Optimal investment and consumption in a market with Markovian switching coefficients and borrowing

Abstract

We consider the problem of optimal investment and consumption in a market with Markovian switching coefficients. We further assume that the borrowing interest rate is higher than the lending interest rate. The power utility from consumption and terminal wealth is used as an optimality criterion. Due to different borrowing and lending interest rates, the resulting optimal stochastic control problem has a nonlinear system dynamics with Markovian switching. We obtain an explicit closed-form solution to this problem as a linear state-feedback control the gain of which is determined by a system of coupled Bernoulli backward ordinary differential equations.