Optimal stopping of 2-vector risk process

Let us consider the insurance company having an initial capital a > 0 which insures two kind of risks. The i-th risk makes the stream of insurance premiums with constant rate ci and pays out successive claims, which are representing by i.i.d. random variables Xi,1, Xi,2, . . . with cumulative distribution function Hi. The losses related to the i-th risk occur according to the renewal process {Ni(t), t ≥ 0}, where Ni(t), i = 1, 2, are the number of claims up to the time t in the stream of the risk i. The renewal processes are mutually independent and they are independent on the sequence of claims. The 2vector point process (N1(t), N2(t)), t ≥ 0 can be represented also by a sequence of random variables Tn taking values in [0,∞] such that T0 = 0, Tn <∞ then Tn < Tn+1 for n ∈ N, and a sequence of {1, 2}-valued random variables Zn for n ∈ N∪{0} (see Brémaud [2] Ch. II). Let us assume that there are i.i.d. r.v. Xi,n with continuous, cumulative distribution function Fi. Define the balance between premium and collection of payoffs covering the claims Ri(t) = cit− ∑Ni(t) s=1 Xi,s in the risk i. The considered processes allow to describe the reserves of the insurance companies. The management decisions concerning the portfolio of risks change parameters of the processes. The main idea is to find the good moment to manage the assets. Various approaches are taken into account which are subject of the research (see Azuce and Roberts [1], Muciek [5], Rolski et al. [7], Karpowicz and Szajowski [4], Muciek and Szajowski [6]). The presented model concerns its application to the reinsurance of the companies. There are two companies which insure the risks modeled by the first and the second process, respectively. They agree to re-insure each other. It means that both companies share the risks in some proportion. There are some limitation of the behavior of the companies and the aims of each are measured by their utility functions. We are looking for the strategies which form an equilibrium. The first paper on the competitive stopping of the discrete time stochastic processes has been presented by Dynkin [3]. Many extensions, modification and Dynkin’s game applications have been investigated. Only few of them are strictly related to the risk process.