Application of cumulant techniques to multiplicative stochastic processes

Abstract

The use of cumulant techniques for analyzing time dependent, stochastic matrix expressions of the form $\mathrm{〈T}{\displaystyle {lim}_{\leftarrow }}\exp {\mathrm{[\int }}_0^t\mathbf{B̃}\mathrm{(S)d\hspace{0.17em}S]〉}$ is explained. Because cumulants are complicated expressions when B̃(t) does not commute with itself at unequal times, we explicitly work out cumulant expressions up to fourth order. The fourth order terms can be used to demonstrate that noncommutivity prevents the generalization, to time‐dependent, stochastic matrices which do not commute with themselves at unequal times, of the result which applies to commuting stochastic processes that states: If the stochastic process is Gaussian, then its cumulant expansion truncates after the second cumulant. Furthermore, it is argued that if the stochastic matrix process is both Gaussian and purely random then the cumulant expansion does truncate after the second cumulant, after all. The significance of this result with respect to the application of approximation involving cumulants is mentioned.