This article provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. This analysis has been motivated by a large and growing use of matrix-valued affine processes in finance, including multi-asset option pricing with stochastic volatility and correlation structures, and fixed-income models with stochastically correlated risk factors and default intensities.

Let (W, W') be an exchangeable pair. Assume that

E(W − W'|W) = g(W) + r(W),

where g(W) is a dominated term and r(W) is negligible. Let G(t) = ∫₀^tg(s) ds and define p(t) = c₁e^−c₀G(t), where c₀ is a properly chosen constant and c₁ = 1 / ∫_−∞^∞e^−c₀G(t) dt. Let Y be a random variable with the probability density function p. It is proved that W converges to Y in distribution when the conditional second moment of (W − W') given W satisfies a law of large numbers. A Berry–Esseen type bound is also given. We use this technique to obtain a Berry–Esseen error bound of order $1/\sqrt{n}$ in the noncentral limit theorem for the magnetization in the Curie–Weiss ferromagnet at the critical temperature. Exponential approximation with application to the spectrum of the Bernoulli–Laplace Markov chain is also discussed.

This paper presents a heavy-traffic analysis of the behavior of a single-server queue under an Earliest-Deadline-First (EDF) scheduling policy in which customers have deadlines and are served only until their deadlines elapse. The performance of the system is measured by the fraction of reneged work (the residual work lost due to elapsed deadlines) which is shown to be minimized by the EDF policy. The evolution of the lead time distribution of customers in queue is described by a measure-valued process. The heavy traffic limit of this (properly scaled) process is shown to be a deterministic function of the limit of the scaled workload process which, in turn, is identified to be a doubly reflected Brownian motion. This paper complements previous work by Doytchinov, Lehoczky and Shreve on the EDF discipline in which customers are served to completion even after their deadlines elapse. The fraction of reneged work in a heavily loaded system and the fraction of late work in the corresponding system without reneging are compared using explicit formulas based on the heavy traffic approximations. The formulas are validated by simulation results.

This paper derives the asymptotic behavior of realized power variation of pure-jump Itô semimartingales as the sampling frequency within a fixed interval increases to infinity. We prove convergence in probability and an associated central limit theorem for the realized power variation as a function of its power. We apply the limit theorems to propose an efficient adaptive estimator for the activity of discretely-sampled Itô semimartingale over a fixed interval.

We investigate the Longstaff–Schwartz algorithm for American option pricing assuming that both the number of regressors and the number of Monte Carlo paths tend to infinity. Our main results concern extensions, respectively, applications of results by Glasserman and Yu [Ann. Appl. Probab. 14 (2004) 2090–2119] and Stentoft [Manag. Sci. 50 (2004) 1193–1203] to several Lévy models, in particular the geometric Meixner model. A convenient setting to analyze this convergence problem is provided by the Lévy–Sheffer systems introduced by Schoutens and Teugels.

We study Atlas-type models of equity markets with local characteristics that depend on both name and rank, and in ways that induce a stable capital distribution. Ergodic properties and rankings of processes are examined with reference to the theory of reflected Brownian motions in polyhedral domains. In the context of such models we discuss properties of various investment strategies, including the so-called growth-optimal and universal portfolios.

Two of the most popular approximations for the distribution of the steady-state waiting time, W_∞, of the M/G/1 queue are the so-called heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the traffic intensity, ρ, is close to 1 and the processing times have finite variance, the heavy-traffic approximation states that the distribution of W_∞ is roughly exponential at scale O((1 − ρ)⁻¹), while the heavy tailed asymptotic describes power law decay in the tail of the distribution of W_∞ for a fixed traffic intensity. In this paper, we assume a regularly varying processing time distribution and obtain a sharp threshold in terms of the tail value, or equivalently in terms of (1 − ρ), that describes the point at which the tail behavior transitions from the heavy-traffic regime to the heavy-tailed asymptotic. We also provide new approximations that are either uniform in the traffic intensity, or uniform on the positive axis, that avoid the need to use different expressions on the two regions defined by the threshold.

A series of recent articles introduced a method to construct stochastic partial differential equations (SPDEs) which are invariant with respect to the distribution of a given conditioned diffusion. These works are restricted to the case of elliptic diffusions where the drift has a gradient structure and the resulting SPDE is of second-order parabolic type.

The present article extends this methodology to allow the construction of SPDEs which are invariant with respect to the distribution of a class of hypoelliptic diffusion processes, subject to a bridge conditioning, leading to SPDEs which are of fourth-order parabolic type. This allows the treatment of more realistic physical models, for example, one can use the resulting SPDE to study transitions between meta-stable states in mechanical systems with friction and noise. In this situation the restriction of the drift being a gradient can also be lifted.

The goal of cancer genome sequencing projects is to determine the genetic alterations that cause common cancers. Many malignancies arise during the clonal expansion of a benign tumor which motivates the study of recurrent selective sweeps in an exponentially growing population. To better understand this process, Beerenwinkel et al. [PLoS Comput. Biol. 3 (2007) 2239–2246] consider a Wright–Fisher model in which cells from an exponentially growing population accumulate advantageous mutations. Simulations show a traveling wave in which the time of the first k-fold mutant, T_k, is approximately linear in k and heuristics are used to obtain formulas for ET_k. Here, we consider the analogous problem for the Moran model and prove that as the mutation rate μ → 0, T_k ∼ c_k log(1 / μ), where the c_k can be computed explicitly. In addition, we derive a limiting result on a log scale for the size of X_k(t) = the number of cells with k mutations at time t.

We consider a processor sharing queue where the number of jobs served at any time is limited to K, with the excess jobs waiting in a buffer. We use random counting measures on the positive axis to model this system. The limit of this measure-valued process is obtained under diffusion scaling and heavy traffic conditions. As a consequence, the limit of the system size process is proved to be a piece-wise reflected Brownian motion.