Optimal asset allocation under search frictions and stochastic interest rate

In this paper, we investigate an optimal asset allocation problem in a financial market consisting of one risk-free asset, one liquid risky asset and one illiquid risky asset. The liquidity risk stems from the asset that cannot be traded continuously, and the trading opportunities are captured by a Poisson process with constant intensity. Also, it is assumed that the interest rate is stochastically varying and follows the Cox-Ingersoll-Ross model. The performance functional of the decision maker is selected as the expected logarithmic utility of the total wealth at terminal time. The dynamic programming principle coupled with the Hamilton-Jacobi-Bellman equation has been adopted to solve this stochastic optimal control problem. In order to reduce the dimension of the problem, we introduce the proportion of the wealth invested in the illiquid risky asset and derive the semi-analytical form of the value function using a separation principle. A finite difference method is employed to solve the controlled partial differential equation satisfied by a function which is an important component of the value function. The numerical examples and their economic interpretations are then discussed.