Optimal Dynamic Risk Control for Insurers with State-Dependent Income

In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.

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Optimal Dynamic Risk Control for Insurers with State-Dependent Income

Journal of Applied Probability ◽

10.1017/s0021900200011335 ◽

2014 ◽

Vol 51 (02) ◽

pp. 417-435 ◽

Cited By ~ 1

Author(s):

Ming Zhou ◽

Jun Cai

Keyword(s):

Risk Model ◽

Closed Form Solution ◽

Jump Process ◽

Form Solution ◽

Markov Jump ◽

Surplus Process ◽

State Dependent ◽

Absolute Ruin ◽

Insurance Portfolio ◽

Optimal Dynamic

In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.

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Automated State-Dependent Importance Sampling for Markov Jump Processes via Sampling from the Zero-Variance Distribution

Journal of Applied Probability ◽

10.1239/jap/1409932671 ◽

2014 ◽

Vol 51 (3) ◽

pp. 741-755

Author(s):

Adam W. Grace ◽

Dirk P. Kroese ◽

Werner Sandmann

Keyword(s):

Importance Sampling ◽

Rare Event ◽

Jump Process ◽

Small Sample ◽

Telecommunication Networks ◽

Jump Processes ◽

Markov Jump ◽

Markov Jump Processes ◽

State Dependent ◽

Wide Range

Many complex systems can be modeled via Markov jump processes. Applications include chemical reactions, population dynamics, and telecommunication networks. Rare-event estimation for such models can be difficult and is often computationally expensive, because typically many (or very long) paths of the Markov jump process need to be simulated in order to observe the rare event. We present a state-dependent importance sampling approach to this problem that is adaptive and uses Markov chain Monte Carlo to sample from the zero-variance importance sampling distribution. The method is applicable to a wide range of Markov jump processes and achieves high accuracy, while requiring only a small sample to obtain the importance parameters. We demonstrate its efficiency through benchmark examples in queueing theory and stochastic chemical kinetics.

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PORTFOLIO SELECTION BY MINIMIZING THE PRESENT VALUE OF CAPITAL INJECTION COSTS

Astin Bulletin ◽

10.1017/asb.2014.22 ◽

2014 ◽

Vol 45 (1) ◽

pp. 207-238 ◽

Cited By ~ 5

Author(s):

Ming Zhou ◽

Kam C. Yuen

Keyword(s):

Portfolio Selection ◽

Risk Model ◽

Numerical Study ◽

Optimal Investment ◽

Investment Policy ◽

Model Parameters ◽

Capital Injection ◽

Surplus Process ◽

Insurance Portfolio ◽

Special Cases

AbstractThis paper considers the portfolio selection and capital injection problem for a diffusion risk model within the classical Black–Scholes financial market. It is assumed that the original surplus process of an insurance portfolio is described by a drifted Brownian motion, and that the surplus can be invested in a risky asset and a risk-free asset. When the surplus hits zero, the company can inject capital to keep the surplus positive. In addition, it is assumed that both fixed and proportional costs are incurred upon each capital injection. Our objective is to minimize the expected value of the discounted capital injection costs by controlling the investment policy and the capital injection policy. We first prove the continuity of the value function and a verification theorem for the corresponding Hamilton–Jacobi–Bellman (HJB) equation. We then show that the optimal investment policy is a solution to a terminal value problem of an ordinary differential equation. In particular, explicit solutions are derived in some special cases and a series solution is obtained for the general case. Also, we propose a numerical method to solve the optimal investment and capital injection policies. Finally, a numerical study is carried out to illustrate the effect of the model parameters on the optimal policies.

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Optimal dynamic reinsurance with common shock dependence and state-dependent risk aversion

International Journal of Financial Engineering ◽

10.1142/s242478631950004x ◽

2019 ◽

Vol 06 (01) ◽

pp. 1950004

Author(s):

Caibin Zhang ◽

Zhibin Liang ◽

Kam Chuen Yuen

Keyword(s):

Risk Aversion ◽

Risk Model ◽

Optimal Strategies ◽

Model Parameters ◽

Compound Poisson ◽

State Dependent ◽

Optimal Dynamic ◽

Game Theoretic ◽

Time Inconsistent ◽

The Impact

This paper studies an optimal dynamic proportional reinsurance in a risk model with two dependent classes of insurance business. Under the criterion of maximizing the mean–variance utility of the terminal wealth with state-dependent risk aversion, we formulate the time-inconsistent problem within a game theoretic framework. By the technique of stochastic control theory, explicit expressions of the optimal results are derived not only for diffusion risk model but also for compound Poisson risk model. Furthermore, the similar problem with constant risk aversion is studied as well. Finally, some numerical examples are presented to show the impact of model parameters on the optimal strategies for both compound Poisson and diffusion cases.

