Chemical Kinetics, Markov Chains, and the Imaginary It\^{o} Interpretation

The abstract chemical reaction $$A+A \to \emptyset,$$ understood as a Markov chain in continuous time, has been studied in the physical literature for several years. It has been claimed that this reaction can be described by means of the stochastic differential equation $$d \phi = - \phi^2 dt + i \, \phi \, dW_t,$$ where $i$ is the imaginary unit. This affirmation is, at least, intriguing, and has led to controversy and criticisms in the literature. The goal of this work is to give partial evidence that such a description may be possible.