zbMATH — the first resource for mathematics

Optimal stopping of controlled jump diffusion processes: A viscosity solution approach. (English) Zbl 0899.60039
Summary: This paper concerns the optimal stopping time problem in a finite horizon of a controlled jump diffusion process. We prove that the value function is continuous and is a viscosity solution of the integro-differential variational inequality arising from the associated dynamic programming. We also establish comparison principles, which yield uniqueness results. Moreover, the viscosity solution approach allows us to extend maximum principles for linear parabolic integro-differential operators in \({\mathcal C}^0([0, T]\times \mathbb{R}^n)\) and to obtain \({\mathcal C}^{1,2}([0, T)\times \mathbb{R}^n)\) existence result for the associated Cauchy problem in the nondegenerate case.

60G40 Stopping times; optimal stopping problems; gambling theory