# American Institute of Mathematical Sciences

## Journals

JIMO
Journal of Industrial & Management Optimization 2006, 2(2): 165-175 doi: 10.3934/jimo.2006.2.165
In this paper, ruin probabilities are examined in a discrete time risk model in which the interest rates follow a Markov chain with a denumerable state space and the net losses(the claim amount minus the premium income) are assumed to have a dependent AR(1) structure. An upper bound for ultimate ruin probability is obtained by martingale approach. Recursive equations for both finite time ruin probabilities and ultimate ruin probability are derived. By integrating the inductive method and the recursive equation, an upper bound is given for both finite time ruin probabilities and ultimate ruin probability.
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JIMO
Journal of Industrial & Management Optimization 2016, 12(2): 529-541 doi: 10.3934/jimo.2016.12.529
In this paper, we discuss a Markov chain approximation method to price European options, American options and barrier options in a Markovian regime-switching environment. The model parameters are modulated by a continuous-time, finite-state, observable Markov chain, whose states represent the states of an economy. After selecting an equivalent martingale measure by the regime-switching Esscher transform, we construct a discrete-time, inhomogeneous Markov chain to approximate the dynamics of the logarithmic stock price process. Numerical examples and empirical analysis are used to illustrate the practical implementation of the method.
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JIMO
Journal of Industrial & Management Optimization 2018, 13(5): 1-19 doi: 10.3934/jimo.2018015

This paper investigates the optimal strategies for liability management and dividend payment in an insurance company. The surplus process is jointly determined by the reinsurance policies, liability levels, future claims and unanticipated shocks. The decision maker aims to maximize the total expected discounted utility of dividend payment in infinite time horizon. To describe the extreme scenarios when catastrophic events occur, a jump-diffusion Cox-Ingersoll-Ross process is adopted to capture the substantial claim rate hikes. Using dynamic programming principle, the value function is the solution of a second-order integro-differential Hamilton-Jacobi-Bellman equation. The subsolution--supersolution method is used to verify the existence of classical solutions of the Hamilton-Jacobi-Bellman equation. The optimal liability ratio and dividend payment strategies are obtained explicitly in the cases where the utility functions are logarithm and power functions. A numerical example is provided to illustrate the methodologies and some interesting economic insights.

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JIMO
Journal of Industrial & Management Optimization 2015, 11(2): 461-478 doi: 10.3934/jimo.2015.11.461
We assume that the asset value process of some company is directly related to its stock price dynamics, which can be modeled by geometric Brownian motion. The company can control its asset by paying dividends and injecting capitals, of course both procedures imply proportional and fixed costs for the company. To maximize the expected present value of the dividend payments minus the capital injections until the time of bankruptcy, which is defined as the first time when the asset value falls below the regulation requirement $m$, we seek to find the joint optimal dividend payment and capital injection strategy. By solving the Quasi-variational inequalities, the optimal control problem is addressed, which depends on the parameters of the model and the costs. The sensitivities of transaction costs (such as tax, consulting fees) to the optimal strategy, the expected growth rate and volatility of the firm asset value are also examined, some interesting economic insights are included.
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JIMO
Journal of Industrial & Management Optimization 2013, 9(2): 411-429 doi: 10.3934/jimo.2013.9.411
This paper extends the model in Riesner (2007) to a Markov modulated Lévy process. The parameters of the Lévy process switch over time according to the different states of an economy, which is described by a finite-state continuous time Markov chain. Employing the local risk minimization method, we find an optimal hedging strategy for a general payment process. Finally, we give an example for single unit-linked insurance contracts with guarantee to display the specific locally risk-minimizing hedging strategy.
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JIMO
Journal of Industrial & Management Optimization 2014, 10(4): 1235-1259 doi: 10.3934/jimo.2014.10.1235
In the framework of dual risk model, Yao et al. [18](Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs. European Journal of Operational Research, 211, 568-576) show how to determine optimal dividend and capital injection strategy when the dividend rate is unrestricted and the bankruptcy is forbidden. In this paper, we further include constrain on dividend rate and allow for bankruptcy when it is in deficit. We seek the optimal strategy for maximizing the expected discounted dividends minus the discounted capital injections before bankruptcy. Explicit solutions for strategy and value function are obtained when income jumps follow a hyper-exponential distribution, the corresponding limit results are presented, some known results are extended.
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JIMO
Journal of Industrial & Management Optimization 2010, 6(4): 761-777 doi: 10.3934/jimo.2010.6.761
We consider the optimal control problem with dividend payments and issuance of equity in a dual risk model. Such a model might be appropriate for a company that specializes in inventions and discoveries, which pays costs continuously and has occasional profits. Assuming proportional transaction costs, we aim at finding optimal strategy which maximizes the expected present value of the dividends payout minus the discounted costs of issuing new equity before bankruptcy. By adopting some of the techniques and methodologies in L$\phi$kka and Zervos (2008), we construct two categories of suboptimal models, one is the ordinary dual model without issuance of equity, the other one assumes that, by issuing new equity, the company never goes bankrupt. We identify the value functions and the optimal strategies corresponding to the suboptimal models in two different cases. For exponentially distributed jump sizes, closed-form solutions are obtained.
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JIMO
Journal of Industrial & Management Optimization 2014, 10(3): 795-815 doi: 10.3934/jimo.2014.10.795
This paper focuses on stochastic investment games between two investors with incorporating the influence of the macro economical environment that modeled by a Markov chain with $d$ states. There are two correlated assets are available to two investors, each investor can only invest into one of assets and his opponent choose to invest the other one. The dynamic of the two assets are driven by two drifted Brownian motion with coefficients specified by the functions of the Markov chain. Thus the system considered in this paper is controlled SDEs with random coefficients. Only one payoff function is available to both investors, one investor wants to maximize the expected payoff function, while his opponent wants to minimize the quantity at the same time. As results, the existence of the saddle point of the game, a couple of equations satisfied by the value functions and optimal policies for both investors are derived. Based on finite-difference method and weak convergence theory, a vector-valued Markov chain is constructed for approximating the underlying risky process weakly, which enables us to obtain the value function and optimal policies numerically. To some extend, we can view this paper as a further research of the problems proposed in Wan [23].
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JIMO
Journal of Industrial & Management Optimization 2008, 4(4): 801-815 doi: 10.3934/jimo.2008.4.801
In this paper, optimal problems for the insurer who can invest on risky market and purchase reinsurance are considered. The surplus process of the insurer is a kind of perturbed classical risk model with stochastic premium income. The investment return generating process of the risky market is a drifted Brownian motion plus a compound Poisson process. The objective function in this paper is to maximize the expected utility of wealth of the insurer at terminal time, say $T$. By solving the Hamilton-Jacobi-Bellman equations related to our optimal control problems, the closed form expression for optimal strategy and the value function is derived, which indicates that the value function for an insurer to purchase both investment and reinsurance is always better than the one for the insurer to purchase only either investment or reinsurance.
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JIMO
Journal of Industrial & Management Optimization 2018, 14(1): 371-395 doi: 10.3934/jimo.2017051

This article deals with an optimal dividend, reinsurance and capital injection control problem in the diffusion risk model. Under the objective of maximizing the insurance company's value, we aim at finding the joint optimal control strategy. We assume that there exist both the fixed and proportional costs in control processes and the excess-of-loss reinsurance is "expensive". We derive the closed-form solutions of the value function and optimal strategy by using stochastic control methods. Some economic interpretations of the obtained results are also given.

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