# Optimal investment models with stochastic volatility: the time inhomogeneous case

### Abstract

In a recent paper by Pham [11] a multidimensional model with stochastic volatility and portfolio constraints has been proposed, solving a class of investment problems. One feature which is common with these problems is that the resultant Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) is highly nonlinear. Therefore, a transform is primordial to express the value function in terms of a semilinear PDE with quadratic growth on the derivative term. Some proofs for the existence of smooth solution to this equation have been provided for this equation by Pham [11]. In that paper they illustrated some common stochastic volatility examples in which most of the parameters are time-homogeneous. However, there are cases where time-dependent parameters are needed, such as in the calibrating financial models. Therefore, in this paper we extend the work of Pham [11] to the time-inhomogeneous case.

**Keywords**: Semilinear partial differential equation, stochastic volatility, smooth solution, Hamilton-Jacobi-Bellman equation, time-dependent utility function, utility optimisation.