An operator approach for Markov chain weak approximations with an application to infinite activity L\'{e}vy driven SDEs

Weak approximations have been developed to calculate the expectation value of functionals of stochastic differential equations, and various numerical discretization schemes (Euler, Milshtein) have been studied by many authors. We present a general framework based on semigroup expansions for the construction of higher-order discretization schemes and analyze its rate of convergence. We also apply it to approximate general L\'{e}vy driven stochastic differential equations.Comment: Published in at http://dx.doi.org/10.1214/08-AAP568 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Efficient simulation of stochastic chemical kinetics with the Stochastic Bulirsch-Stoer extrapolation method

BackgroundBiochemical systems with relatively low numbers of components must be simulated stochastically in order to capture their inherent noise. Although there has recently been considerable work on discrete stochastic solvers, there is still a need for numerical methods that are both fast and accurate. The Bulirsch-Stoer method is an established method for solving ordinary differential equations that possesses both of these qualities.ResultsIn this paper, we present the Stochastic Bulirsch-Stoer method, a new numerical method for simulating discrete chemical reaction systems, inspired by its deterministic counterpart. It is able to achieve an excellent efficiency due to the fact that it is based on an approach with high deterministic order, allowing for larger stepsizes and leading to fast simulations. We compare it to the Euler ?-leap, as well as two more recent ?-leap methods, on a number of example problems, and find that as well as being very accurate, our method is the most robust, in terms of efficiency, of all the methods considered in this paper. The problems it is most suited for are those with increased populations that would be too slow to simulate using Gillespie’s stochastic simulation algorithm. For such problems, it is likely to achieve higher weak order in the moments.ConclusionsThe Stochastic Bulirsch-Stoer method is a novel stochastic solver that can be used for fast and accurate simulations. Crucially, compared to other similar methods, it better retains its high accuracy when the timesteps are increased. Thus the Stochastic Bulirsch-Stoer method is both computationally efficient and robust. These are key properties for any stochastic numerical method, as they must typically run many thousands of simulations

Weak order for the discretization of the stochastic heat equation driven by impulsive noise

Considering a linear parabolic stochastic partial differential equation driven by impulsive space time noise, dX_t+AX_t dt= Q^{1/2}dZ_t, X_0=x_0\in H, t\in [0,T], we approximate the distribution of X_T. (Z_t)_{t\in[0,T]} is an impulsive cylindrical process and Q describes the spatial covariance structure of the noise; Tr(A^{-\alpha})0 and A^\beta Q is bounded for some \beta\in(\alpha-1,\alpha]. A discretization (X_h^n)_{n\in\{0,1,...,N\}} is defined via the finite element method in space (parameter h>0) and a \theta-method in time (parameter \Delta t=T/N). For \phi\in C^2_b(H;R) we show an integral representation for the error |E\phi(X^N_h)-E\phi(X_T)| and prove that |E\phi(X^N_h)-E\phi(X_T)|=O(h^{2\gamma}+(\Delta t)^{\gamma}) where \gamma<1-\alpha+\beta.Comment: 29 pages; Section 1 extended, new results in Appendix

Higher-order numerical methods for stochastic simulation of\ud chemical reaction systems

In this paper, using the framework of extrapolation, we present an approach for obtaining higher-order -leap methods for the Monte Carlo simulation of stochastic chemical kinetics. Specifically, Richardson extrapolation is applied to the expectations of functionals obtained by a fixed-step -leap algorithm. We prove that this procedure gives rise to second-order approximations for the first two moments obtained by the chemical master equation for zeroth- and first-order chemical systems. Numerical simulations verify that this is also the case for higher-order chemical systems of biological importance. This approach, as in the case of ordinary and stochastic differential equations, can be repeated to obtain even higher-order approximations. We illustrate the results of a second extrapolation on two systems. The biggest barrier for observing higher-order convergence is the Monte Carlo error; we discuss different strategies for reducing it

On the asymptotic stability and numerical analysis of solutions to nonlinear stochastic differential equations with jumps

This paper is concerned with the stability and numerical analysis of solution to highly nonlinear stochastic differential equations with jumps. By the Itô formula, stochastic inequality and semi-martingale convergence theorem, we study the asymptotic stability in the pth moment and almost sure exponential stability of solutions under the local Lipschitz condition and nonlinear growth condition. On the other hand, we also show the convergence in probability of numerical schemes under nonlinear growth condition. Finally, an example is provided to illustrate the theor