A new approach to simulating stochastic delayed systems

•A method is proposed for deriving stochastic delay differential equations (SDDEs) from delayed chemical master equations.•Equivalence of alternative SDDE formulations is established in terms of distribution and sample paths.•An algorithm for deriving a computationally efficient form of SDDEs is developed.•The new method is shown to deliver excellent agreement with direct stochastic simulations, while having a much superior speed of performance. In this paper we present a new method for deriving Itô stochastic delay differential equations (SDDEs) from delayed chemical master equations (DCMEs). Considering alternative formulations of SDDEs that can be derived from the same DCME, we prove that they are equivalent both in distribution, and in sample paths they produce. This allows us to formulate an algorithmic approach to deriving equivalent Itô SDDEs with a smaller number of noise variables, which increases the computational speed of simulating stochastic delayed systems. The new method is illustrated on a simple model of two interacting species and a model with bistability, and in both cases it shows excellent agreement with the results of direct stochastic simulations, while also demonstrating a much superior speed of performance.