Optimal proportional reinsurance with common shock dependence to minimise the probability of drawdown

Abstract

In this paper, we study the optimal proportional reinsurance 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, and the criterion is to minimise the probability of drawdown, namely, the probability that the value of the surplus process reaches some fixed proportion of its maximum value to date. By the method of maximising the ratio of drift of a diffusion divided to its volatility squared, and the technique of stochastic control theory and the corresponding Hamilton–Jacobi–Bellman equation, we investigate the optimisation problem in two different cases. Furthermore, we constrain the reinsurance proportion in the interval [0,1] for each case, and derive the explicit expressions of the optimal proportional reinsurance strategy and the minimum probability of drawdown. Finally, some numerical examples are presented to show the impact of model parameters on the optimal results.

Optimal reinsurance and investment in danger‐zone and safe‐region

“…Furthermore, in combination with Proposition 1, we can see that either drawdown occurs with probability (u, m) = P( < ∞) or the optimal controlled surplus value lies strictly between m and u s , for all time, with probability of 1 − (u, m). The similar conclusion is also derived in the works of Bayraktar and Zhang, 23 Angoshtari et al, 5 and Han et al 6,7…”

“…Firstly, note that when the surplus is relatively low, the insurer prefers to pay more attention to reducing the risk; but when the surplus becomes relatively high, the insurer may be more interested in reaching a goal as quickly as possible. Thus, it is meaningful to consider the objectives of survival and growth in two complementary regions, and our optimal results for both aspects of the problems will therefore complement the results in Han et al 6,7 Secondly, we assume that the insurer takes both investment and reinsurance into consideration and the price process of risky asset is correlated to the claim process. Short-selling is prohibited and the reinsurance proportion is constrained into [0, 1].…”

“…Note that when = 0, minimizing the probability of drawdown is equal to minimizing the probability of ruin. Angoshtari et al 5 and Han et al 6,7 minimized the probability of drawdown over an infinite-time horizon and showed that the strategy which minimizes the probability of ruin also minimizes the probability of drawdown. Besides, Angoshtari et al 8 and Chen et al 9 computed the optimal investment strategy to minimize the probability of lifetime drawdown for an individual investor.…”

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

“…Recently, and Yuen et al (2015) consider the optimal proportional reinsurance strategy in a risk model with two or more dependent classes of insurance business under the criterion of maximizing the expected exponential utility, and Bi et al (2016) study the optimal MV investment and reinsurance problem with bankruptcy prohibition. For more researches about dependent risk, it can be found in Ming et al (2016), , Liang et al (2018), Han et al (2018) and the references therein.…”

Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs

“…They found that the optimal strategy for a random (or finite) maturity setting such as lifetime drawdown was very different from that of the corresponding ruin problem. Furthermore, Han et al [16,18] considered the problem of optimal reinsurance which minimized the probability of drawdown for the risk model with thinning dependence/common shock structure. For any other works involving drawdown, we can refer to Grossman and Zhou [15], Cvitanić and Karatzas [9], and Elie and Touzi [11].…”