Henri Schurz

Numerical solution of SDE through computer experiments

1: Background on Probability and Statistics.- 1.1 Probability and Distributions.- 1.2 Random Number Generators.- 1.3 Moments and Conditional Expectations.- 1.4 Random Sequences.- 1.5 Testing Random Numbers.- 1.6 Markov Chains as Basic Stochastic Processes.- 1.7 Wiener Processes.- 2: Stochastic Differential Equations.- 2.1 Stochastic Integration.- 2.2 Stochastic Differential Equations.- 2.3 Stochastic Taylor Expansions.- 3: Introduction to Discrete Time Approximation.- 3.1 Numerical Methods for Ordinary Differential Equations.- 3.2 A Stochastic Discrete Time Simulation.- 3.3 Pathwise Approximation and Strong Convergence.- 3.4 Approximation of Moments and Weak Convergence.- 3.5 Numerical Stability.- 4: Strong Approximations.- 4.1 Strong Taylor Schemes.- 4.2 Explicit Strong Schemes.- 4.3 Implicit Strong Approximations.- 4.4 Simulation Studies.- 5: Weak Approximations.- 5.1 Weak Taylor Schemes.- 5.2 Explicit Weak Schemes and Extrapolation Methods.- 5.3 Implicit Weak Approximations.- 5.4 Simulation Studies.- 5.5 Variance Reducing Approximations.- 6: Applications.- 6.1 Visualization of Stochastic Dynamics.- 6.2 Testing Parametric Estimators.- 6.3 Filtering.- 6.4 Functional Integrals and Invariant Measures.- 6.5 Stochastic Stability and Bifurcation.- 6.6 Simulation in Finance.- References.- List of PC-Exercises.- Frequently Used Notations.

Balanced Implicit Methods for Stiff Stochastic Systems

This paper introduces some implicitness in stochastic terms of numerical methods for solving stiff stochastic differential equations and especially a class of fully implicit methods, the balanced methods. Their order of strong convergence is proved. Numerical experiments compare the stability properties of these schemes with explicit ones.

ON EFFECTS OF DISCRETIZATION ON ESTIMATORS OF DRIFT PARAMETERS FOR DIFFUSION PROCESSES

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.

The Invariance of Asymptotic Laws of Linear Stochastic Systems under Discretization

The stochastic trapezoidal rule provides the only equidistant discretization scheme from the family of implicit Euler methods (see [12]) which possesses the same asymptotic (stationary) law as underlying continuous time, linear and autonomous stochastic systems with white or coloured noise. This identity holds even when integration time goes to infinity, independent of used integration step size ! Especially, the asymptotic behaviour of first two moments of corresponding probability distributions is rigorously examined and compared in this paper. The coincidence of asymptotic moments is shown for autonomous systems with multiplicative (parametric) and additive noise using fixed point principles and the theory of positive operators. The key result turns out to be useful for adequate implementation of stochastic algorithms applied to numerical solution of autonomous stochastic differential equations. In particular, it has practical importance when accurate long time integration is required such as in the process of estimation of Lyapunov exponents or stationary measures for oscillators in mechanical engineering.

Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise

Several results concerning asymptotical mean square stability of equilibria of specific linear stochastic systems are presented and proven. These discrete time systems can be interpreted as numerical solution of stochastic differential equations driven by Wiener noise. Effects of the presented mean square calculus are shown by the Kubo oscillator perturbed by white noise and a simplified system of noisy Brusselator equations

THE NUMERICAL SOLUTION OF NONLINEAR STOCHASTIC DYNAMICAL SYSTEMS: A BRIEF INTRODUCTION

The numerical analysis of stochastic differential equations, currently undergoing rapid development, differs significantly from its deterministic counterpart due to the peculiarities of stochastic calculus. This article presents a brief, pedagogical introduction to the subject from the perspective of stochastic dynamical systems. The key tool is the stochastic Taylor expansion. Strong, pathwise approximations are distinguished from weak, functional approximations, and their role in stability with Lyapunov exponents and stiffness is discussed.

Preservation of probabilistic laws through Euler methods for ornstein-uhlenbeck process

There is a lack of appropriate replication of the asymptotical behaviour of stationary stochastic differential equations solved by numerical methods. The paper illustrates this fact with the stationary Ornstein-Uhlenbeck process and family of implicit Euler methods. For description of occuring bias, notions of asymptotical p-th. mean, mean, mean square and equilibrium preservation are introduced, due to stochasticity of stationary law. Only the trapezoidal formula among these methods is optimal in the sense of replication of exact asymptotical behaviour. We also discuss the general probabilistic law of linear Euler methods. The results can be useful for implementation of stochastic-numerical algorithms (e.g. for linear-implicit methods) in several disciplines of Natural and Environmental Sciences

Parameter Optimization of relaxed Ordered Subsets Pre-computed Back Projection (BP) based Penalized-Likelihood (OS-PPL) Reconstruction in Limited-angle X-ray Tomography

This paper presents a two-step strategy to provide a quality-predictable image reconstruction. A Pre-computed Back Projection based Penalized-Likelihood (PPL) method is proposed in the strategy to generate consistent image quality. To solve PPL efficiently, relaxed Ordered Subsets (OS) is applied. A training sets based evaluation is performed to quantify the effect of the undetermined parameters in OS, which lets the results as consistent as possible with the theoretical one.

Global asymptotic stability of solutions of cubic stochastic difference equations

Global almost sure asymptotic stability of solutions of some nonlinear stochastic difference equations with cubic-type main part in their drift and diffusive part driven by square-integrable martingale differences is proven under appropriate conditions in ℝ1. As an application of this result, the asymptotic stability of stochastic numerical methods, such as partially drift-implicit θ-methods with variable step sizes for ordinary stochastic differential equations driven by standard Wiener processes, is discussed.

Higher Order Approximate Markov Chain Filters

The aim of this paper is to construct higher order approximate discrete time filters for continuous time finite-state Markov chains with observations that are perturbed by the noise of a Wiener process.