We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

During the last decade, a big progress has been achieved in the analysis and numerical treatment of Initial Value Problems (IVPs) in Differential Algebraic Equations (DAEs) and Ordinary Differential Equations (ODEs). In spite of the rich variety of results available in the literature, there are still many specific problems that require special attention. Two of such, which are considered in this work, are the optimization of order of accuracy and reduction of cost of functional evaluations of Explicit Runge - Kutta (ERK) methods.

Traditionally, the maximum attainable order p of an s-stage ERK method for advancing the solution of an IVP is such that
p(s)
 4
 
In 1999, Goeken presented an s-stage ERK Method of order p(s)=s +1,s>2.
However, this work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher 
U n and Jonhson [94]; and the classical ERK methods. The order p of 
the new scheme called Multiderivative Explicit Runge-Kutta (MERK) Methods is such that p(s) 2. The stability, convergence and implementation for the optimization of IVPs in DAEs and ODEs systems are also considered.

Numerical solution of initial value problems in differential - algebraic equations

Part I. Deterministic Differential Equations: 1. Linear analysis 2. Galerkin approximation and finite elements 3. Time-dependent differential equations Part II. Stochastic Processes and Random Fields: 4. Probability theory 5. Stochastic processes 6. Stationary Gaussian processes 7. Random fields Part III. Stochastic Differential Equations: 8. Stochastic ordinary differential equations (SODEs) 9. Elliptic PDEs with random data 10. Semilinear stochastic PDEs.

/pdf/an-introduction-to-computational-stochastic-pdes-74xqtzrc5j.pdf

An Introduction to Computational Stochastic PDEs

This article reviews progress and challenges in model driven EEG/fMRI fusion with a focus on brain oscillations. Fusion is the combination of both imaging modalities based on a cascade of forward models from ensemble of post-synaptic potentials (ePSP) to net primary current densities (nPCD) to EEG; and from ePSP to vasomotor feed forward signal (VFFSS) to BOLD. In absence of a model, data driven fusion creates maps of correlations between EEG and BOLD or between estimates of nPCD and VFFS. A consistent finding has been that of positive correlations between EEG alpha power and BOLD in both frontal cortices and thalamus and of negative ones for the occipital region. For model driven fusion we formulate a neural mass EEG/fMRI model coupled to a metabolic hemodynamic model. For exploratory simulations we show that the Local Linearization (LL) method for integrating stochastic differential equations is appropriate for highly nonlinear dynamics. It has been successfully applied to small and medium sized networks, reproducing the described EEG/BOLD correlations. A new LL-algebraic method allows simulations with hundreds of thousands of neural populations, with connectivities and conduction delays estimated from diffusion weighted MRI. For parameter and state estimation, Kalman filtering combined with the LL method estimates the innovations or prediction errors. From these the likelihood of models given data are obtained. The LL-innovation estimation method has been already applied to small and medium scale models. With improved Bayesian computations the practical estimation of very large scale EEG/fMRI models shall soon be possible.

/pdf/model-driven-eeg-fmri-fusion-of-brain-oscillations-220y6ok0wu.pdf

Model driven EEG/fMRI fusion of brain oscillations

Stochastic differential algebraic equations of index 1 and applications in circuit simulation

In this article the numerical approximation of solutions of Ito stochastic differential equations is considered, in particular for equations with a small parameter $\epsilon$ in the noise coefficient We construct stochastic linear multistep methods and develop the fundamental numerical analysis concerning their mean-square consistency, numerical stability in the mean-square sense and mean-square convergence For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency Further, for the small noise case we obtain expansions of the local error in terms of the step size and the small parameter $\epsilon$ Simulation results using several explicit and implicit stochastic linear $k$-step schemes, $k=1,\;2$, illustrate the theoretical findings

/pdf/multistep-methods-for-sdes-and-their-application-to-problems-5doetky5b9.pdf

Multistep methods for SDEs and their application to problems with small noise

A strategy for controlling the stepsize in the numerical integration of stochastic differential equations (SDEs) is presented. It is based on estimating the pth mean of local errors. The strategy leads to stepsize sequences that are identical for all computed paths. For the family of Euler schemes for SDEs with small noise, we derive computable estimates for the dominating term of the pth mean of local errors and show that the strategy becomes efficient for reasonable stepsizes. Numerical experience is reported for test examples including scalar SDEs and a stochastic circuit model.

/pdf/stepsize-control-for-mean-square-numerical-methods-for-4zfxz2zpe7.pdf

Stepsize Control for Mean-Square Numerical Methods for Stochastic Differential Equations with Small Noise

How Floquet Theory Applies to Index 1 Differential Algebraic Equations

We deal with linear multi-step methods for SDEs and study when the numerical approximation shares asymptotic properties in the mean-square sense of the exact solution. As in deterministic numerical analysis we use a linear time-invariant test equation and perform a linear stability analysis. Standard approaches used either to analyse deterministic multi-step methods or stochastic one-step methods do not carry over to stochastic multi-step schemes. In order to obtain sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods we construct and apply Lyapunov-type functionals. In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams–Bashforth- and Adams–Moulton-methods, the Milne–Simpson method and the BDF method.

/pdf/asymptotic-mean-square-stability-of-two-step-methods-for-1poshs4irh.pdf

Renate Winkler

Papers

Stochastic differential algebraic equations of index 1 and applications in circuit simulation

Multistep methods for SDEs and their application to problems with small noise

Stepsize Control for Mean-Square Numerical Methods for Stochastic Differential Equations with Small Noise

How Floquet Theory Applies to Index 1 Differential Algebraic Equations

Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations