Applications of Stochastic Differential Game Theory for Markov Jump Linear Systems to Finance and Insurance
This chapter mainly introduces applications of stochastic differential game theory for Markov jump linear systems to finance and insurance. Firstly, a risk minimization problem is considered in a continuous-time Markovian regime switching financial model
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Applications of Stochastic Differential Game Theory for Markov Jump Linear Systems to Finance and Insurance
This chapter mainly introduces applications of stochastic differential game theory for Markov jump linear systems to finance and insurance. Firstly, a risk minimization problem is considered in a continuous-time Markovian regime switching financial model modulated by a continuous-time, finite-state, Markov chain. And then, European option valuation under Markovian regime-switching models is studied. Lastly, a game theoretic approach for optimal investment-reinsurance problem of an insurance company under Markovian regime-switching models is introduced in this chapter.
7.1
Introduction
In recent years, Markovian regime-switching models have attracted much attention by researchers and practitioners in economics and finance. Econometric applications of Markovian regime-switching were pioneered by the original work of Reinhard (1984) in which different states of the Markovian chain represent different stages of the economic state, known as the risk model with Markov-modulation by Asmussen (1989) [1]. The Markov-modulation can explain changes in macroeconomic conditions, changes in political systems, influence of major financial news, different stages of business cycles and so on. Presently, portfolio selection and option pricing models with Markov-modulation have been discussed by many researcher, and this has been an important problem from both theoretical and practical perspectives. Moreover, game theory reflects rational thinking modes of players, which, especially stochastic differential game, has been an important method for economic analyzation [2, 3]. So, by means of stochastic differential game, this chapter discusses problems of portfolio risk minimization, option pricing and optimal investment of an insurance company under Markovian regime-switching models. Considering the market as a “virtual” game player, a two-player, zero-sum, stochastic differential game © Springer International Publishing Switzerland 2017 C.-k. Zhang et al., Non-cooperative Stochastic Differential Game Theory of Generalized Markov Jump Linear Systems, Studies in Systems, Decision and Control 67, DOI 10.1007/978-3-319-40587-2_7
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7 Applications of Stochastic Differential Game Theory …
model between investors and markets is built. A verification theorem for the Hamilton_Jacobi_Bellman (HJB) solution of the game is provided.
7.2
Portfolio Risk Minimization and Differential Games
Risk management is an important issue in the modern banking and finance industries. Some recent financial crises, including the Asian financial crisis, the collapse of Long-Term Capital Management, the turmoil at Barings and Orange Country, raise the concern of regulators about the risk taking activities of banks and financial institutions and their practice of risk management. Recently, Value at Risk (VaR) has emerged as a standard and popular tool for risk measurement and management. VaR tells us the extreme loss of a portfolio over a fixed time peri
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