This paper builds a convergence analysis of explicit tau-leaping schemes for simulat-ing chemical reactions from the viewpoint of stochastic differential equations. Mathematically, thechemical reaction process is a pure jump process on a lattice with state-dependent intensity. Thestochastic differential equation form of the chemical master equation can be given via Poisson ran-dom measures. Based on this form, different types of tau-leaping schemes can be proposed. In orderto make the problem well-posed, a modified explicit tau-leaping scheme is considered. It is shownthat the mean square strong convergence is of order 1/2 and the weak convergence is of order 1 forthis modified scheme. The novelty of the analysis is to handle the non-Lipschitz property of thecoefficients and jumps on the integer lattice.

In this paper, we present a class of explicit numerical methods for stiff Itˆo stochasticdifferential equations (SDEs). These methods are as simple to program and to use as the well-knownEuler-Maruyama method, but much more efficient for stiff SDEs.For such problems, it is wellknown that standard explicit methods face step-size reduction. While semi-implicit methods canavoid these problems at the cost of solving (possibly large)nonlinear systems, we show that the step-size reduction phenomena can be reduced significantly for explicit methods by using stabilizationtechniques. Stabilized explicit numerical methods calledS-ROCK (for stochastic orthogonal Runge-Kutta Chebyshev) have been introduced in [C. R. Acad. Sci. Paris, vol. 345, no. 10, 2007] asan alternative to (semi-) implicit methods for the solutionof stiff stochastic systems. In this paperwe discuss a genuine Itˆo version of the S-ROCK methods whichavoid the use of transformationformulas from Stratonovich to Itˆo calculus. This is important for many applications. We present twofamilies of methods for one-dimensional and multi-dimensional Wiener processes. We show that forstiff problems, significant improvement over classical explicit methods can be obtained. Convergenceand stability properties of the methods are discussed and numerical examples as well as applicationsto the simulation of stiff chemical Langevin equations are presented.

We aim to construct higher order tau-leaping methods for numerically simulating stochastic chemical kinetic systems in this paper. By adding a random correction to the primitive tau-leaping scheme in each time step, we greatly improve the accuracy of the tau-leaping approximations. This gain in accuracy actually comes from the reduction in the local truncation error of the scheme in the order of τ, the marching time step size. While the local truncation error of the primitive tau-leaping method is

Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiffordinary differential equations. Making use of special properties of Chebyshev-like poly-nomials, these methods have favorable stability properties compared to standard explicitmethods while remaining explicit. A new class of such methods, called ROCK, introducedin [Numer. Math., 90, 1-18, 2001] has recently been extended to stiff stochastic differentialequations under the name S-ROCK [C. R. Acad. Sci. Paris, 345(10), 2007 and Commun.Math. Sci, 6(4), 2008]. In this paper we discuss the extension of the S-ROCK methodsto systems with discrete noise and propose a new class of methods for such problems, theτ-ROCK methods. One motivation for such methods is the simulation of multi-scale orstiff chemical kinetic systems and such systems are the focus of this paper, but our newmethods could potentially be interesting for other stiff systems with discrete noise. Twoversions of theτ-ROCK methods are discussed and their stability behavior is analyzed ona test problem. Compared to theτ-leaping method, a significant speed-up can be achievedfor some stiff kinetic systems. The behavior of the proposed methods are tested on severalnumerical experiments.

There are two scalings for the convergence analysis of tau-leaping methods in the literature. This paper attempts to resolve this debate in the paper. We point out the shortcomings of both scalings. We systematically develop the weak Ito-Taylor expansion based on the infinitesimal generator of the chemical kinetic system and generalize the rooted tree theory for ODEs and SDEs driven by Brownian motion to rooted directed graph theory for the jump processes. We formulate the local truncation error analysis based on the large volume scaling. We find that even in this framework the midpoint tau-leaping does not improve the weak local order for the covariance compared with the explicit tau-leaping. We propose a procedure to explain the numerical order behavior by abandoning the dependence on the volume constant V from the leading error term. The numerical examples validate our arguments. We also give a general global weak convergence analysis for the explicit tau-leaping type methods in the large volume scaling.