Portfolio optimization for jump-diffusion risky assets with regime switching: A time-consistent approach

Abstract

<p style='text-indent:20px;'>In this paper, an optimal portfolio selection problem with mean-variance utility is considered for a financial market consisting of one risk-free asset and two risky assets, whose price processes are modulated by jump-diffusion model, the two jump number processes are correlated through a common shock, and the Brownian motions are supposed to be dependent. Moreover, it is assumed that not only the risk aversion coefficient but also the market parameters such as the appreciation and volatility rates as well as the jump amplitude depend on a Markov chain with finite states. In addition, short selling is supposed to be prohibited. Using the technique of stochastic control theory and the corresponding extended Hamilton-Jacobi-Bellman equation, the explicit expressions of the optimal strategies and value function are obtained within a game theoretic framework, and the existence and uniqueness of the solutions are proved as well. In the end, some numerical examples are presented to show the impact of the parameters on the optimal strategies, and some further discussions on the case of <inline-formula><tex-math id="M1">\begin{document}$ n\geq 3 $\end{document}</tex-math></inline-formula> risky assets are given to demonstrate the important effect of the correlation coefficient of the Brownian motions on the optimal results.</p>