Ruin probabilities for competing claim processes

Let C 1, C 2,…,C m be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x + ct - C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators C i . Formulae for the probability that ruin is caused by C i are derived. These formulae can be extended to perturbed risk processes of the type X(t) = x + ct - C(t) + Z(t), where Z is a Lévy process with mean 0 and no positive jumps.

[1]  S. Zacks,et al.  Combinatorial Methods in the Theory of Stochastic Processes , 1968 .

[2]  Hans U. Gerber,et al.  Risk theory for the compound Poisson process that is perturbed by diffusion , 1991 .

[3]  Tomasz Rolski,et al.  Stochastic Processes for Insurance and Finance , 2001 .

[4]  Hansjörg Furrer,et al.  Risk processes perturbed by α-stable Lévy motion , 1998 .

[5]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[6]  Ken-iti Sato.  Lévy Processes and Infinitely Divisible Distributions , 1999 .

[7]  F. Dufresne,et al.  Risk Theory with the Gamma Process , 1991, ASTIN Bulletin.

[8]  Paul Embrechts,et al.  Stochastic processes in insurance and finance , 2001 .

[9]  Hanspeter Schmidli.  Distribution of the first ladder height of a stationary risk process perturbed by α-stable Lévy motion , 2001 .

[11]  Matthias Winkel,et al.  Electronic Foreign-Exchange Markets and Passage Events of Independent Subordinators , 2005 .

[12]  Miljenko Huzak,et al.  Ruin probabilities and decompositions for general perturbed risk processes , 2004, math/0407125.

[13]  V. Zolotarev.  The First Passage Time of a Level and the Behavior at Infinity for a Class of Processes with Independent Increments , 1964 .

[14]  R. Doney,et al.  Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity , 2002 .

[15]  H. D. Miller.  Combinatorial methods in the theory of stochastic processes , 1968, Comput. J..

[16]  Hailiang Yang,et al.  Spectrally negative Lévy processes with applications in risk theory , 2001, Advances in Applied Probability.