Abstract: This paper is concerned with a portfolio optimization problem with liability and stochastic interest rate model, where risk-free interest rate is assumed to follow the Vasicek interest rate model, while liability process follows Brownian motion with drift. Moreover, it was assumed that liability dynamic is correlated with stock price and interest rate. Dynamic programming principle was applied to obtain Hamilton-Jacobi-Bellman(HJB) equation for the value function and further Legendre transform was used to derive its dual equation. Power utility and exponential utility function were chosen for the analysis. Finally, the closed-form solutions to the optimal investment strategies were obtained by using separate variable and variable change technique and numerical examples were presented to illustrate the impact of market parameters on the optimal policies.  

Key words: Vasicek interest rate model, liability process, dynamic programming principle, Legendre transform, dynamic portfolio selection