Abstract
We consider a nonzero-sum stochastic differential portfolio game problem in a continuous-time Markov regime switching environment when the price dynamics of the risky assets are governed by a Markov-modulated geometric Brownian motion (GBM). The market parameters, including the bank interest rate and the appreciation and volatility rates of the risky assets, switch over time according to a continuous-time Markov chain. We formulate the nonzero-sum stochastic differential portfolio game problem as two utility maximization problems of the sum process between two investors’ terminal wealth. We derive a pair of regime switching Hamilton-Jacobi-Bellman (HJB) equations and two systems of coupled HJB equations at different regimes. We obtain explicit optimal portfolio strategies and Feynman-Kac representations of the two value functions. Furthermore, we solve the system of coupled HJB equations explicitly in a special case where there are only two states in the Markov chain. Finally we provide comparative statics and numerical simulation analysis of optimal portfolio strategies and investigate the impact of regime switching on optimal portfolio strategies.
Highlights
The optimal portfolio selection has been studied extensively in modern finance
In order to investigate how the Markov regime switching influences optimal investment strategies and value functions and provide meaningful comparative statics analysis, we present one special case of the nonzero-sum stochastic differential portfolio game problem established in Sections 3 and 4
In this paper we dealt with a nonzero-sum stochastic differential portfolio game problem between two investors in a continuous-time Markov regime switching model
Summary
The optimal portfolio selection has been studied extensively in modern finance. This research is of great importance from both theoretical and practical purposes. Browne [20] formulated various versions of a zerosum stochastic differential game to investigate dynamic optimal investment problems between two “small” investors in continuous time He provided the existence conditions of Nash equilibrium and gave explicit representations for the value functions and optimal portfolio strategies. Elliott and Siu [26] extended the model to a continuous-time Markovian regime switching setting and continued to study the risk minimization portfolio selection problem by using stochastic differential game. They provided explicit Nash equilibria and derived closed-form solutions to value functions It seems that the literature has not well studied the optimal portfolio interactive decision making problem under stochastic differential game in a continuous-time Markovian regime switching setting. We summarize the findings and outline some potential topics for future research
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