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This paper assumes that the insurance company has the option to buy reinsurance, therefore, the time to buy reinsurance and the proportion of reinsurance become the biggest concern of the insurance company, namely, the optimal reinsurance strategy. Then considers the factors which may exert an influence on the optimal reinsurance strategy (1)the surplus process of insurance company, (2)optimally chooses a time to begin negotiating with a reinsurer to buy quota-share, or proportional reinsurance, which causes an implementation delay(denoted byΔ≥0), (3)chooses the optimal proportion of reinsurance at the beginning of the negotiation period, and(4)pays a fixed transaction cost when the contract is signed(Δunits of time after negotiation begins). The ultimate goal is to choose an optimal reinsurance strategy that maximizes the expectation of total wealth utility at the time of bankruptcy. This setup leads to a combined problem of optimal stopping and stochastic control, however, inspired from bibliog-raphy [6](S.Dayanik.I.Karatzas, On the optimal stopping problem for one-dimension diffusions, Stochastic Processes and their Applications,107(2)(2003) 173-21), usual variational inequality method is not adopted in this paper. Instead, certain function H is constructed for studying its concave and convex property, and then it can determine the optimal time graphically. This paper has also modified the criterion of optimal reinsurance strategy from simple linear function, which was adopted in the bibliography [1](Yoshida-honmachi.Sakyo-ku,kyoto.Ann Arbor, Optimal reinsurance strategy under fixed cost and delay, Stochastic Processes and their applications, (4) (2008),2-4 and 17-19), to quadratic utility function. In the end, an extreme value function and an optimal reinsurance strategy are achieved, therefore, this paper is a generalization of bibliography [1].
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