A risk process is def'med as [1]: u(t) = u + ct --s(t), where u is the starting capital of an insurance company, c > 0 is a constant that represents the arrival rate of insurance claims, s(t) is a stochastic process that expresses the total amount paid to insurance claims in time t. In the classical risk model, the number of payments in the interval [0, t] has a Poisson distribution (p.t)kexp( --ta)lk ! k > 1, and the amount of each payment follows the distribution function F(t), t > 0, with mean r (here F(+0) = 0).A quantitative measure of long-term stability of the risk process is the probability of ruin q(u), which is regarded as a function of the starting capital, q(u) = P(u(0 < 0 for some t > 0). Stability is ensured by the condition that q(u) remains less than some prespecified value e (for instance, 9 = 0.001).Denote by p(u) = I -q(u) the probability of survival. This function satisfies the renewal equation [1] it ~u) = a(l + a) .-~ + r -~,,,f t,(u -, z)O-."(~))a., o (1) where p = ~r)-t c --I. If we define the distr~ution function G(z) = r -t I0 (1 --F(t))dt, z > 0, then Eq. (I) can be rewritten as I p(u) =p(l,+p) 7 ~ + (1 § ~t f p(u -z)dC,(z). (2) 0 The unique solution of this equation is obtained in the form t,(u) = q ~(1 -q)tGb(u),t-0where q = p/(1 + p), and * is the convolution operation.In this article, we present a Monte-Carlo estimate for the probability q(u) based on the representation (3) of the solution of Eq. (2). We show how this estimate can be used to solve the equation q(u) = e for the unknown parameter u given c and for the unknown parameter c (which also determines the ruin probability, although this is not made explicit in our notation) given u.
