Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.

The normal inverse Gaussian process has been used to model both stock returns and interest rate processes. Although several numerical methods are available to compute, for instance, VaR and derivatives values, these are in a relatively undeveloped state compared to the techniques available in the Gaussian case. This paper shows how a Monte Carlo valuation method may be used with the NIG process, incorporating stratified sampling together with an inverse Gaussian bridge. The method is illustrated by pricing average rate options. We find the method is up to around 200 times faster than plain Monte Carlo. These efficiency gains are similar to those found in a related paper, Ribeiro and Webber (02) [20], which develops an analogous method for the variancegamma process. ∗Corresponding author. Claudia Ribeiro gratefully acknowledges the support of Fundação para a Ciência e a Tecnologia and Faculdade de Economia, Universidade do Porto. We are grateful for helpful discussions with participants at QMF Sydney 2002, and to Lynda McCarthy for detailed comments on the manuscript. 1 1

This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a subordinator, that is an increasing process with stationary independent increments. Examples include the two-parameter family of Poisson-Dirichlet models derived from the Poisson process of jumps of a stable subordinator. Applications are made to the random partition generated by the lengths of excursions of a Brownian motion or Brownian bridge conditioned on its local time at zero.

Abstract We construct a probability model for rupture times on a recurrent earthquake source. Adding Brownian perturbations to steady tectonic loading produces a stochastic load-state process. Rupture is assumed to occur when this process reaches a critical-failure threshold. An earthquake relaxes the load state to a characteristic ground level and begins a new failure cycle. The load-state process is a Brownian relaxation oscillator. Intervals between events have a Brownian passage-time distribution that may serve as a temporal model for time-dependent, long-term seismic forecasting. This distribution has the following noteworthy properties: (1) the probability of immediate rerupture is zero; (2) the hazard rate increases steadily from zero at t � 0 to a finite maximum near the mean recurrence time and then decreases asymptotically to a quasi-stationary level, in which the conditional probability of an event becomes time independent; and (3) the quasi-stationary failure rate is greater than, equal to, or less than the mean failure rate because the coefficient of variation is less than, equal to, or greater than 1 / �2 � 0.707. In addition, the model provides expressions for the hazard rate and probability of rupture on faults for which only a bound can be placed on the time of the last rupture. The Brownian relaxation oscillator provides a connection between observable event times and a formal state variable that reflects the macromechanics of stress and strain accumulation. Analysis of this process reveals that the quasi-stationary distance to failure has a gamma distribution, and residual life has a related exponential distribution. It also enables calculation of “interaction ” effects due to external perturbations to the state, such as stress-transfer effects from earthquakes outside the target source. The influence of interaction effects on recurrence times is transient and strongly dependent on when in the loading cycle step perturbations occur. Transient effects may be much stronger than would be predicted by the “clock change ” method and characteristically decay inversely with elapsed time after the perturbation.

This paper shows how the invariance of the arc sine distribution on (0; 1) under a family of rational maps is related on the one hand to various integral identities with probabilistic interpretations involving random variables derived from Brownian motion with arc sine, Gaussian, Cauchy and other distributions, and on the other hand to results in the analytic theory of iterated rational maps. 1 Introduction L'evy[20, 21] showed that a random variable A with the arc sine law P (A 2 da) = da ß p a p 1 \Gamma a (0 ! a ! 1) (1) Research supported in part by N.S.F. Grant 97-03961 can be constructed in numerous ways as a function of the path of a one-dimensional Brownian motion (B t ; 0 t 1), or more simply as A = 1 2 (1 \Gamma cos \Theta) d = 1 2 (1 \Gamma cos 2\Theta) = cos 2 \Theta (2) where \Theta has uniform distribution on [0; 2ß] and d = denotes equality in distribution. See [31, 7] and papers cited there for various extensions of L'evy's results. In connection...

