In this paper we develop a constructive approach to studying continuously and discretely sampled functionals of Lévy processes. Estimates for the rate of convergence of the discretely sampled functionals to the continuously sampled functionals are derived, reducing the study of the latter to that of the former. Laguerre reduction series for the discretely sampled functionals are developed, reducing their study to that of the moment generating function of the pertinent Lévy processes and to that of the moments of these processes in particular. The results are applied to questions of contingent claim valuation, such as the explicit valuation of Asian options, and illustrated in the case of generalized inverse Gaussian Lévy processes.

In this paper we develop a constructive structure theory for a class of exponential functionals of Brownian motion which includes Asian option values. This is done in two stages of differing natures. As a first step, the functionals are represented as Laguerre reduction series obtained from main results of Schröder (2006), this paper's companion paper. These reduction series are new and given in terms of the negative moments of the integral of geometric Brownian motion, whose structure theory is developed in a second step. Providing a new angle on these processes, this is done by establishing connections with theta functions. Integral representations and computable formulae for the negative moments are thus derived and then shown to furnish highly efficient ways for computing the negative moments. Application of this paper's Laguerre reduction series in numerical examples suggests that one of the most efficient methods for the explicit valuation of Asian options is obtained. The paper also provides mathematical background results referred to in Schröder (2005c).

In this paper we develop methods for reducing the study, the computation, and the construction of stochastic functionals to those of standard concepts such as the moments of the pertinent random variables. Principally, our methods are based on the notion of ladder height densities and their Laguerre expansions, and our results provide a unifying framework for the distinct approaches of Dufresne (2000) and Schröder (2005).

The estimation of P(Sn>u) by simulation, where Sn is the sum of independent, identically distributed random varibles Y1,…,Yn, is of importance in many applications. We propose two simulation estimators based upon the identity P(Sn>u)=nP(Sn>u, Mn=Yn), where Mn=max(Y1,…,Yn). One estimator uses importance sampling (for Yn only), and the other uses conditional Monte Carlo conditioning upon Y1,…,Yn−1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.

Particle filters are Monte Carlo methods that aim to approximate the optimal filter of a partially observed Markov chain. In this paper, we study the case in which the transition kernel of the Markov chain depends on unknown parameters: we construct a particle filter for the simultaneous estimation of the parameter and the partially observed Markov chain (adaptive estimation) and we prove the convergence of this filter to the correct optimal filter, as time and the number of particles go to infinity. The filter presented here generalizes Del Moral's Monte Carlo particle filter.

The titular question is investigated for fairly general semimartingale investment and asset price processes. A discrete-time consideration suggests a stochastic differential equation and an integral expression for the time value in the continuous-time framework. It is shown that the two are equivalent if the jump part of the price process converges. The integral expression, which is the answer to the titular question, is the sum of all investments accumulated with returns on the asset (a stochastic integral) plus a term that accounts for the possible covariation between the two processes. The arbitrage-free price of the time value is the expected value of the sum (i.e. integral) of all investments discounted with the locally risk-free asset.

Let {νε, ε>0} be a family of probabilities for which the decay is governed by a large deviation principle, and consider the simulation of νε0(A) for some fixed measurable set A and some ε0>0. We investigate the circumstances under which an exponentially twisted importance sampling distribution yields an asymptotically efficient estimator. Varadhan's lemma yields necessary and sufficient conditions, and these are shown to improve on certain conditions of Sadowsky. This is illustrated by an example to which Sadowsky's conditions do not apply, yet for which an efficient twist exists.

Consider a continuous-time Markov chain evolving in a random environment. We study certain forms of interaction between the process of interest and the environmental process, under which the stationary joint distribution is tractable. Moreover, we obtain necessary and sufficient conditions for a product-form stationary distribution. A number of examples that illustrate the applicability of our results in queueing and population growth models are also included.

We give explicit upper bounds for convergence rates when approximating both one- and two-sided general curvilinear boundary crossing probabilities for the Wiener process by similar probabilities for close boundaries of simpler form, for which computation of the boundary crossing probabilities is feasible. In particular, we partially generalize and improve results obtained by Pötzelberger and Wang in the case when the approximating boundaries are piecewise linear. Applications to barrier option pricing are also discussed.

The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

We study numerical integration based on Markov chains. Our focus is on establishing error bounds uniformly on classes of integrands. Since in general state space the concept of uniform ergodicity is too restrictive to cover important cases, we analyze the error of V-uniformly ergodic Markov chains. We place emphasis on the interplay between ergodicity properties of the transition kernel, the initial distributions and the classes of integrands. Our analysis is based on arguments from interpolation theory.

