On Investment Consumption Modeling with Jump Process Extensions for Productive Sectors

Abstract

We consider the optimal investment and consumption behavior of an agricultural sector that generates income from productive capital and interest payments and that has expenditures due to interest payments for debt. The productivity of capital, the interest rate, and the capital depreciation rate are uncertain. We adapt an existing model from the literature and generalize it to a stochastic depreciation of capital stock. Moreover, we assume a discontinuous evolution of the productivity of capital. The uncertainties are modeled as stochastic processes with discontinuous paths using well-known Wiener processes and simple Poisson processes. Accordingly, the investor’s net wealth follows a jump diffusion stochastic differential equation. The goal of maximizing the utility of consumption over an infinite time horizon leads to a stochastic optimal control problem. We choose a hyperbolic absolute risk aversion utility function. To determine the optimal investment and consumption rate, the associated Hamilton–Jacobi–Bellman equation is solved by a separation method, and a verification theorem is applied.