Monte-Carlo estimate of the probability of ruin in a compound poisson model of risk theory

Abstract

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.

Stochastic successive approximation method for assessing the insolvency risk of an insurance company

“…In Theorem 1, we strengthen the result of [13] by proving that, with probability one, we have here the convergence uniform with respect to u ³ 0 . We can also use the Bernoulli scheme for estimation of nonruin probability by formulas (10) and (11).…”

“…This problem continues to draw attention of researchers [5-8, 11-16, 18-33]. In particular, in [13], based on representation (4), a point ruin probability estimate of the Monte-Carlo type is constructed. A result of this article is a strengthening of this result, namely, the proof of uniform convergence of the estimate with probability one.…”

“…(estimate (14) was proposed in [13]), where { } x nk are independent random quantities that are equally distributed for all n N = 1,... , and k = 1 2 , ,... with distribution function (5); a random quantity n n u ( ) finite with probability one for each u is determined by the condition x…”

“…Representation (4) is the source of numerous approaches to the estimation of the nonruin probability j( ) u [12][13][14][15][16]. For example, by virtue of B. Levi's theorem [34, p. 305], series (4) can be rewritten in the form…”

“…To obtain a point ruin probability estimate (for a fixed opening capital), a similar method was applied in [12][13][14][15], and earlier it was used in [16] for reliability estimation. We note that, as distinct from [13], the explicit form of requirement distribution was not used in [12][13][14][15] but its nonparametric (empirical) estimate was applied. In point estimation (for fixed values of the opening capital), the convergence of the method follows with probability one from the law of large numbers.…”