One approach to the computation of the price of an Asian option involves the Hartman–Watson distribution. However, numerical problems for its density occur for small values. This motivates the asymptotic study of its distribution function.

We introduce adaptive-simulation schemes for estimating performance measures for stochastic systems based on the method of control variates. We consider several possible methods for adaptively tuning the control-variate estimators, and describe their asymptotic properties. Under certain assumptions, including the existence of a ‘perfect control variate’, all of the estimators considered converge faster than the canonical rate of n −1/2, where n is the simulation run length. Perfect control variates for a variety of stochastic processes can be constructed from ‘approximating martingales’. We prove a central limit theorem for an adaptive estimator that converges at rate A similar estimator converges at rate n −1. An exponential rate of convergence is also possible under suitable conditions.

Discrete time-series models are commonly used to represent economic and physical data. In decision making and system control, the first-passage time and level-crossing probabilities of these processes against certain threshold levels are important quantities. In this paper, we apply an integral-equation approach together with the state-space representations of time-series models to evaluate level-crossing probabilities for the AR(p) and ARMA(1,1) models and the mean first passage time for AR(p) processes. We also extend Novikov's martingale approach to ARMA(p,q) processes. Numerical schemes are used to solve the integral equations for specific examples.

Recently, several authors have studied the transient and the equilibrium behaviour of stochastic population processes with total catastrophes. These models are reasonable for modelling populations that are exposed to extreme disastrous phenomena. However, under mild disastrous conditions, the appropriate model is a stochastic process subject to binomial catastrophes. In the present paper we consider a special such model in which a population evolves according to a compound Poisson process and catastrophes occur according to a renewal process. Every individual of the population survives after a catastrophe with probability p, independently of anything else, i.e. the population size is reduced according to a binomial distribution. We study the equilibrium distribution of this process and we derive an algorithmic procedure for its approximate computation. Bounds on the error of this approximation are also included.

In this paper, we study Markov fluid queues where the net fluid rate to a single-buffer system varies with respect to the state of an underlying continuous-time Markov chain. We present a novel algorithmic approach to solve numerically for the steady-state solution of such queues. Using this approach, both infinite- and finite-buffer cases are studied. We show that the solution of the infinite-buffer case is reduced to the solution of a generalized spectral divide-and-conquer (SDC) problem applied on a certain matrix pencil. Moreover, this SDC problem does not require the individual computation of any eigenvalues and eigenvectors. Via the solution for the SDC problem, a matrix-exponential representation for the steady-state queue-length distribution is obtained. The finite-buffer case, on the other hand, requires a similar but different decomposition, the so-called additive decomposition (AD). Using the AD, we obtain a modified matrix-exponential representation for the steady-state queue-length distribution. The proposed approach for the finite-buffer case is shown not to have the numerical stability problems reported in the literature.

We consider minimum relative entropy calibration of a given prior distribution to a finite set of moment constraints. We show that the calibration algorithm is stable (in the Prokhorov metric) under a perturbation of the prior and the calibrated distributions converge in variation to the measure from which the moments have been taken as more constraints are added. These facts are used to explain the limiting properties of the minimum relative entropy Monte Carlo calibration algorithm.

We address the problem of tracking the time-varying linear subspaces (of a larger system) under a Bayesian framework. Variations in subspaces are treated as a piecewise-geodesic process on a complex Grassmann manifold and a Markov prior is imposed on it. This prior model, together with an observation model, gives rise to a hidden Markov model on a Grassmann manifold, and admits Bayesian inferences. A sequential Monte Carlo method is used for sampling from the time-varying posterior and the samples are used to estimate the underlying process. Simulation results are presented for principal subspace tracking in array signal processing.

This paper introduces a method of generating real harmonizable multifractional Lévy motions (RHMLMs). The simulation of these fields is closely related to that of infinitely divisible laws or Lévy processes. In the case where the control measure of the RHMLM is finite, generalized shot-noise series are used. An estimation of the error is also given. Otherwise, the RHMLM Xh is split into two independent RHMLMs, Xε,1 and Xε,2. More precisely, Xε,2 is an RHMLM whose control measure is finite. It can then be rewritten as a generalized shot-noise series. The asymptotic behaviour of Xε,1 as ε → 0+ is further elaborated. Sufficient conditions to approximate Xε,1 by a multifractional Brownian motion are given. The error rate in terms of Berry-Esseen bounds is then discussed. Finally, some examples of simulation are given.

We improve on previous finite time estimates for the simulated annealing algorithm which were obtained from a Cheeger-like approach. Our approach is based on a Poincaré inequality.