scholarly journals Optimal Reinsurance Problem under Fixed Cost and Exponential Preferences

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 295
Author(s):  
Matteo Brachetta ◽  
Claudia Ceci
Keyword(s):  
Optimal Stopping ◽  
Risk Model ◽  
Stopping Time ◽  
Fixed Cost ◽  
Model Parameters ◽  
Insurance Company ◽  
Optimal Reinsurance ◽  
Retention Level ◽  
Step Procedure ◽  
Time Problem

We investigate an optimal reinsurance problem for an insurance company taking into account subscription costs: that is, a constant fixed cost is paid when the reinsurance contract is signed. Differently from the classical reinsurance problem, where the insurer has to choose an optimal retention level according to some given criterion, in this paper, the insurer needs to optimally choose both the starting time of the reinsurance contract and the retention level to apply. The criterion is the maximization of the insurer’s expected utility of terminal wealth. This leads to a mixed optimal control/optimal stopping time problem, which is solved by a two-step procedure: first considering the pure-reinsurance stochastic control problem and next discussing a time-inhomogeneous optimal stopping problem with discontinuous reward. Using the classical Cramér–Lundberg approximation risk model, we prove that the optimal strategy is deterministic and depends on the model parameters. In particular, we show that there exists a maximum fixed cost that the insurer is willing to pay for the contract activation. Finally, we provide some economical interpretations and numerical simulations.

Optimal stopping of a risk process: model with interest rates

2002 ◽  
Vol 39 (2) ◽  
pp. 261-270 ◽  
Author(s):  
Bogdan Krzysztof Muciek
Keyword(s):  
Interest Rates ◽  
Optimal Stopping ◽  
Process Model ◽  
Stopping Time ◽  
Risk Process ◽  
Risk Theory ◽  
Dynamic Programming Method ◽  
Insurance Company ◽  
Initial Capital ◽  
The Times

The following problem in risk theory is considered. An insurance company, endowed with an initial capital a ≥ 0, receives premiums and pays out claims that occur according to a renewal process {N(t), t ≥ 0}. The times between consecutive claims are i.i.d. The sequence of successive claims is a sequence of i.i.d. random variables. The capital of the company is invested at interest rate α ∊ [0,1], claims increase at rate β ∊ [0,1]. The aim is to find the stopping time that maximizes the capital of the company. A dynamic programming method is used to find the optimal stopping time and to specify the expected capital at that time.

Optimal stopping of a risk process: model with interest rates

2002 ◽  
Vol 39 (02) ◽  
pp. 261-270 ◽  
Author(s):  
Bogdan Krzysztof Muciek
Keyword(s):  
Interest Rates ◽  
Optimal Stopping ◽  
Process Model ◽  
Stopping Time ◽  
Risk Process ◽  
Risk Theory ◽  
Dynamic Programming Method ◽  
Insurance Company ◽  
Initial Capital ◽  
The Times

The following problem in risk theory is considered. An insurance company, endowed with an initial capital a ≥ 0, receives premiums and pays out claims that occur according to a renewal process {N(t), t ≥ 0}. The times between consecutive claims are i.i.d. The sequence of successive claims is a sequence of i.i.d. random variables. The capital of the company is invested at interest rate α ∊ [0,1], claims increase at rate β ∊ [0,1]. The aim is to find the stopping time that maximizes the capital of the company. A dynamic programming method is used to find the optimal stopping time and to specify the expected capital at that time.

scholarly journals Optimal per-loss reinsurance and investment to minimize the probability of drawdown

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xia Han ◽  
Zhibin Liang ◽  
Yu Yuan ◽  
Caibin Zhang
Keyword(s):  
Closed Form ◽  
Financial Market ◽  
Value Function ◽  
Risk Model ◽  
Investment Strategy ◽  
Insurance Company ◽  
Optimal Reinsurance ◽  
Claim Number ◽  
Common Shock ◽  
Insurance Business

<p style='text-indent:20px;'>In this paper, we study an optimal reinsurance-investment problem in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. We assume that the insurer can purchase per-loss reinsurance for each line of business and invest its surplus in a financial market consisting of a risk-free asset and a risky asset. Under the criterion of minimizing the probability of drawdown, the closed-form expressions for the optimal reinsurance-investment strategy and the corresponding value function are obtained. We show that the optimal reinsurance strategy is in the form of pure excess-of-loss reinsurance strategy under the expected value principle, and under the variance premium principle, the optimal reinsurance strategy is in the form of pure quota-share reinsurance. Furthermore, we extend our model to the case where the insurance company involves <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ (n\geq3) $\end{document}</tex-math></inline-formula> dependent classes of insurance business and the optimal results are derived explicitly as well.</p>

An explicit solution to an optimal stopping problem with regime switching

2001 ◽  
Vol 38 (2) ◽  
pp. 464-481 ◽  
Author(s):  
Xin Guo
Keyword(s):  
Optimal Stopping ◽  
Regime Switching ◽  
Hidden Markov ◽  
Stopping Time ◽  
Closed Form Solution ◽  
Form Solution ◽  
Jump Discontinuities ◽  
Time Problem ◽  
Standard Application ◽  
Hidden Markov Process

We investigate an optimal stopping time problem which arises from pricing Russian options (i.e. perpetual look-back options) on a stock whose price fluctuations are modelled by adjoining a hidden Markov process to the classical Black-Scholes geometric Brownian motion model. By extending the technique of smooth fit to allow jump discontinuities, we obtain an explicit closed-form solution. It gives a non-standard application of the well-known smooth fit principle where the optimal strategy involves jumping over the optimal boundary and by an arbitrary overshoot. Based on the optimal stopping analysis, an arbitrage-free price for Russian options under the hidden Markov model is derived.

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