Applications Of Markov-Modulated Processes In Insurance And Finance

Compared with the classical risk model or finance model, the Markov-modulated model or Markovian regime-switching model seems to provide a better fit to the reality data of insurance and finance. In risk theory, the Markov-modulated risk model can capture the feature that insurance policies may need to change if the environment, such as weather condition, economical or political environment, etc, changes. For example, in car insurance, weather condition plays a major role in the occurrence of accidents. The claim size distribution and the intensity of the claim arrival process in different weather conditions will be very different. Therefore the insurance policies of insurance companies, such as premium rate, will be different in different weather conditions. In finance theory, the celebrated Black-Scholes-Merton financial market is based on a geometric Brownian motion to capture the price dynamics of the underlying security. However, numerous empirical studies reveal that this assumption for assets price dynamics cannot provide realistic description for some important empirical behavior of financial returns such as the heavy-tailedness of the return's distribution and the time-varying volatility of return. Hardy [75] shows that the Markovian regime-switching model provides a significantly better fit to the data than do other popular models. Therefore, the class of Markov-modulated or Markovian regime-switching model becomes more and more important in insurance and finance.On the basis of these background, my doctoral dissertation is mainly devoted to considering the applications of Markov-modulated or Markovian regime-switching model in insurance and finance. The dissertation is organized as follows.A brief overview of the history of Markov-modulated or Markovian regime-switching model in risk theory and finance theory, as well as the main contents of this dissertation, are given in the first chapter.The second chapter is the theoretical foundation of this dissertation. In this chapter, we review some basic definitions and results on double martingales, continuous time Markov chain and Markov-modulated Lévy processes. The main contribution of this chapter is that we introduce the Markov chain following the framework of Elliott et al. [51] and characterize the Markov chain by associating it with a marked point process. Based on the marked point process characterization of Markov chain, we introduce the j^th Markov jump martingale which plays a major rule in completing the Markovian regime-switching market in Chapter 4. Besides, we also derive some new results, such as the integral representation of the function of Markov chain and the It(o|^) differential rule for the generalized Markov-modulated stochastic process, which are very useful in characterizing the measure change of Markov-modulated Brownian motion in Chapter 4 and proving the verification theorem for the optimization problems in the sequent chapters.The remaining chapters are devoted to solving the following five problems of the Markov-modulated or Markovian regime-switching model in insurance and finance.Gerber-Shiu discounted penalty function of Markov-modulated compound Poisson risk model. Risk theory received a substantial boost with the article of Gerber and Shiu [66] in 1998, in which the Gerber-Shiu discounted penalty function was introduced. Since the basic actuarial variables such as the time of ruin, the surplus immediately prior to ruin, the deficit at ruin, are embedded in the Gerber-Shiu discounted penalty function, a large number of papers in the ruin literature are devoted to obtaining closed forms of Gerber-Shiu discounted penalty functions in a variety of risk models, see Wang and Wu [163] for the perturbed compound Poisson risk process with constant interest, Zhang et al. [178] for classical risk model with a two-step premium rate, Landriault and Willmot [99], Willmot [165], Willmot and Dickson [166] for the renewal risk model, Garrido and Morales [63], Morales [122], Morales and Olivares [123] for the Lévy risk model. Although the Markov-modulated risk model was first introduced by Janssen [88] and later treated by Janssen and Reinhard [89], Reinhard [140], Asmussen [5, 7], B(a|¨)uerle [14], Wu [169] ,Wu and Wei [167], Ng and Yang [128, 129] and so on, Albrecher and Boxma [2] were the first to consider the Gerber-Shiu discounted penalty function for the Markov-dependent risk model. Later Ng and Yang [129] also considered the Gerber-Shiu discounted penalty functions for the Markov-modulated compound Poisson risk model. However, they only gave a system of integro-differential equations for Gerber-Shiu functions and did not discuss the solution of this system of integro-differential equations. Inspired by the work of Albrecher and Boxma [2] and Ng and Yang [129], we study the Gerber-Shiu discounted penalty functions for the Markov-modulated compound Poisson risk model and obtain closed forms of Gerber-Shiu discounted penalty functions for this model, see Chapter 3 of my doctoral dissertation. Later Lu and Tsai [110] considered the Gerber-Shiu discounted penalty functions for the Markov-modulated compound Poisson risk model perturbed by diffusion.Chapter 3 is devoted to obtaining closed forms of the Gerber-Shiu discounted penalty functions for the Markov-modulated compound Poisson risk model. We first present a system of integro-differential equations of the Gerber-Shiu discounted penalty functions for arbitrary premium rate. By using Laplace transform to solve the integro-differential equations, we obtain explicit expressions of the Laplace transforms of Gerber-Shiu discounted penalty functions. Closed forms of the discounted joint density function of surplus prior to and after ruin for the initial surplus 0 are obtained. Besides, we derive explicit formulas of Gerber-Shiu discounted penalty functions for the K_n-family claim size distributions in the two-state case through introducing the operator T_r. Finally, numerical examples are presented to illustrate our results.Completing the Markovian regime-switching market via double martingales. Completeness of the financial market is an important concept in the finance literature and the theory of the complete market is more mature than that of the incomplete market. Therefore, if we can find a method to complete the incomplete market, we can use the results of the complete market to solve the problem in the original incomplete market. Karatzas et al. [95] introduced a way to complete the incomplete market by adding the fictitious stocks for the original incomplete market. By using the results of utility maximization problem in the complete market and the relationship between the complete market and the original incomplete market, Karatzas et al. [95] solved the utility maximization problem in the original incomplete market. The Markovian regime-switching market consists of one bond and one stock, see Chapter 4 for the description of this market. It is well-known that the Markovian regime-switching market is incomplete (see Elliott and Swishchuk [48]). So we want to find a method to complete the Markovian regime-switching market and study the portfolio optimization in the completed Markovian regime-switching market. The idea of completing the Markovian regime-switching is inspired by Corcuera et al. [34], Corcuera et al. [35], Niu [130] and Guo [71]. By adding the power-jump assets (related to the suitably compensated power-jump processes of the underlying Lévy process), Corcuera et al. [34] completed the geometric Lévy market, while, Corcuera et al. [35] considered the optimal investment problems in the completed geometric Lévy market with the new power-jump assets. Niu [130] studied the optimal investment problems in a simple Lévy market which was completed by adding the power-jump assets introduced by Corcuera et al. [34]. Guo [71] introduced the change-of-state contract to complete the Markovian regime-switching market. However, this method is not good enough for the solution of optimal investment problems. Therefore we need to find some other method to complete the market and consider the optimal investment problems.In Chapter 4 and Chapter 5, we find two new methods to complete the Markovian regime-switching market via the theory of double martingales and marked point process. The problem of portfolio optimization for maximizing expected utility in the completed Markovian regime-switching market is well solved.In Chapter 4, we first give a characterization of all structure-preserving equivalent martingale measure, under which the Markov-modulated Brownian motion remains a Markov-modulated Brownian motion but with changed parameters. Then by using the martingale representation technique of double martin- gales introduced by Elliott [44], we complete the Markovian regime-switching market by adding the j^th Markovian jump assets. We prove that the enlarged Markovian regime-switching market(i.e. the Markovian regime-switching market completed by adding the j^th Markovian jump assets.) is complete and the explicit hedging policy for a non-negative square-integrable contingent in the enlarged Markovian regime-switching market is also obtained. We also prove that the enlarged Markovian regime-switching market is arbitrage-free and the unique equivalent martingale measure is determined by the explicit expression of likelihood ratio process.In Chapter 5, we make a minor modification to the method of completing the Markovian regime-switching market in Chapter 4. One potential problem of the j^th Markovian jump assets introduced in Chapter 4 is that it can take negative values with positive probability. To exclude this potential problem, we consider a set of j^th Markovian geometric jump assets, which always take non-negative values. We prove that the Markovian regime-switching market enlarged by a set of j^th Markovian geometric jump assets is complete and arbitrage-free. The portfolio selection problem in the enlarged market in the case of a power utility and a logarithmic utility is well solved. In the case of logarithmic utility we adopt the direct differentiation approach to derive the closed-form solution of the optimal portfolio strategy, while for the power utility, we adopt the dynamic programming approach to derive the optimal portfolio strategies. Closed-form solutions for the optimal portfolio strategies and the value function are obtained in both cases. Besides, we also give a Feynman-Kac representation of the value function in both utility cases.Maximizing the expected utility with proportional reinsurance and investment in the extended Markovian regime-switching market with multiple risky assets under no-shorting constraint. Expected utility, as an important objective function in the financial and actuarial literature, has attracted a great deal of interest, see Bai and Guo [10, 11], Brendle [23], Browne [25], Cvitanic and Karatzas [37], Karatzas et al. [95], Lakner [97], Pham and Quenez [132], Yang and Zhang [172] and so on. Browne [25] considered the optimal investment policies for maximizing exponential utility and minimizing the probability of ruin. Bai and Guo [10] extended the result of Browne [25] to the case of the financial market with multiple risky asset and no-shorting constraint. Bai and Guo [11] also considered the utility maximization in the case of partial information. Yang and Zhang [172] studied the optimal investment policies of an insurer with jump-diffusion risk process. The explicit expressions for the optimal strategy and the value function are given.In contrast to above papers, we consider the problem of maximizing the expected utility in the Markov-modulated case, i.e., the surplus of the insurance company and the prices of the financial market are modulated by a continuous time Markov chain. The financial market considered in Chapter 6 and Chapter 7 is the extended Markovian regime-switching market. There are two main differences between the extended Markovian regime-switching market and the Markovian regime-switching market considered in Chapter 4 and Chapter 5: firstly, the extended Markovian regime-switching market consists of one bond and d stocks while the Markovian regime-switching market considered in Chapter 4 and Chapter 5 only consists of one bond and one stock; secondly, in the extended Markovian regime-switching market the bank interest rates, stocks' appreciation rates, and volatility rates not only are modulated by a continuous Markov chain, but also depend on time t.In Chapter 6, we adopt the direct differentiation approach and dynamic programming approach to solve the portfolio selection