Legendre transform-dual solution for investment and consumption problem under the Vasicek model

This paper studies an investment and consumption problem with stochastic interest rate, where interest rate is governed by the Vasicek model. The financial market is composed of one risk-free asset and one risky asset, in which stock price dynamics is assumed to be generally correlated with interest rate dynamics. The aim is to maximize expected utility of consumption and terminal wealth in the finite horizon. Legendre transform is used to deal with this investment and consumption problem and the explicit solutions of the optimal investment and consumption strategies with power and logarithm preference are achieved. Finally, the authors add a numerical example to analyze the effect of market parameters on the optimal investment and consumption strategy and provide some economic implications.