zbMATH — the first resource for mathematics

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.

62P05 Applications of statistics to actuarial sciences and financial mathematics
Full Text: DOI
[1] Beekman, J.A., Two stochastic processes, (1974), Almqvist & Wiksell Stockholm · Zbl 0137.35601
[2] Bowers, N.L.; Gerber, H.U.; Hickman, J.C.; Jones, D.A.; Nesbitt, C.J., Actuarial mathematics, (1987), Society of Actuaries Itasca, IL
[3] Dalbaen, F.; Haezendonck, J., Inversed martingales in risk theory, Insurance: mathematics and economics, 4, 201-206, (1985) · Zbl 0571.62093
[4] Feller, W., An introduction to probability theory and its applications, 2, (1966), Wiley New York · Zbl 0138.10207
[5] Prabhu, N.U., Queues and inventories, (1965), Wiley New York · Zbl 0131.16904
[6] Segerdahl, C.O., A survey of results in the collective theory of risk, (), 276-299 · Zbl 0122.15501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.