problems in the extended Markovian regime-switching market for the logarithmic and power utility respectively. Closed-form solutions for the optimal portfolio strategies and the Feynman-Kac representation of value functions in both logarithmic and power utility with no shorting constraint or not are obtained.In Chapter 7, we focus on the optimal proportional reinsurance and investment in the extended Markovian regime-switching market. The main differences from the classical optimal proportional reinsurance and investment problems (see Browne [25]) are: firstly, the surplus of the insurance company is modeled by a Markov-modulated Brownian motion; secondly, the financial market is the extended Markovian regime-switching market; thirdly, no shorting constraint is considered in our problem. We adopt the dynamic programming approach to obtain the optimal proportional reinsurance and investment strategy for maximizing exponential utility function from terminal wealth under the no-shorting constraint. We first present the Hamilton-Jacobi-Bellman (HJB) equation satisfied by the optimal value function and then construct the solution of the HJB equation by the usual separable variable approach. At last, we verify that the solution of the HJB equation is indeed the optimal value function. Closed-form of the optimal proportional reinsurance and investment strategies and the Feynman-Kac representation of the optimal value function are obtained. Besides, we find that the first part of Bai and Guo [10] is a special case of our results. We verify the result of Bai and Guo [10] through our model.Mean-Variance problem in the Markov-switching jump-diffusion market. Compared with the utility maximization problems, mean-variance problem enables an investor to seek the highest return after specifying his/her acceptable risk level that is quantified by the variance of the return. The mean-variance problem was originally proposed by Markowitz [113, 114] for portfolio construction in a single period. Now the mean-variance approach has become the foundation of modern finance theory and has inspired numerous extensions and applications. The continuous time mean-variance problem in the geometric Brownian motion market has been considered by many authors, see Bielecki et al. [20], Li et al. [102], Lim [103], Lim and Zhou [104], Xiong and Zhou [170], Zhou and Li [181], and so on. However, the papers on the mean-variance problem in the Markovian regime-switching market are quite few. So far as we know, there are only three papers on the mean-variance problem in the Markovian regime-switching market, see Zhou and Yin [182], Yin and Zhou [174] and Chen et al. [31]. The above three papers only consider mean-variance problem in the case of the Markovian regime-switching market without jump. So in Chapter 8, we consider mean-variance problem in the Markov-switching jump-diffusion market.In Chapter 8, we first present the description of the Markov-switching jump- diffusion market and reduce the mean-variance problem to a stochastic linear-quadratic (LQ) problem by using the method of Zhou and Li [181]. The feasibility of the mean-variance problem is discussed and some necessary and sufficient conditions for the feasibility in Markov-switching market are given. Through dynamic programming technique, we solve the stochastic LQ problem and obtain the closed form of the value function. Prom the relationship between the stochastic LQ problem and the original mean-variance problem and the well-known Lagrange duality theorem, we obtain efficient portfolio and efficient frontier in a closed form. We also obtain the minimum variance, i.e. the minimum possible terminal variance, along with the portfolio that attains the minimum variance. Besides, we also prove that the mutual fund theorem in the Markov-switching market still holds.Optimal reinsurance and investment in a hidden Markov model. Recently, more and more papers consider the hidden Markov model in insurance and finance, see Bauerle and Rieder [17], Elliott and van der Hoek [49], Elliott et al. [51], Lakner [97], Nagai and Runggaldier [126], Pham and Quenez [132], Rieder and Bauerle [141], Sass [146], Sass and Haussmann [147], and so on. Lakner [97] considered the optimal investment and consumption problem when appreciate rate are unobservable variable through martingale approach and obtained explicit results for log and power utility. The same methodology was applied to the case when the appreciate rate process is a linear diffusion in Lakner [98]. Bauerle and Rieder [17], Sass and Haussmann [147] considered the problem of maximizing the expected utility of the terminal wealth in the case that the drift rate of the stock is Markov-modulated and can not be observed by the investor. Bauerle and Rieder [17] studied the hidden Markov jump intensity model. Using a classical result from filter theory, the hidden problem was reduced to the complete observation problem. Nagai and Runggaldier [126] considered the utility maximization for market models with hidden markov factors through PDE approach.Lakner [97] considered the optimal investment and consumption problem when appreciate rate are unobservable variable through martingale approach and obtained explicit results for log and power utility. The same methodology was applied to the case when the appreciate rate process is a linear diffusion in Lakner [98]. Pham and Quenez [132] addressed the maximization problem of expected utility from terminal wealth in a financial market where price process of risky assets follows a stochastic volatility model and only the vector of stock prices are observed by the investors.Based on these papers, in Chapter 9, we consider an optimal reinsurance and investment problems in a multiple risky assets market with appreciation rate driven by a hidden Markov chain. The surplus of the insurance company is modeled by a Brownian motion with drift and the objective function is the expected exponential utility. By using the filter theory, we establish the separation principle and reduce the problem to the completely observed case. Through the dynamic programming approach and the Girsanov change of measure, we characterize the value function as the unique viscosity solution of a linear parabolic partial differential equation and obtain the Feynman-Kac representation of the value function.