Optimal Investment and Consumption With Stochastic Factor and Delay

Abstract

We analyse an optimal portfolio and consumption problem with stochastic factor and delay over a finite time horizon. The financial market includes a risk-free asset, a risky asset and a stochastic factor. The price process of the risky asset is modelled as a stochastic differential delay equation whose coefficients vary according to the stochastic factor; the drift also depends on its historical performance. Employing the stochastic dynamic programming approach, we establish the associated Hamilton–Jacobi–Bellman equation. Then we solve the optimal investment and consumption strategies for the power utility function. We also consider a special case in which the price process of the stochastic factor degenerates into a Cox–Ingersoll–Ross model. Finally, the effects of the delay variable on the optimal strategies are discussed and some numerical examples are presented to illustrate the results.

An optimal portfolio choice problem with delays

“…Our model becomes the original Merton model if we do not consider the effects of delays of the risky asset. More precisely, if we let µ 2 ≡ 0 and µ 3 ≡ 0 in the value function (40) and the wealth process (6), our model coincides with the original Merton model . Remark 3.17 (Merton's Problem with CARA).…”

“…First we consider the original Merton problem (without delays). Our model becomes the original Merton model if we do not consider the effects of delays of the risky asset, that is, if we let µ 2 ≡ 0 and µ 3 ≡ 0 in the value function (11) and the wealth process (6). Remark 3.9 (Merton's Problem with CRRA).…”

“…Recently, [3] examined the optimal consumption and investment problem for a risky asset with delays, and provided a closed-form solution for the Merton problem with a power-type utility function under certain assumptions. Since then, there have been numerous extensions to their work, including multi-dimensional cases [11], the incorporation of stochastic factors [1,6], infinite time horizons [9], and complete memory [10], among others.…”

“…Moreover, we simulate sample paths of the total wealth of the investor X(t) in (6), the exponential moving average of the total wealth Y (t) in (1) and the parallel movement of historical wealth Z(t) in (2) for different values of various parameters h, λ and µ 3 for each utility function. For both a power-type and an exponentialtype utility function, we find that µ 3 has a greater impact on the wealth X(t) than h and λ.…”