Convergence in multiscale financial models with non-Gaussian stochastic volatility (Q2808055)

The authors consider stochastic control systems affected by a fast mean reverting volatility \(Y(t)\) driven by a pure jump Lévy process. The system is associated with the payoff functional containing a continuous function of quadratic growth that is the subject of the maximization among admissible control functions. The value function of this optimal control problem is considered as the solution of the integro-differential Hamilton-Jacobi-Bellman equation. The novelty of the paper is combining multiple scales and stochastic volatility with jumps. It is assumed that \(Y(t)\) evolves at a faster time scale \(\frac{t}{\epsilon}\) than the assets and asymptotics as \(\epsilon\rightarrow0\) is studied. So, this is a singular perturbation problem that is studied by the methods of partial differential equations within the theory of viscosity solutions.