Stochastic Taylor Expansion Research Articles

Higher order numerical schemes for affinely controlled nonlinear systems

A systematic method for the derivation of high order schemes for affinely controlled nonlinear systems is developed. Using an adaptation of the stochastic Taylor expansion for control systems we construct Taylor schemes of arbitrary high order and indicate how derivative free Runge-Kutta type schemes can be obtained. Furthermore an approximation technique for the multiple control integrals appearing in the schemes is proposed.

Sharp Laplace Asymptotics For a Parabolic SPDE

In this note we prove a precise asymptotic estimate for Laplace type functionals for a parabolic SPDE. We use a large deviation principle, the stochastic Taylor expansion, some exponential inequalities and support theorems for our stochastic partial differential equation

Weak Approximations and Extrapolations of Stochastic Differential Equations with Jumps

Numerical discretization schemes are developed to approximate functionals of stochastic differential equations with jumps, and the convergence is shown to have an appropriate order. For the Euler scheme and the second order weak scheme, the leading coefficient of their global errors are determined by the stochastic Taylor expansion. Based on the error expression, the extrapolation technique can be applied to get a higher order convergence. Numerical examples are provided to compare various weak schemes and extrapolations.

Almost sure convergence of stochastic taylor expansions for functions of real-valued two-parameter continuous brownian semimartingales

We show the convergence (almost sure and in quadratic mean) of the Taylor expansion

Approximation of Multiple Stochastic Integrals and Its Application to Stochastic Differential Equations

As multiple stochastic integrals are not very easy to simulate, we would like to treat them as solutions of systems of stochastic differential equations and solve them successively and recursively approximated by the stochastic Taylor expansion as a Chen series in terms of a Philip Hall basis or Lyndon basis. We can save sufficient values of multiple stochastic integrals with independent sample paths in a look-up table for future use. The table can be used to implement high order schemes to solve stochastic differential equations numerically. A numerical example will be shown to illustrate the efficiency.

Strong convergence of stochastic taylor expansions of two-parameter random fields

(1997). Strong convergence of stochastic taylor expansions of two-parameter random fields. Stochastic Analysis and Applications: Vol. 15, No. 2, pp. 137-149.

Multiple stochastic integrals appearing in the stochastic Taylor expansions

Multiple stochastic integrals appearing in the stochastic Taylor expansions

Higher-order implicit strong numerical schemes for stochastic differential equations

Higher-order implicit numerical methods which are suitable for stiff stochastic differential equations are proposed. These are based on a stochastic Taylor expansion and converge strongly to the corresponding solution of the stochastic differential equation as the time step size converges to zero. The regions of absolute stability of these implicit and related explicit methods are also examined.

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.

Numerical methods for stochastic differential equations

The numerical analysis of stochastic differential equations differs significantly from that of ordinary differential equations due to the peculiarities of stochastic calculus. This book provides an introduction to stochastic calculus and stochastic differential equations, both theory and applications. The main emphasise is placed on the numerical methods needed to solve such equations. It assumes an undergraduate background in mathematical methods typical of engineers and physicists, through many chapters begin with a descriptive summary which may be accessible to others who only require numerical recipes. To help the reader develop an intuitive understanding of the underlying mathematicals and hand-on numerical skills exercises and over 100 PC Exercises (PC-personal computer) are included. The stochastic Taylor expansion provides the key tool for the systematic derivation and investigation of discrete time numerical methods for stochastic differential equations. The book presents many new results on higher order methods for strong sample path approximations and for weak functional approximations, including implicit, predictor-corrector, extrapolation and variance-reduction methods. Besides serving as a basic text on such methods. the book offers the reader ready access to a large number of potential research problems in a field that is just beginning to expand rapidly and is widely applicable.

Relations between multiple ito and stratonovich integrals

In this paper we investigate relations within and between sets of multiple Ito and Stratonovich integrals which are useful in the application of stochastic Taylor expansions of Ito processes. We obtain formulae expressing multiple Ito integrals in terms of multiple Stratonovich integrals and vice versa and also formulae relating multiple stochastic integrals of the same type

Stratonovich and Ito Stochastic Taylor Expansions

AbstractThe Stratonovich stochastic Taylor formula for diffusion processes is stated and proved. It has a simpler structure and is a more natural generalization of the deterministic Taylor formula than the Ito stochastic Taylor formula.

Orbital dynamics in a stochastic atmosphere

An algorithm is developed to analyze the orbital dynamics of a near-Earth satellite subjected to the random perturbations of a atmosphere. The orbital equations include the perturbation caused by an oblate geopotential and a drag model where the atmospheric density varies exponentially with height and temporally with the solar-activity and semiannual cycles. A random process representative of short-term density fluctuations modulates this deterministic density function. The solution begins with the analytical averaging of both the gravity and the drag perturbations, following the principle of the Method of Averaging. The drag averages are expanded in a series of newly defined functions called the two-dimensional hyperbolic Bessel functions/' Their nice analytical properties greatly simplify the next part of the solution where the equations of motion are further transformed into a stochastic Taylor expansion.'' Unlike the standard linear approach used in the treatment of systems, this technique preserves the full nonlinearity of the equations of deterministic motion and, therefore, avoids the usual restriction to small excursions from an initial or reference state. The analytical nature of the drag averages provides insight into the orbital effects of random density fluctuations and results in an efficient algorithm that prescribes the evolution of the mean and variance of the orbital state. Based on this algorithm, a computer program has been developed to generate confidence intervals that bound, in a probabilistic sense, the trajectory of a satellite. Typical numerical results assuming a 300-km-high near-circular orbit are illustrated. Although orbital error analysis is the main application of the paper, its analytical results can also be used in the development of an orbit determination and density estimation algorithm.