Ross Maller

Ruin probabilities and overshoots for general Lévy insurance risk processes

We formulate the insurance risk process in a general Levy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to −∞ a.s. and the positive tail of the Levy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Kluppelberg [Stochastic Process. Appl. 64 (1996) 103–125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207–226] for ruin probabilities and the overshoot in random walk and compound Poisson models are shown to have analogues in the general setup. The identities we derive open the way to further investigation of general renewal-type properties of Levy processes.

Continuous Time Volatility Modelling: COGARCH versus Ornstein–Uhlenbeck Models

We compare the probabilistic properties of the non-Gaussian Ornstein-Uhlenbeck based stochastic volatility model of Barndorff-Nielsen and Shephard (2001) with those of the COGARCH process. The latter is a continuous time GARCH process introduced by the authors (2004). Many features are shown to be shared by both processes, but differences are pointed out as well. Furthermore, it is shown that the COGARCH process has Pareto like tails under weak regularity conditions.

Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity

We prove some limiting results for a Lévy process Xt as t↓0 or t→∞, with a view to their ultimate application in boundary crossing problems for continuous time processes. In the present paper we are mostly concerned with ideas related to relative stability and attraction to the normal distribution on the one hand and divergence to large values of the Lévy process on the other. The aim is to find analytical conditions for these kinds of behaviour which are in terms of the characteristics of the process, rather than its distribution. Some surprising results occur, especially for the case t↓0; for example, we may have Xt/t →P+∞ (t↓0) (weak divergence to +∞), whereas Xt/t→∞ a.s. (t↓0) is impossible (both are possible when t→∞), and the former can occur when the negative Lévy spectral component dominates the positive, in a certain sense. “Almost sure stability” of Xt, i.e., Xt tending to a nonzero constant a.s. as t→∞ or as t↓0, after normalisation by a non-stochastic measurable function, reduces to the same type of convergence but with normalisation by t, thus is equivalent to “strong law” behaviour. Boundary crossing problems which are amenable to the methods we develop arise in areas such as sequential analysis and option pricing problems in finance.

On continuity properties of the law of integrals of levy processes

Let (ξ, η) be a bivariate Levy process such that the integral \(\int_0^\infty {e^{ - \xi _{t - } } d\eta _t }\)converges almost surely. We characterise, in terms of their Levy measures, those Levy processes for which (the distribution of) this integral has atoms. We then turn attention to almost surely convergent integrals of the form I := ∫ 0 ∞ g(ξ t ) dt, where g is a deterministic function. We give sufficient conditions ensuring that I has no atoms, and under further conditions derive that I has a Lebesgue density. The results are also extended to certain integrals of the form ∫ 0 ∞ g(ξ t ) dY t , where Y is an almost surely strictly increasing stochastic process, independent of ξ.

A Multinomial Approximation for American Option Prices in Levy Process Models

This paper gives a tree-based method for pricing American options in models where the stock price follows a general exponential Levy process. A multinomial model for approximating the stock price process, which can be viewed as generalizing the binomial model of Cox, Ross, and Rubinstein (1979) for geometric Brownian motion, is developed. Under mild conditions, it is proved that the stock price process and the prices of American-type options on the stock, calculated from the multinomial model, converge to the corresponding prices under the continuous time Levy process model. Explicit illustrations are given for the variance gamma model and the normal inverse Gaussian process when the option is an American put, but the procedure is applicable to a much wider class of derivatives including some path-dependent options. Our approach overcomes some practical difficulties that have previously been encountered when the Levy process has infinite activity.

Generalised Ornstein-Uhlenbeck Processes and the Convergence of Lévy Integrals

Exponential functionals of the form \( \int_{0}^t \mathrm{e}^{-\xi_{s-}} \mathrm{d}\eta_s\) constructed from a two dimensional Levy process \((\xi,\eta)\) are of interest and application in many areas. In particular, the question of the convergence of the integral \( \int_{0}^\infty \mathrm{e}^{-\xi_{t-}} \mathrm{d}\eta_t\) arises in recent investigations such as those of Barndorff-Nielsen and Shephard [3] in financial econometrics, and in those of Carmona, Petit and Yor [9], and Yor [40, 41], where it is related among other things to the existence of an invariant measure for a generalised Ornstein-Uhlenbeck process. We give a complete solution to the convergence question for integrals of the form \(\int_0^\infty g(\xi_{t-}) \mathrm{d}\eta_t\), when g(t) = e-t and \(\eta_t\) is general, or \(g(\cdot)\) is a nonincreasing function and \(\mathrm{d}\eta_t = \mathrm{d} t\), and some other related results. The necessary and sufficient conditions for convergence are stated in terms of the canonical characteristics of the Levy process. Some applications in various areas (compound Poisson processes, subordinated perpetuities, the Doleans-Dade exponential) are also outlined.

Estimating the numbers of prison terms in criminal careers from one-step probabilities of recidivism

A method of using estimates of “one-step” probabilities of recidivism, i.e., conditional probabilities of individuals returning to prison for the jth time given release for the (j-1)st time, to estimate the numbers of prison terms expected to be accumulated by the individuals, is presented. The method is illustrated by calculating the expected numbers of prison terms separately for racial and gender groups in a large data base of Western Australian prisoners. The recidivism probabilities for these data were estimated by fitting Weibull “mixture” models to the (possibly censored) times to recidivate. The probabilities increase strongly asj increases from 1 to 6, then level off. Large differences between them are due to racial and gender group and these are reflected in the differing expected prison career durations for these groups. The effect of interventions which might lower recidivism is discussed in the light of the method as applied to these estimates.

Ornstein-Uhlenbeck processes and extensions

This paper surveys a class of Generalised Ornstein-Uhlenbeck (GOU) processes associated with Levy processes, which has been recently much analysed in view of its applications in the financial modelling area, among others. We motivate the Levy GOU by reviewing the framework already well understood for the “ordinary” (Gaussian) Ornstein-Uhlenbeck process, driven by Brownian motion; thus, defining it in terms of a stochastic differential equation (SDE), as the solution of this SDE, or as a time changed Brownian motion. Each of these approaches has an analogue for the GOU. Only the second approach, where the process is defined in terms of a stochastic integral, has been at all closely studied, and we take this as our definition of the GOU (see Eq. (12) below).

Relative stability and the strong law of large numbers

SummaryLetX1,X2,..., be i.i.d. random variables andSn=X1+X2+⋯. +Xn. In this paper we simplify Rogozins condition forSn/Bn