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Mathematical fun with ruin theory. (English) Zbl 0657.62121
Some classical results of ruin theory are derived by probabilistic methods, which have an interest of their own. Let \(X_ 1\), \(X_ 2\),... be positive, independent and identically distributed random variables with common mean \(\mu\). Let \(S_ k=X_ 1+...+X_ k\), \(x>0\) and \(0<a<1/\mu\). Then the expectations of the series \[ \sum^{\infty}_{k=0}a^ k(S_ k+x)^{k-1}e^{-a(S_ k+x)}/k!\quad and\quad \sum^{\infty}_{k=0}A^ k(S_ k+x)^ ke^{-a(S_ k+x)}/k! \] are surprisingly simple: the first is 1/x, and the second is 1/(1-a\(\mu)\). Another basic element is the following classical result. Let \(U(t)=ct-S(t)\) denote the difference between premiums received and claims paid by time t. Then, given that \(U(t)=x\) (i.e. that the process U crosses the level x at time t), the conditional probability that the level x has not been attained before time t is identical to the conditional probability that U has never been negative before time t, and this probability is simply x/(ct); this result can be easily obtained with a martingale argument that is due to F. Delbaen and J. Haezendonck [ibid. 4, 201-206 (1985; Zbl 0571.62093)].
Applications to ruin theory include the probability of ruin with no initial surplus, and the distribution H of the first surplus below the initial level. In the case of an arbitrary initial surplus, a series representation is given for the probability of ruin. If all claims are of a constant size, we get well-known expressions for the probability of ruin. Finally it is shown how the convolutions of H are related to the \(S_ k's\). In a special case the well-known formula for the convolution of uniform distributions is obtained.

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
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