Abstract
Summary. When the output of an atomistic simulation (such as the Gillespie stochastic simulation algorithm, SSA) can be approximated as a diffusion process, we may be interested in the dynamic features of the deterministic (drift) component of this diffusion. We perform traditional scientific computing tasks (integration, steady state and closed orbit computation, and stability analysis) on such a drift component using a SSA simulation of the Cyclic Lotka-Volterra system as our illustrative example. The results of short bursts of appropriately initialized SSA simulations are used to fit local diffusion models using Aït-Sahalia's transition density expansions [1, 2, 3] in a maximum likelihood framework. These estimates are then coupled with standard numerical algorithms (such as Newton-Raphson or numerical integration routines) to help design subsequent SSA experiments. A brief discussion of the validity of the local diffusion approximation of the SSA simulation (a jump process) is included.
Introduction
Reactive particle dynamic models arise in scientific fields ranging from physical and chemical processes to systems biology [33, 34, 37, 41, 38, 19, 17] . Incorporating successive levels of detail in the modeling quickly leads to models that are analytically intractable, necessitating computational exploration. Gillespie's Stochastic Simulation Algorithm (SSA) and its variants [41, 20, 19] have gained popularity in recent years for modeling so-called mixed reacting systems; the approach provides a middle ground between detailed molecular dynamics and lumped, Ordinary Differential Equation (ODE) descriptions of chemical kinetics, incorporating fluctuations. Knowing the kinetic scheme underlying such a simulation allows one to write, at the infinite particle limit, the corresponding kinetic ODE. At intermediate particle numbers (N mol ), the SSA has been approximated with the continuous "chemical Langevin equation" [19] .
In what follows we will assume that the results of an SSA simulation can be successfully approximated through a continuous diffusion process. Explicit knowledge of the drift and noise components of such a process allows one to easily analyze certain features of the overall behavior; one might, for example, be interested in the bifurcation behavior of the "underlying" drift component of the model, including the number and stability of its steady states and their parametric dependence. In our work we assume that the only available simulation tool is a "black box" SSA simulator, in which the mechanistic rules have been correctly incorporated, but which we, as users, do not know: we can only observe the SSA simulator output. We want to perform a quantitative computational study of the underlying drift component. Since we cannot derive it in closed form (not knowing the evolution rules), we want to perform this study using the least possible simulation with the SSA code. The approach we use follows the so-called "equation-free" framework [28, 27] : in this framework traditional numerical algorithms become protocols for designing short bursts of numerical experiments with the SSA code. The quantities necessary for numerical computation with the unavailable model (time derivatives, the action of Jacobians) are estimated locally by processing the "fine scale" SSA simulations. In this work we extract such numerical information via parametric local diffusion models using the transition density expansions proposed by Aït-Sahalia [2, 3] . The numerical procedures we illustrate can also be used, in principle, for different types of "fine scale" models if their output happens to be well approximated by diffusion processes.
The article is organized as follows: In Section 2 we describe our illustrative model system. We then quickly outline the basic ideas underlying equationfree numerics (Section 3), and discuss our estimation procedure (Section 4). Our computational results are presented in Section 5, and we conclude with a discussion including goodness-of-fit issues.
The Lattice Lotka-Volterra Model
Our Cyclic Lotka-Volterra [36, 14] illustrative example consists of a threespecies (X, Y and S) nonlinear kinetic scheme of the following form [36] :
In the remainder of the paper we will refer to it simply as LV. In the deterministic limit, this kinetic scheme gives rise to a set of three coupled nonlinear ODEs for the evolution of the concentrations X, Y and S.

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