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Ruin Probabilities for Two Collaborating Insurance Companies

Michna Zbigniew

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 2

360--386

We find a formula for the supremum distribution of spectrally positive or negative Lévy processes with a broken linear drift. This gives formulas for ruin probabilities if two insurance companies (or two branches of the same company) divide between them both claims and premia in some specified proportions or if the premium rate for a given insurance portfolio is changed at a certain time. As an example we consider a gamma Lévy process, an -stable Lévy process and Brownian motion. Moreover we obtain identities for the Laplace transform of the distribution of the supremum of Lévy processes with a randomly broken drift (random time of the premium rate change) and on random intervals (random time when the insurance portfolio is closed).

On the supremum from gaussian processes over infinite horizon

Dębicki Krzysztof, Michna Zbigniew, Rolski Tomasz

Probability and Mathematical Statistics

1998

Vol. 18, Fasc. 1

83--100

In the paper we study the asymptotic of the tail of distribution function P(A(X,c) > x) for x→∞, where A(X, c) is the supremum of X (t)—ct over [0, ∞). In particular, X(t) is the fractional Brownian motion, a nonlinearly scaled Brownian motion or some integrated stationary Gaussian processes. For the fractional Brownian motion we give a stronger result than a recent one of Duffield and O’Connell [5].