Mihael Perman

Size-biased sampling of Poisson point processes and excursions

SummarySome general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function. These formulae explain why in certain cases (gamma and stable) the size-biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking (residual allocation) scheme defined by independent beta random variables. An application is made to length biased sampling of excursions of a Markov process away from a recurrent point of its statespace, with emphasis on the Brownian and Bessel cases when the associated inverse local time is a stable subordinator. Results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by Brownian motion.

Ruin probabilities and decompositions for general perturbed risk processes

We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, and the perturba- tion being a spectrally negative Levy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival proba- bility of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.

Order statistics for jumps of normalised subordinators

A subordinator is a process with independent, stationary, non-negative increments. On the unit interval we can view this process as the distribution function of a random measure, and, dividing this random measure by its total mass, we get a random discrete probability distribution. Formulae for the joint distribution of the n largest atoms in this distribution are derived. They are used to derive some results about the Poisson-Dirichlet process. Subordinators arise as inverse local times of diffusions and the atoms in the random measure associated with them correspond to the lengths of excursions of the diffusion away form 0. For Brownian motion, or more generally, for Bessel processes of dimension [delta], 0

An excursion approach to maxima of the Brownian bridge.

Distributions of functionals of Brownian bridge arise as limiting distributions in non-parametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. The idea of rescaling and conditioning on the local time has been used widely in the literature. In this paper it is used to give a unified derivation of a number of known distributions, and a few new ones. Particular cases of calculations include the distribution of the Kolmogorov-Smirnov statistic and the Kuiper statistic.

Optimal hedging strategies in equity-linked products

Abstract Equity-linked insurance policies are one of the most widespread insurance products. In many cases such contracts have guarantees like a minimum return over the lifetime of the policy. Liabilities arising from such guarantees must be hedged by suitable investments. There are restrictions on hedging strategies in many jurisdictions but with the more flexible regulatory framework of Solvency 2 there are alternative ways to hedge certain guaranteed products using derivative securities. In this paper we investigate when it is optimal to switch to hedging liabilities with derivative securities in the framework of the Cox–Ross–Rubinstein model. This leads to optimal stopping problems that can be solved explicitly. Mortality is also incorporated in the model. The results may indicate the level of reserves necessary to meet obligations with the desired level of confidence. In particular the strategy may be applicable in adverse market conditions.

Quantile approximations in auto-regressive portfolio models

This paper develops an analytical approximation for the distribution function of a terminal value of a periodic series of buy-and-hold investments placed over a fixed time horizon for the case when log-returns of assets follow a p-th order vector auto-regressive process. The derivation is based on a first order Taylor conditioned approximation with a suitably chosen conditioning variable. The results indicate a remarkably good fit between the approximating procedure and simulations based on realistic parameters.

Analytical Approximation for a Multi-Period Portfolio Problem With Vector Autoregressive Returns

This paper develops an analytical approximation, based on conditioning on the first order Taylor series expansion, for the distribution function of a terminal value of a series of constant mix portofolio investments placed over fixed time horizon for the case when log-returns of assets follow a p-th order vector autoregressive process. The results of the numerical simulation based on realistic parameters of the process of returns indicate extremely good fit between the approximating procedure and Monte Carlo simulation.