History
The process that lead to the algorithm recognizes several important steps. In 1931, Andrei Kolmogorov introduced the differential equations corresponding to the time-evolution of stochastic processes that proceed by jumps, today known as Kolmogorov equations (Markov jump process) (a simplified version is known as master equation in the natural sciences). It was William Feller, in 1940, who found the conditions under which the Kolmogorov equations admitted (proper) probabilities as solutions. In his Theorem I (1940 work) he establishes that the time-to-the-next-jump was exponentially distributed and the probability of the next event is proportional to the rate. As such, he established the relation of Kolmogorov's equations with stochastic processes. Later, Doob (1942, 1945) extended Feller's solutions beyond the case of pure-jump processes. The method was implemented in computers by David George Kendall (1950) using the Manchester Mark 1 computer and later used by Maurice S. Bartlett (1953) in his studies of epidemics outbreaks. Gillespie (1977) worked ignoring this history as he writes "It should be emphasized, though, that the master equation itself plays no role whatsoever in either the derivation or the implementation of the stochastic simulation algorithm". Gillespie then proceeds through a heuristic argument to introduce the algorithm.
Read more about this topic: Gillespie Algorithm
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