Numerical methods for strong solutions of stochastic differential equations: an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

Subgeometric rates of convergence for a class of continuous-time Markov process

Let (Phi(t))(t is an element of R+) be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure pi. We investigate the rates of convergence of the transition function P-t(x, (.)) to pi; specifically, we find conditions under which r(t) vertical bar vertical bar P-t (x, (.)) - pi vertical bar vertical bar -> 0 as t -> infinity, for suitable subgeometric rate functions r(t), where vertical bar vertical bar - vertical bar vertical bar denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

A side order of stem cells: The SP phenotype

A defining property of murine hematopoietic stein cells (HSCs) is low fluorescence after staining with Hoechst 33342 and Rhodamine 123. These dyes have proven to be remarkably powerful tools in the purification and characterization of HSCs when used alone or in combination with antibodies directed against stem cell epitopes. Hoechst low cells are described as side population (SP) cells by virtue of their typical profiles in Hoechst red versus Hoechst blue bivariate fluorescent-activated cell sorting dot plots. Recently, excitement has been generated by the findings that putative stem cells from solid tissues may also possess this SP phenotype. SP cells have now been isolated from a wide variety of mammalian tissues based on this same dye efflux phenomenon, and in many cases this cell population has been shown to contain apparently multipotent stem cells. What is yet to be clearly addressed is whether cell fusion accounts for this perceived SP multipotency. Indeed, if low fluorescence after Hoechst staining is a phenotype shared by hematopoietic and organ-specific stem cells, do all resident tissue SP cells have bone marrow origins or might the SP phenotype be a property common to all stem cells? Subject to further analysis, the SP phenotype may prove invaluable for the initial isolation of resident tissue stem cells in the absence of definitive cell-surface markers and may have broad-ranging applications in stem cell biology, from the purification of novel stem cell populations to the development of autologous stem cell therapies.