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An ordinary di erential equations approach for nite time ruin probabilities, including interest rates
 

Summary: An ordinary di erential equations approach for nite
time ruin probabilities, including interest rates
Florin Avram 
June 20, 2002
Provisional draft
Abstract
Markovian modelling via a process with jumps (for example a Levy process) requires
solving partial integro-di erential equations, which typically do not admit analytic
solutions, with the notable exception of the case when the jump distributions are phase{
type and the parameters (drift, volatility, etc) are constant or Markov modulated. In
this paper we explore the implications of an idea of Asmussen [12], [13], which traces
the origin of all these explicit answers to the availability of an associated continuous
\embedding" process, obtained simply by levelling out the positive jumps to sample
path segments with slope +1 and the negative jumps to sample path segments with
slope 1; and which has the same rst passage probabilities. Moreover, in the case of
phase{type jumps, the embedding process is a continuous semi-Markov process whose
generator is an ordinary (vector) di erential operator, more convenient analytically
and numerically than the original partial integro-di erential operator. In the presence
of constant coeÆcients, this reduces ruin problems to exponentiating matrices, which is
further simpli ed by the fact that the eigenvectors of the matrices involved are typically

  

Source: Avram, Florin - Laboratoire de Mathématiques Appliquées, Université de Pau et des Pays de l'Adour

 

Collections: Mathematics