Abstract not found

Abstract not found

In many applications, the primary objective of numerical simulation of time-evolving systems is the prediction of macroscopic, or coarse-grained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic approaches have been introduced to extract effective, lower-dimensional, models for the macroscopic dynamics; the starting point is the full, detailed, evolution equations. In many cases the effective low-dimensional dynamics may be stochastic, even when the original starting point is deterministic. This review surveys a number of these new approaches to the problem of extracting effective dynamics, highlighting similarities and differences between them. The importance of model problems for the evaluation of these new approaches is stressed, and a number of model problems are described. When the macroscopic dynamics is stochastic, these model problems are either obtained through a clear separation of time-scales, leading to a stochastic effect of the fast dynamics on the slow dynamics, or by considering high dimensional ordinary differential equations which, when projected onto a low dimensional subspace, exhibit stochastic behaviour through the presence of a broad frequency spectrum. Models whose stochastic microscopic behaviour leads to deterministic macroscopic dynamics are also introduced. The algorithms we overview include SVD-based methods for nonlinear problems, model reduction for linear control systems, optimal prediction techniques, asymptotics-based mode elimination, coarse timestepping methods and transfer-operator based methodologies.

We propose a functional estimation procedure for homogeneous stochastic differential equations based on a discrete sample of observations and with minimal requirements on the data generating process. We show how to identify the drift and diffusion function in situations where one or the other function is considered a nuisance parameter. The asymptotic behavior of the estimators is examined as the observation frequency increases and as the time span lengthens. We prove almost sure consistency and weak convergence to mixtures of normal laws, where the mixing variates depend on the chronological local time of the underlying diffusion process, that is the random time spent by the process in the vicinity of a generic spatial point. The estimation method and asymptotic results apply to both stationary and nonstationary recurrent processes.

Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability.

In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical Black-Scholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. The underlying might be the price of a stock or an index say and a constant volatility corresponds to a fixed standard deviation for the random fluctuations in the returns of the underlying. Modern market phenomena makes it important to analyze the situation when this volatility is not fixed but rather is heterogeneous and varies with time. In previous work, see for instance [5], we considered the situation when the volatility is fast mean reverting. Using a singular perturbation expansion we derived an approximation for option prices. We also provided a calibration method using observed option prices as represented by the so-called term structure of implied volatility. Our analysis of market data, however, shows the need for introducing also a slowly varying factor in the model for the stochastic volatility. The combination of regular and singular perturbations approach that we set forth in this paper deals with this case. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parametrization of the

We describe the dynamical and bifurcational behavior of two mutually inhibitory, leaky, neural units subject to external stimulus, random noise, and ‘priming biases. ’ The model describes a simple forced choice experiment and accounts for varying levels of expectation and control. By projecting the model’s dynamics onto slow manifolds, using judicious linear approximations, and solving for one-dimensional (reduced) probability densities, analytical estimates are developed for reaction time distributions and shown to compare satisfactorily with ‘full ’ numerical data. A sensitivity analysis is performed and the effects of parameters assessed. The predictions are also compared with behavioral data. These results may help correlate low-dimensional models of stochastic neural networks with cognitive test data, and hence assist in parameter choices and model building. 1 Introduction and

The Black-Scholes theory of option pricing has been considered for many years as an important but very approximate zeroth-order description of actual market behavior. We generalize the functional form of the diffusion of these systems and also consider multi-factor models including stochastic volatility. Daily Eurodollar futures prices and implied volatilities are fit to determine exponents of functional behavior of diffusions using methods of global optimization, Adaptive Simulated Annealing (ASA), to generate tight fits across moving time windows of Eurodollar contracts. These short-time fitted distributions are then developed into long-time distributions using a robust non-Monte Carlo path-integral algorithm, PATHINT, to generate prices and derivatives commonly used by option traders.

Abstract. We study the quickest detection problem of a sudden change in the arrival rate of a Poisson process from a known value to an unknown and unobservable value at an unknown and unobservable disorder time. Our objective is to design an alarm time which is adapted to the history of the arrival process and detects the disorder time as soon as possible. In previous solvable versions of the Poisson disorder problem, the arrival rate after the disorder has been assumed a known constant. In reality, however, we may at most have some prior information on the likely values of the new arrival rate before the disorder actually happens, and insufficient estimates of the new rate after the disorder happens. Consequently, we assume in this paper that the new arrival rate after the disorder is a random variable. The detection problem is shown to admit a finite-dimensional Markovian sufficient statis-tic if the new rate has a discrete distribution with finitely-many atoms. Furthermore, the detection problem is cast as a discounted optimal stopping problem with running cost for a finite-dimensional piecewise-deterministic Markov process. This optimal stopping problem is studied in detail in the special case where the new

The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plug-and-play property. Our work builds on recently developed plug-and-play inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae. 1. Introduction. A