Integro-PDE in Hilbert Spaces: Existence of Viscosity Solutions

Existence of a viscosity solution to a non-local Hamilton-Jacobi-Bellman equation in a Hilbert space is established. We prove that the value function of an associated stochastic control problem is a viscosity solution. We provide a complete proof of the Dynamic Programming Principle for the stochastic control problem. We also illustrate the theory with Bellman equations associated to a controlled wave equation and controlled Musiela equation of mathematical finance both perturbed by Lévy processes.