Approximation of Optimal Reinsurance and Dividend Payout Policies

We consider the stochastic process of the liquid assets of an insurance company assuming that the management can control this process in two ways: first, the risk exposure can be reduced by affecting reinsurance, but this decreases the premium income; and second, a dividend has to be paid out to the shareholders. The aim is to maximize the expected discounted dividend payout until the time of bankruptcy. The classical approach is to model the liquid assets or risk reserve process of the company as a piecewise deterministic Markov process. However, within this setting the control problem is very hard. Recently several papers have modeled this problem as a controlled diffusion, presuming that the policy obtained is in some sense good for the piecewise deterministic problem as well. We will clarify this statement in our paper. More precisely, we will first show that the value function of the controlled diffusion provides an asymptotic upper bound for the value functions of the piecewise deterministic problems under diffusion scaling. Finally it will be shown that the upper bound is achieved in the limit under the optimal feedback control of the diffusion problem. This property is called asymptotic optimality.

[1]  Hanspeter Schmidli,et al.  Optimal Proportional Reinsurance Policies in a Dynamic Setting , 2001 .

[2]  Upendra Dave,et al.  Applied Probability and Queues , 1987 .

[3]  Paul Embrechts,et al.  Ruin estimation for a general insurance risk model , 1994, Advances in Applied Probability.

[4]  H. Kushner Heavy Traffic Analysis of Controlled Queueing and Communication Networks , 2001 .

[5]  Paul Embrechts,et al.  Martingales and insurance risk , 1989 .

[6]  Sid Browne Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time , 1997, Math. Oper. Res..

[7]  Søren Asmussen,et al.  Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation , 2000, Finance Stochastics.

[8]  W. Fleming,et al.  Deterministic and Stochastic Optimal Control , 1975 .


[10]  J. Michael Harrison,et al.  Ruin problems with compounding assets , 1977 .

[11]  Larry A. Shepp,et al.  Risk vs. profit potential: A model for corporate strategy , 1996 .

[12]  Michael I. Taksar,et al.  Optimal risk control for a large corporation in the presence of returns on investments , 2001, Finance Stochastics.

[13]  H. Schmidli On minimizing the ruin probability by investment and reinsurance , 2002 .

[14]  Alexander Pollatsek,et al.  A theory of risk , 1970 .

[15]  M. Schäl On piecewise deterministic Markov control processes: Control of jumps and of risk processes in insurance , 1998 .

[16]  A. W. Kemp,et al.  Applied Probability and Queues , 1989 .

[17]  J. Grandell Aspects of Risk Theory , 1991 .

[18]  Karl Borch THE CAPITAL STRUCTURE OF A FIRM , 1969 .

[19]  Michael I. Taksar,et al.  Controlling Risk Exposure and Dividends Payout Schemes:Insurance Company Example , 1999 .

[20]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Statistics of random processes , 1977 .

[21]  Hans Bühlmann Mathematical Methods in Risk Theory , 1970 .

[22]  Bjarne Højgaard,et al.  Optimal proportional reinsurance policies for diffusion models , 1998 .

[23]  Schmidli Hanspeter Diffusion approximations for a risk process with the possibility of borrowing and investment , 1994 .

[24]  Sid Browne Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin , 1995, Math. Oper. Res..

[26]  Søren Asmussen,et al.  Controlled diffusion models for optimal dividend pay-out , 1997 .

[27]  H. Kushner Numerical Methods for Stochastic Control Problems in Continuous Time , 2000 .

[28]  Jan Grandell,et al.  A class of approximations of ruin probabilities , 1977 .

[29]  C. Maglaras Discrete-review policies for scheduling stochastic networks: trajectory tracking and fluid-scale asymptotic optimality , 2000 .

[30]  V. Borkar Controlled diffusion processes , 2005, math/0511077.

[31]  P. Protter,et al.  Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations , 1991 .

[32]  Michael Vogt,et al.  Optimal Dynamic XL Reinsurance , 2003, ASTIN Bulletin.

[33]  Hans U. Gerber Games of Economic Survival with Discrete- and Continuous-Income Processes , 1972, Oper. Res..

[34]  Mark H. Davis Markov Models and Optimization , 1995 .

[35]  S. Sethi,et al.  A Stochastic Extension of the Miller-Modigliani Framework , 1991 .

[36]  Hans U. Gerber,et al.  An introduction to mathematical risk theory , 1982 .

[37]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .