Existence of solutions for the nonlinear partial differential equation arising in the optimal investment problem

Abstract

We are concerned with the solvablity of certain nonlinear partial differential equation (PDE), which is derived from the optimal investment problem under the random risk process. The equation describes the evolution of the Arrow-Pratt coefficient of absolute risk aversion with respect to the optimal value function. Employing the fixed point approach combined with the convergence argument we show the existence of solutions.