Sharp conditions for certain ruin in a risk process with stochastic return on investments. (English) Zbl 0932.60044

The paper analyzes the wealth process \(Y\) of an asset holder whose cumulative random income flow \(P\) is continuously compounded at a random return rate \(R\) and devalued by a random inflation \(I\). Processes \(P\), \(R\) and \(I\) are Lévy semimartingales. \(P\) is independent of \((I,R)\). Jumps in \(I\) and \(R\) are a.s. greater than \(-1\). Using the Lévy-Itô representations of \(P\) and \(R\) and imposing certain integrability conditions on the compensators of the random measures of their jumps, the author gives sufficient conditions for “certain ruin” (the wealth \(Y\) becoming negative in a finite time), given that the drift part of the return process \(R\) is non-positive. For the case of a positive drift of \(R\), a formula for probability of ruin in a finite time is proposed.
Reviewer: A.Derviz (Praha)


60G35 Signal detection and filtering (aspects of stochastic processes)
60H20 Stochastic integral equations
91B30 Risk theory, insurance (MSC2010)
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