Higher-order numerical methods for stochastic simulation of\ud chemical reaction systems

In this paper, using the framework of extrapolation, we present an approach for obtaining higher-order -leap methods for the Monte Carlo simulation of stochastic chemical kinetics. Specifically, Richardson extrapolation is applied to the expectations of functionals obtained by a fixed-step -leap algorithm. We prove that this procedure gives rise to second-order approximations for the first two moments obtained by the chemical master equation for zeroth- and first-order chemical systems. Numerical simulations verify that this is also the case for higher-order chemical systems of biological importance. This approach, as in the case of ordinary and stochastic differential equations, can be repeated to obtain even higher-order approximations. We illustrate the results of a second extrapolation on two systems. The biggest barrier for observing higher-order convergence is the Monte Carlo error; we discuss different strategies for reducing it

Parametric and non-parametric gradient matching for network inference:a comparison

Abstract Background Reverse engineering of gene regulatory networks from time series gene-expression data is a challenging problem, not only because of the vast sets of candidate interactions but also due to the stochastic nature of gene expression. We limit our analysis to nonlinear differential equation based inference methods. In order to avoid the computational cost of large-scale simulations, a two-step Gaussian process interpolation based gradient matching approach has been proposed to solve differential equations approximately. Results We apply a gradient matching inference approach to a large number of candidate models, including parametric differential equations or their corresponding non-parametric representations, we evaluate the network inference performance under various settings for different inference objectives. We use model averaging, based on the Bayesian Information Criterion (BIC), to combine the different inferences. The performance of different inference approaches is evaluated using area under the precision-recall curves. Conclusions We found that parametric methods can provide comparable, and often improved inference compared to non-parametric methods; the latter, however, require no kinetic information and are computationally more efficient

Sequential Change-Point Detection for Diffusion Processes

In this work the problem of sequential detection of changes in the drift parameter of diffusion processes is considered under the assumption that the processes can be observed continuously. A corresponding monitoring procedure is suggested and its asymptotic behaviour under the null hypothesis as well as under the alternative is investigated. The proposed procedure is similar to the CUSUM one for discrete-time processes. For constructing the test statistic, the one-step method of Le Cam is applied. In order to prove limit theorems in change-point analysis, typically strong approximations by Gaussian processes are the key tools. Two main results of the thesis are the strong invariance principles (with rate) for certain stochastic integrals and for the estimator process. Based on these approximations, two methods of proof are developed for the weak convergence of the test statistic under the null hypothesis. Moreover, the asymptotic normality of the stopping time under the alternative is proven. The thesis is completed by studying some examples of stochastic differential equations which can be treated by the presented methodology

Monte-Carlo methods for backward stochastic differential equations : segment-wise dynamic programming and fast rates for lower bounds

In this thesis, we study two different algorithms using Monte-Carlo methods for solving backward stochastic differential equations. In the first chapter, we present a new algorithm where the backward stochastic differential equation is discretized to a dynamic programming equation alternating between a multi-step forward approach on segments of the time grid and a one-step scheme between segments. Conditional expectations are computed via least squares regression on function spaces. We optimize the length of the segments in dependence on the dimension and smoothness of the backward stochastic differential equation and compute the complexity needed to achieve a desired accuracy in the limit as the number of time points in the discretization goes to infinity. In the second chapter, we consider a discretized backward stochastic differential equation in form of a dynamic programming equation and study an algorithm for constructing lower bounds for its value at time zero. The algorithm uses a pre-computed approximate solution of this equation to sample a control process which is used to derive the lower bound for the solution. We derive asymptotic error bounds and compute the complexity required to achieve a desired accuracy in dependence on the input approximation. The results of both algorithms are illustrated by numerical examples.Diese Arbeit beschäftigt sich mit zwei Algorithmen zur Lösung von rückwärtsstochastischen Differentialgleichungen mit Hilfe von Monte-Carlo Methoden. Im ersten Kapitel wird ein neuer Algorithmus vorgestellt, der auf einem Diskretisierungsverfahren beruht, welches zwischen einer Mehr-Schritt Darstellung auf Zeitsegmenten und einem Ein-Schritt Verfahren zwischen den Segmenten alterniert. Auftretende bedingte Erwartungswerte werden dabei als Projektionen auf endlich dimensionale Funktionenräume berechnet. Die Wahl der Segmentlänge wird in Abhängigkeit der Glattheit und der Dimension der Differentialgleichung optimiert und es wird der asymptotische Rechenaufwand ermittelt, welcher notwendig ist um eine vorgegebene Genauigkeit zu erzielen. Im zweiten Kapitel wird ein Algorithmus zur Konstruktion von unteren Schranken der Lösung rückwärtsstochastischer Differentialgleichungen zum Startzeitpunkt untersucht. Hierfür wird mit Hilfe einer vorab berechneten Approximation der Lösung ein Kontrollprozess simuliert mit dessen Hilfe schließlich die Schranke berechnet wird. Es werden asymptotische Fehlerschranken sowie der erforderliche Rechenaufwand zur Erzielung einer vorgegebenen Genauigkeit hergeleitet. Die theoretischen Ergebnisse bezüglich der beiden Algorithmen werden mit numerischen Beispielen illustriert