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Automated State-Dependent Importance Sampling for Markov Jump Processes via Sampling from the Zero-Variance Distribution

Journal of Applied Probability ◽

10.1017/s0021900200011645 ◽

2014 ◽

Vol 51 (03) ◽

pp. 741-755

Author(s):

Adam W. Grace ◽

Dirk P. Kroese ◽

Werner Sandmann

Keyword(s):

Importance Sampling ◽

Rare Event ◽

Jump Process ◽

Small Sample ◽

Telecommunication Networks ◽

Jump Processes ◽

Markov Jump ◽

Markov Jump Processes ◽

State Dependent ◽

Wide Range

Many complex systems can be modeled via Markov jump processes. Applications include chemical reactions, population dynamics, and telecommunication networks. Rare-event estimation for such models can be difficult and is often computationally expensive, because typically many (or very long) paths of the Markov jump process need to be simulated in order to observe the rare event. We present a state-dependent importance sampling approach to this problem that is adaptive and uses Markov chain Monte Carlo to sample from the zero-variance importance sampling distribution. The method is applicable to a wide range of Markov jump processes and achieves high accuracy, while requiring only a small sample to obtain the importance parameters. We demonstrate its efficiency through benchmark examples in queueing theory and stochastic chemical kinetics.

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The Decompositions of the Discounted Penalty Functions and Dividends-Penalty Identity in a Markov-Modulated Risk Model

Astin Bulletin ◽

10.1017/s0515036100015051 ◽

2008 ◽

Vol 38 (1) ◽

pp. 53-71 ◽

Cited By ~ 8

Author(s):

Shuanming Li ◽

Yi Lu

Keyword(s):

Risk Model ◽

Jump Process ◽

Risk Process ◽

Penalty Functions ◽

Arrival Process ◽

Claim Amount ◽

Markov Jump ◽

Expected Discounted Penalty Function ◽

Markov Modulated ◽

Expected Discounted Penalty

In this paper, we study the expected discounted penalty functions and their decompositions in a Markov-modulated risk process in which the rate for the Poisson claim arrivals and the distribution of the claim amounts vary in time depending on the state of an underlying (external) Markov jump process. The main feature of the model is the flexibility modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate. Explicit formulas for the expected discounted penalty function at ruin, given the initial surplus, and the initial and terminal environment states, are obtained when the initial surplus is zero or when all the claim amount distributions are from the rational family. We also investigate the distributions of the maximum surplus before ruin and the maximum severity of ruin. The dividends-penalty identity is derived when the model is modified by applying a barrier dividend strategy.

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The Decompositions of the Discounted Penalty Functions and Dividends-Penalty Identity in a Markov-Modulated Risk Model

Astin Bulletin ◽

10.2143/ast.38.1.2030402 ◽

2008 ◽

Vol 38 (01) ◽

pp. 53-71 ◽

Cited By ~ 22

Author(s):

Shuanming Li ◽

Yi Lu

Keyword(s):

Risk Model ◽

Jump Process ◽

Risk Process ◽

Penalty Functions ◽

Arrival Process ◽

Claim Amount ◽

Markov Jump ◽

Expected Discounted Penalty Function ◽

Markov Modulated ◽

Expected Discounted Penalty

In this paper, we study the expected discounted penalty functions and their decompositions in a Markov-modulated risk process in which the rate for the Poisson claim arrivals and the distribution of the claim amounts vary in time depending on the state of an underlying (external) Markov jump process. The main feature of the model is the flexibility modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate. Explicit formulas for the expected discounted penalty function at ruin, given the initial surplus, and the initial and terminal environment states, are obtained when the initial surplus is zero or when all the claim amount distributions are from the rational family. We also investigate the distributions of the maximum surplus before ruin and the maximum severity of ruin. The dividends-penalty identity is derived when the model is modified by applying a barrier dividend strategy.

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