Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market
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Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market E. Savku1,2
· G.-W Weber1,3
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We apply dynamic programming principle to discuss two optimal investment problems by using zero-sum and nonzero-sum stochastic game approaches in a continuous-time Markov regime-switching environment within the frame work of behavioral finance. We represent different states of an economy and, consequently, investors’ floating levels of psychological reactions by a D-state Markov chain. The first application is a zero-sum game between an investor and the market, and the second one formulates a nonzero-sum stochastic differential portfolio game as the sensitivity of two investors’ terminal gains. We derive regime-switching Hamilton–Jacobi–Bellman–Isaacs equations and obtain explicit optimal portfolio strategies with Feynman–Kac representations of value functions. We illustrate our results in a two-state special case and observe the impact of regime switches by comparative results. Keywords Control · Stochastic processes · Behavioral finance · Game theory · Dynamic programming
1 Introduction Behavioral finance is a highlighted area of research focused on how psychological influences can affect the market outcomes. Hence, a key aspect in behavioral finance theory is to show that the main point in market fluctuations is not the events themselves, but the human reactions to those events. For several applications of this theory [see (De Bondt et al. 2008; Kourtidis et al. 2011; Subrahmanyam 2008; Statman 2008; Shefrin and Statman 1993; Wu et al. 2012; Loewenstein 2000); references therein]. For example, in Kourtidis et al. (2011), the authors analyze the market participants’ behavior in the financial market by the behavioral factors of over-confidence, risk tolerance, social influence and self-monitoring. Broadly, behavioral
B
E. Savku [email protected]
1
Institute of Applied Mathematics, Middle East Technical University, 06800 Ankara, Turkey
2
École Polytechnique, CMAP, 91128 Palaiseau, France
3
Faculty of Management Engineering, Poznan University of Technology, ul. Jacka Rychlewskiego 2, 11, 60-965 Poznan, Poland
123
Annals of Operations Research
finance theories provide clearer explanations of substantial market anomalies like bubbles, crashes, and deep recessions. In Masood et al. (2017), the authors states that the market participants tend to overreact or underreact to information and events depending on the general market atmosphere by using an empirical data and conclude that the market participants tend to overreact during a crisis period and underreact during a bubble period. At this point, the importance of a regime-switching model arises, which can describe the abrupt changes, their consequences in economy, and the phenomenon arise as the new behavior of financial variables. Regime switching models can catch this novel tendency of financial markets, which often persists for several
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