Abstract.A practical and accessible introduction to numerical methods for stochastic differential equations is given. The reader is assumed to be familiar with Euler's method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable; however, no knowledge of advanced probability theory or stochastic processes is assumed. The article is built around 10 MATLAB programs, and the topics covered include stochastic integration, the Euler-Maruyama method, Milstein's method, strong and weak convergence, linear stability, and the stochastic chain rule.

The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces. Application of these Markov chain results leads to straightforward proofs of ergodicity for a variety of SDEs, in particular for problems with degenerate noise and for problems with locally Lipschitz vector fields. The key points which need to be verified are the existence of a Lyapunov function inducing returns to a compact set, a uniformly reachable point from within that set, and some smoothness of the probability densities; the last two points imply a minorization condition. Together the minorization condition and Lyapunov structure give geometric ergodicity. Applications include the Langevin equation, the Lorenz equation with degenerate noise and gradient systems. The ergodic theorems proved are strong, yielding exponential convergence of expectations for classes of measurable functions restricted only by the condition that they grow no faster than the Lyapunov function. The same Markov chain theory is then used to study time-discrete approximations of these SDEs. It is shown that the minorization condition is robust under approximation. For globally Lipschitz vector fields this is also true of the Lyapunov condition. However in the locally Lipschitz case the Lyapunov condition fails for explicit methods such as Euler-Maruyama; it is, in general, only inherited by specially constructed implicit discretizations. Examples of such discretization based on backward Euler methods are given, and approximation of the Langevin equation studied in some detail.

A fundamental issue in neuroscience is the relation between structure and function. However, gross landmarks do not correspond well to microstructural borders and cytoarchitecture cannot be visualized in a living brain used for functional studies. Here, we used diffusion-weighted and functional MRI to test structurefunction relations directly. Distinct neocortical regions were defined as volumes having similar connectivity profiles and borders identified where connectivity changed. Without using prior information, we found an abrupt profile change where the border between supplementary motor area (SMA) and pre-SMA is expected. Consistent with this anatomical assignment, putative SMA and pre-SMA connected to motor and prefrontal regions, respectively. Excellent spatial correlations were found between volumes defined by using connectivity alone and volumes activated during tasks designed to involve SMA or pre-SMA selectively. This finding demonstrates a strong relationship between structure and function in medial frontal cortex and offers a strategy for testing such correspondences elsewhere in the brain.S ince early attempts to parcellate human and nonhuman cortex into structurally distinct subdivisions, the hypothesis that structural borders correspond to functional borders has been widely held (1-3). However, this hypothesis has been tested only rarely. Structural features such as sulci and gyri are commonly used to define anatomical regions in functional imaging, neurophysiology, and lesion studies, yet they have only a limited correspondence to more fine-grained structural organization such as cytoarchitecture (4-6). Microstructural borders based, for example, on measurements of cyto-, myelo-, or receptor architecture (7-9), can only be defined post mortem, and the methodological demands of such studies preclude investigation of the regional functional specializations in the same animals. Detailed testing of the relationship between these anatomicallybased measures and function based on comparisons between subjects is limited by the apparently substantial interindividual variations in microstructural anatomical boundaries (4-6).A structural feature that has not previously been used to define areal boundaries in the human neocortex is connectivity to other brain regions. Whereas features such as cytoarchitecture, myeloarchitecture, and receptor distributions distinguish the processing capabilities of a region, connectional anatomy constrains the nature of the information available to a region and the influence that it can exert over other regions in a distributed network. Therefore, not only does structural variation reflect functional organization, but local structural organization also determines local functional specialization. Data on brain connectivity in macaque monkeys show that cytoarchitectonically and functionally distinct regions of prefrontal cortex have distinct connectivity ''fingerprints'' (10). Differences in connectivity that parallel differences in cytoarchitecture have been used to define su...

Abstract. Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly relevant. In this work we prove strong convergence results under less restrictive conditions. First, we give a convergence result for Euler-Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p > 2. As an application of this general theory we show that an implicit variant of Euler-Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler-Maruyama approximation to a perturbed SDE of the same form. Second, we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.