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Daniel T. Gillespie developed stochastic simulation methods for reacting molecular populations. In the Gillespie algorithm, a reaction propensity function specifies an event rate, and the algorithm samples both the next reaction time and the reaction channel rather than evolving only mean concentrations.^[ref]

If the total propensity is $a_0>0$, the waiting time is exponentially distributed with rate $a_0$, and channel $j$ is selected with probability $a_j/a_0$ at the current state. His tau-leaping approach groups reaction events to accelerate simulation, introducing a controlled approximation rather than retaining the exact event-by-event law.^[ref]

A birth-death process with unit upward and downward jumps and rates $b(x)$ and $d(x)$ has infinitesimal generatorIf $f$ varies slowly over one jump, Taylor expansion through second order gives $Lf\simeq (b-d)f^{\prime }+(b+d)f^{\prime \prime }/2$. Thus its diffusion approximation has drift coefficient $u=b-d$ and, in the conventiondiffusion coefficient $D=(b+d)/2$. Equivalently it solves $d{X}_t=u(X_t)dt+\sqrt{b(X_t)+d(X_t)}d{B}_t$, where $B$ is Brownian motion. The sum measures the variance of the short-time increment: births and deaths both add noise even when their mean drifts cancel.

For a population density $x=n/N$ with jump size $1/N$ and birth/death rates $N\beta (x),N\delta (x)$, the drift is $\beta -\delta$ and the diffusion coefficient is $(\beta +\delta )/(2N)$. Fluctuations therefore shrink as $N^{-1/2}$. A useful approximation requires many reaction events while the rates change little; it can fail near an absorbing state or when only a few individuals remain. A continuous diffusion can even visit negative values unless its boundary treatment is specified. The approximation replaces the discrete jump model rather than asserting equality of their laws.^[ref]