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1
Content available remote Two-dimensional ruin probability for subexponential claim size
Foss S., Korshunov D., Palmowski Z., Rolski T.
Probability and Mathematical Statistics
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2017
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Vol. 37, Fasc. 2
319--335
EN
We analyse the asymptotics of ruin probabilities of two insurance companies (or two branches of the same company) that divide between them both claims and premiums in some specified proportions when the initial reserves of both companies tend to infinity, and generic claim size is subexponential.
2
Content available remote A note on first passage probabilities of a Lévy process reflected at a general barrier
Palmowski Z., Świątek P.
Probability and Mathematical Statistics
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2017
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Vol. 37, Fasc. 2
455--469
EN
In this paper we analyze a Lévy process reflected at a general (possibly random) barrier. For this process we prove the Central Limit Theorem for the first passage time.We also give the finite-time first passage probability asymptotics.
3
Content available remote Problem optymalizacyjny de Finettiego dla procesów Lévy’ego
Palmowski Z.
Wiadomości Matematyczne
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2016
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T. 52, Nr 1
1--19
PL
Teoria optymalizacji (dyskretnej i losowej) i teoria sterowania stanowią obszerne dziedziny matematyki, związane z równaniami różniczkowymi, analizą matematyczną i informatyką. Celem tej pracy jest opis bardzo szczególnego stochastycznego problemu optymalizacyjnego, który w ostatnim czasie skupił dużą uwagę uczonych zajmujących się matematyką aktuarialno-finansową oraz pokazanie, jakie metody badawcze i pytania ich dotyczą. Pragniemy zwrócić szczególną uwagę na nowe metody probabilistyczne związane z martyngałami, pozwalające na analizę procesów posiadających skoki.
4
Content available remote Quantile hedging for an insider
Klusik P., Palmowski Z., Zwierz J.
Probability and Mathematical Statistics
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2010
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Vol. 30, Fasc. 2
247--258
EN
In this paper we consider the problem of the quantile hedging from the point of view of a better informed agent acting on the market. The additional knowledge of the agent is modelled by a filtration initially enlarged by some random variable. By using equivalent martingale measures introduced in [1] and [2] we solve the problem for the complete case, by extending the results obtained in [4] to the insider context. Finally, we consider the examples with the explicit calculations within the standard Black–Scholes model.
5
Content available remote Markov processes conditioned to never exit a subspace of the state space
Palmowski Z., Rolski T.
Probability and Mathematical Statistics
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2004
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Vol. 24, Fasc. 2
339--353
EN
In this paper we study Markov processes never exiting (NE) a subspace A of the state space E or, in other words, Markov processes conditioned to stay in the subspace A. We show how the knowledge of the exact asymptotics of the tail distribution of the exit time helps to find the suitable exponential martingale, which, in turn, serves for the change of measure. Under the new probability measure the process is the sought for never exiting one the subspace A. We also find its extended generator and study relationships between the invariant measure (INE) and the quasi-stationary (QS) distribution. We analyze in detail the PDMP processes.
6
Content available remote A note on impotance sampling simulation for germ-grain model
Palmowski Z.
Probability and Mathematical Statistics
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2003
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Vol. 23, Fasc. 2
293--304
EN
In this paper we demonstrate how to use the importance sampling method to simulate rare events in a germ-grain model. We analyze conditions under which two germ-grain models are mutually absolutely continuous. We also find the likelihood set process. We apply these results in simulating the probability that the radius of the occupied component of the origin in continuous percolation is greater than some R. This method is based on the reduction of the variance of estimator.
7
Content available remote Tail probabilities for a risk process with subexponential jumps in a regenerative and diffusion environment
Palmowski Z.
Probability and Mathematical Statistics
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2002
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Vol. 22, Fasc. 2
381-405
EN
In this paper we find a nonexponential Lundberg approximation of the ruin probability in a Cox model, in which a governing process has a regenerative structure and claims are light-tailed or have an intermediate regularly varying distribution. Examples include an intensity process being reflected Brownian motion, square functions of the Ornstein-Uhlenbeck process and splitting reflected Brownian bridges. In particular, we consider a non-Markovian intensity proces.
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