Mathematical fun with ruin theory.

*(English)*Zbl 0657.62121Some 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.

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 |

##### Keywords:

martingales; upcrossings; inclusion-exclusion; risk process; ruin theory; premiums; claims; conditional probability; surplus; probability of ruin; convolution of uniform distributions
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\textit{H. U. Gerber}, Insur. Math. Econ. 7, No. 1, 15--23 (1988; Zbl 0657.62121)

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##### References:

[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 |

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