamilton-Jacobi equations
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