This paper concerns the genealogical structure of a sample of chromosomes sharing a neutral rare allele. We suppose that the mutation giving rise to the allele has only happened once in the history of the entire population, and that the allele is of known frequency q in the population. Within a coalescent framework C. Wiuf and P. Donnelly (1999, Theor. Popul. Biol. 56, 183 201) derived an exact analysis of the conditional genealogy but it is inconvenient for applications. Here, we develop an approximation to the exact distribution of the conditional genealogy, including an approximation to the distribution of the time at which the mutation arose. The approximations are accurate for frequencies q<5 10. In addition, a simple and fast simulation scheme is constructed. We consider a demography parameterized by a d-dimensional vector:=(: 1,..., : d). It is shown that the conditional genealogy and the age of the mutation have distributions that depend on a=q: and q only, and that the effect of q is a linear scaling of times in the genealogy; if q is doubled, the lengths of all branches in the genealogy are doubled. The theory is exemplified in two different demographies of some interest in the study of human evolution: (1) a population of constant size and (2) a population of exponentially decreasing] 2000 Academic Press size (going backward in time). Key Wordsy age of mutation; coalescent theory; genealogy; rare allele; sampling scheme.

K-distributions corresponding to Malkmus' narrow band model are inverse Gaussian distributions. Inverse Gaussian theory developments are therefore directly relevant to gas radiative transfer modeling. The present text illustrates some significant benefits that could be made from this observation : i) k-distribution formulations are simplified, ii) numerical integration procedures can be optimized for each new configuration type, iii) frequently encountered integrals can be solved analyticaly and numerical integrations can be avoided. This last point is illustrated with the compuation of infra-red cooling rates in panetary atmospheres. 1 Introduction Malkmus' narrow band model [1] has become a common tool for modeling gas radiation. It is a two parameters model for the average transmission function of a gas columun of length l over a narrow spectral interval : ø(l) = 1 \Delta Z \Delta exp(\Gammak l)d = exp [OE \Gamma OE (l)] (1) with OE (l) = OE(1 + 2¯l=OE) 1=2 (2) The ...

The authors show how saddlepoint techniques lead to highly accurate approximations for Bayesian predictive densities and cumulative distribution functions in stochastic model settings where the prior is tractable, but not necessarily the likelihood or the predictand distribution. They consider more specifically models involving predictions associated with waiting times for semi-Markov processes whose distributions are indexed by an unknown parameter #. Bayesian prediction for such processes when they are not stationary is also addressed and the inverse-Gaussian based saddlepoint approximation of Wood et al. (1993) is shown to accurately deal with the nonstationarity whereas the normal-based Lugannani & Rice (1980) approximation cannot. Their methods are illustrated by predicting various waiting times associated with M/M/q and M/G/1 queues. They also discuss modifications to the matrix renewal theory needed for computing the moment generating functions that are used in the saddlepoint m...

We show that the generating functions of the generalized Catalan numbers can be identified with the moment generating functions of probability density functions related to the Brownian motion stochastic process. Specifically, the probability density functions are exponential mixtures of inverse Gaussian (EMIG) probability density functions, which arise as the first passage time distributions to the origin of Brownian motion with a negative drift and an exponential initial distribution on the positive halfline. As a consequence of the EMIG representation, we show that the generalized Catalan numbers are the moments of generalized beta distributions. We also study associated convolution sequences arising as the coefficients of the product of two generalized Catalan generating functions. 1

A subpopulation D of rare alleles is considered. The subpopulation is part of a large population that evolves according to a Moran model with selection and growth. Conditional on the current frequency, q, of the rare allele, an approximation to the distribution of the genealogy of D is derived. In particular, the density of the age, T1, of the rare allele is approximated. It is shown that time naturally is measured in units of qN(0) generations, where N(0) is the present day population size, and that the distribution of the genealogy of D depends on the compound parameters \=rqN(0) and _=sqN(0) only. Here, s is the fitness per generation of heterozygote carriers of the rare allele and r is the growth rate per generation of the population. Amongst more, it is shown that for constant population size (\=0) the distribution of D depends on _ only through the absolute value |_|, not the direction of selection.] 2001 Academic Press Key Wordsy birth death process; exponential growth; genealogy; rare allele; selection.