Abstract
This paper considers a portfolio optimization problem with delay. The finance market is consisted of one risk-free asset and one risk asset which price process is modeled by Cox-Ingersoll-Ross stochastic volatility model. In addition, considering the history information related to investment performance, the dynamic of wealth is modeled by stochastic delay differential equation. The investor’s objective is to maximize her expected utility for a linear combination of the terminal wealth and the average performance. By applying stochastic dynamic programming approach, we provide the corresponding Hamilton-Jacobin-Bellman equation and verification theorem, and the closed-form expressions of optimal strategy and optimal value function for CRRA utility are derived. Finally, a numerical example is provided to show our results.
Highlights
In this paper, we consider a portfolio optimization problem with delay, in which the Cox-Ingersoll-Ross (CIR) stochastic volatility model is adopted to describe a non-constant volatility of the risky asset
The finance market is consisted of one risk-free asset and one risk asset which price process is modeled by Cox-Ingersoll-Ross stochastic volatility model
By applying stochastic dynamic programming approach, we provide the corresponding Hamilton-Jacobin-Bellman equation and verification theorem, and the closedform expressions of optimal strategy and optimal value function for CRRA utility are derived
Summary
We consider a portfolio optimization problem with delay, in which the Cox-Ingersoll-Ross (CIR) stochastic volatility model is adopted to describe a non-constant volatility of the risky asset. Elsanousi and Larssen [22] investigate a class of optimal consumption problems where the wealth is given by a stochastic differential delay equation with a parameter, and they obtain the closed-form expressions for the optimal strategies and the value functions in two cases of deterministic parameters and random parameters. Chang et al [23] study an investment and consumption problem of Merton’s type modeled by a stochastic system with delay, and they derive the closed-form expressions for the optimal strategies and the value functions in some situations by adopting stochastic control theory. We consider a new revised portfolio optimization problem in which we formulate the wealth dynamic as a stochastic differential delay equation with volatility driven by CIR model.
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