TY - UNPB

T1 - A simplified derivation of the Linear Noise Approximation

AU - Wallace, Edward WJ

PY - 2010

Y1 - 2010

N2 - Given a discrete stochastic process, for example a chemical reaction system or a birth and death process, we often want to find a continuous stochastic approximation so that the techniques of stochastic differential equations may be brought to bear. One powerful and useful way to do this is the system size expansion of van Kampen to express a trajectory as a small stochastic perturbation to a deterministic trajectory, using a small parameter related to the volume of the system in question. This is usually pursued only up to first order, called the Linear Noise Approximation. The usual derivation of this proceeds via the master equation of the discrete process and derives a Fokker-Planck equation for the stochastic perturbation, both of which are equations for the evolution of probability distributions. Here we present a derivation using stochastic difference equations for the discrete process and leading, via the chemical Langevin equation of Gillespie, directly to a stochastic differential equation for the stochastic perturbation. The new derivation, which does not yield the full system size expansion, draws more explicitly on the intuition of ordinary differential equations so may be more easily digestible for some audiences.

AB - Given a discrete stochastic process, for example a chemical reaction system or a birth and death process, we often want to find a continuous stochastic approximation so that the techniques of stochastic differential equations may be brought to bear. One powerful and useful way to do this is the system size expansion of van Kampen to express a trajectory as a small stochastic perturbation to a deterministic trajectory, using a small parameter related to the volume of the system in question. This is usually pursued only up to first order, called the Linear Noise Approximation. The usual derivation of this proceeds via the master equation of the discrete process and derives a Fokker-Planck equation for the stochastic perturbation, both of which are equations for the evolution of probability distributions. Here we present a derivation using stochastic difference equations for the discrete process and leading, via the chemical Langevin equation of Gillespie, directly to a stochastic differential equation for the stochastic perturbation. The new derivation, which does not yield the full system size expansion, draws more explicitly on the intuition of ordinary differential equations so may be more easily digestible for some audiences.

M3 - Working paper

BT - A simplified derivation of the Linear Noise Approximation

PB - ArXiv

ER -