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.
