Results 1  10
of
15
On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... 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, expl ..."
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Cited by 98 (9 self)
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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.
PoissonKingman Partitions
 of Lecture NotesMonograph Series
, 2002
"... 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 subordin ..."
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Cited by 11 (3 self)
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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 twoparameter family of PoissonDirichlet 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.
A Monte Carlo method for the normal inverse Gaussian option valuation model using an inverse Gaussian bridge
, 2003
"... 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 Ga ..."
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Cited by 10 (0 self)
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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
A Brownian model for recurrent earthquakes
 Bull. Seism. Soc. Am
, 2002
"... Abstract We construct a probability model for rupture times on a recurrent earthquake source. Adding Brownian perturbations to steady tectonic loading produces a stochastic loadstate process. Rupture is assumed to occur when this process reaches a criticalfailure threshold. An earthquake relaxes t ..."
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Cited by 9 (1 self)
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Abstract We construct a probability model for rupture times on a recurrent earthquake source. Adding Brownian perturbations to steady tectonic loading produces a stochastic loadstate process. Rupture is assumed to occur when this process reaches a criticalfailure threshold. An earthquake relaxes the load state to a characteristic ground level and begins a new failure cycle. The loadstate process is a Brownian relaxation oscillator. Intervals between events have a Brownian passagetime distribution that may serve as a temporal model for timedependent, longterm 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 quasistationary level, in which the conditional probability of an event becomes time independent; and (3) the quasistationary 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 quasistationary 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 stresstransfer 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.
Statistical models of spike trains
 In C. Liang, & G. Lord (Eds.), Stochastic methods in neuroscience
, 2009
"... 1 Spiking neurons make inviting targets for analytical methods based on stochastic processes: spike trains carry information in their temporal patterning, yet they are often highly irregular across time and across experimental replications. The bulk of this volume is devoted to mathematical and biop ..."
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Cited by 5 (3 self)
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1 Spiking neurons make inviting targets for analytical methods based on stochastic processes: spike trains carry information in their temporal patterning, yet they are often highly irregular across time and across experimental replications. The bulk of this volume is devoted to mathematical and biophysical models useful in understanding neurophysiological processes. In this chapter we consider statistical models for analyzing spike train data. Strictly speaking, what we would call a statistical model for spike trains is simply a probabilistic description of the sequence of spikes. But it is somewhat misleading to ignore the dataanalytical context of these models. In particular, we want to make use of these probabilistic tools for the purpose of scientific inference. The leap from simple descriptive uses of probability to inferential applications is worth emphasizing for two reasons. First, this leap was one of the great conceptual advances in science, taking roughly two hundred years. It was not until the late 1700s that there emerged any clear notion of inductive (or what we would now call statistical) reasoning; it was not until the first half of the twentieth century that modern methods began to be developed systematically; and it was only in the second half of the twentieth century that these methods
On a Gibbs characterization of normalized generalized Gamma processes
, 707
"... We show that a Gibbs characterization of normalized generalized Gamma processes, recently obtained in Lijoi, Prünster and Walker (2007), can alternatively be derived by exploiting a characterization of exponentially tilted PoissonKingman models stated in Pitman (2003). We also provide a completion ..."
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Cited by 2 (0 self)
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We show that a Gibbs characterization of normalized generalized Gamma processes, recently obtained in Lijoi, Prünster and Walker (2007), can alternatively be derived by exploiting a characterization of exponentially tilted PoissonKingman models stated in Pitman (2003). We also provide a completion of this result investigating the existence of normalized random measures inducing exchangeable Gibbs partitions of type α ∈ (−∞,0].
Some Properties of the Arc Sine Law Related to Its Invariance Under a Family of Rational Maps
, 1999
"... 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 di ..."
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Cited by 1 (1 self)
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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 9703961 can be constructed in numerous ways as a function of the path of a onedimensional 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...
On the genealogy of a sample of neutral rare alleles
"... 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 coal ..."
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Cited by 1 (1 self)
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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 ddimensional 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.
Inverse Gaussian KDistributions
, 1999
"... Kdistributions 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 ..."
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Kdistributions 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) kdistribution 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 infrared 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 ...
Bayesian Prediction of Waiting Times in Stochastic Models
, 2000
"... 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 ..."
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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 semiMarkov processes whose distributions are indexed by an unknown parameter #. Bayesian prediction for such processes when they are not stationary is also addressed and the inverseGaussian based saddlepoint approximation of Wood et al. (1993) is shown to accurately deal with the nonstationarity whereas the normalbased 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...