Zbigniew Palmowski

Parisian ruin probability for spectrally negative Lévy processes

In this note we give, for a spectrally negative Levy process, a compact formula for the Parisian ruin probability, which is defined by the probability that the process exhibits an excursion below zero, with a length that exceeds a certain fixed period r. The formula involves only the scale function of the spectrally negative Levy process and the distribution of the process at time r.

A Martingale Review of some Fluctuation Theory for Spectrally Negative Lévy Processes

We give a review of some fluctuation theory for spectrally negative Levy processes using for the most part martingale theory. The methodology is based on the techniques found in Kyprianou and Palmowski (2003) which deal with similar issues for a general class of Markov additive processes.

Fluctuations of spectrally negative Markov additive processes

For spectrally negative Markov Additive Processes (MAPs) we generalize classical fluctuation identities developed in (1964), (1967), (1975), (1976), (1973), (1990) and (1997) which concern one and two sided exit problems for spectrally negative Levy processes.

The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk

We study the asymptotic probability that a random walk with heavy-tailed increments crosses a high boundary on a random time interval. We use new techniques to extend results of Asmussen (1998) to completely general stopping times, uniformity of convergence over all stopping times, and a wide class of nonlinear boundaries. We give also some examples and counterexamples.

On–off fluid models in heavy traffic environment

We consider fluid models with infinite buffer size. Let {ZN(t)} be the net input rate to the buffer, where {{ZN(t)} is a superposition of N homogeneous alternating on–off flows. Under heavy traffic environment {{ZN(t)} converges in distribution to a centred Gaussian process with covariance function of a single flow. The aim of this paper is to prove the convergence of the stationary buffer content process {XN*(t)} in the fNth model to the buffer content process {XN(t)} in the limiting Gaussian model.

BOUNDS FOR FLUID MODELS DRIVEN BY SEMI-MARKOV INPUTS

In this paper we consider an infinite buffer fluid model whose input is driven by independent semi-Markov processes. The output capacity of the buffer is a constant. We derive upper and lower bounds for the limiting distribution of the stationary buffer content process. We discuss examples and applications where the results can be used to determine bounds on the loss probability in telecommunication networks.

On the exact asymptotics of the busy period in GI/G/1 queues

In this paper we study the busy period in GI/G/1 work-conserving queues. We give the exact asymptotics of the tail distribution of the busy period under the light tail assumptions. We also study the workload process in the M/G/1 system conditioned to stay positive.

A note on martingale inequalities for fluid models

We discuss how to prove exponential upper bounds for simple fluid models driven by a finite state CTMC. In particular, we consider the fluid model of Anick, Mitra and Sondhi, in which the fluid is generated by N independent 0-1 Markovian sources. We also give a result on a generalized eigenvector for fluid models driven by reversible CTMCs.

On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums

This paper concerns an optimal dividend distribution problem for an insurance company with surplus-dependent premium. In the absence of dividend payments, such a risk process is a particular case of so-called piecewise deterministic Markov processes. The control mechanism chooses the size of dividend payments. The objective consists in maximizing the sum of the expected cumulative discounted dividend payments received until the time of ruin and a penalty payment at the time of ruin, which is an increasing function of the size of the shortfall at ruin. A complete solution is presented to the corresponding stochastic control problem. We identify the associated Hamilton–Jacobi–Bellman equation and find necessary and sufficient conditions for optimality of a single dividend-band strategy, in terms of particular Gerber–Shiu functions. A number of concrete examples are analyzed.

Exact and Asymptotic Results for Insurance Risk Models with Surplus-dependent Premiums

In this paper we develop a symbolic technique to obtain asymptotic expressions for ruin probabilities and discounted penalty functions in renewal insurance risk models when the premium income depends on the present surplus of the insurance portfolio. The analysis is based on boundary problems for linear ordinary differential equations with variable coefficients. The algebraic structure of the Greens operators allows us to develop an intuitive way of tackling the asymptotic behavior of the solutions, leading to exponential-type expansions and Cramer-type asymptotics. Furthermore, we obtain closed-form solutions for more specific cases of premium functions in the compound Poisson risk model.