Numerical methods for second-order stochastic differential equations

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.

Enlightening, balancing and transcending entrepreneurial orientation : the 4th annual EO3 research PDW

Entrepreneurial Orientation (EO) has a 30 year history as one of the most used concepts in entrepreneurship research. “Recent attention in formal sessions at the Academy of Management conference programs confirm Entrepreneurial Orientation as a primary construct with a majority of Entrepreneurship Division sponsored sessions devoted to studies using EO related measures”, as reported by the 2010 division program chair, Per Davidson (Roberts, 2010: 9). However, questions continue to be raised concerning over-dependence on parts of one strategic scale, possible inappropriate or under-theorized adaptations, and the lack of theoretical development on application and performance variance in emergent, organizational, and socioeconomic settings. One recent area of investigation in analysis, methods, theory and application concerns an “EO gestalt”, focusing on the family of EO-related measures and theory, rather than on one or more dimensions, in order to explore the theory and process of the Entrepreneurial Orientation phenomenon. The goals of the 4th Annual EO3 PDW are to enlighten researchers on the development of Entrepreneurial Orientation theory and related scales, balance the use of Entrepreneurial Orientation current knowledge with new research frontiers suggested by EO3 scholars’ questions, and transcend boundaries in the discoveries undertaken in the shared interdisciplinary and cross-cultural research agenda currently developing for Entrepreneurial Orientation concepts. Going into its forth year, the EO3 PDW has been pivotal in formalizing discussion, pushing research forward, and gaining insights from experienced and cutting edge scholars, as it provides a point of reference for coalescing research questions and findings surrounding this important concept.

Accelerated leap methods for simulating discrete stochastic chemical kinetics

Biologists are increasingly conscious of the critical role that noise plays in cellular functions such as genetic regulation, often in connection with fluctuations in small numbers of key regulatory molecules. This has inspired the development of models that capture this fundamentally discrete and stochastic nature of cellular biology - most notably the Gillespie stochastic simulation algorithm (SSA). The SSA simulates a temporally homogeneous, discrete-state, continuous-time Markov process, and of course the corresponding probabilities and numbers of each molecular species must all remain positive. While accurately serving this purpose, the SSA can be computationally inefficient due to very small time stepping so faster approximations such as the Poisson and Binomial τ-leap methods have been suggested. This work places these leap methods in the context of numerical methods for the solution of stochastic differential equations (SDEs) driven by Poisson noise. This allows analogues of Euler-Maruyuma, Milstein and even higher order methods to be developed through the Itô-Taylor expansions as well as similar derivative-free Runge-Kutta approaches. Numerical results demonstrate that these novel methods compare favourably with existing techniques for simulating biochemical reactions by more accurately capturing crucial properties such as the mean and variance than existing methods.