  A form of time series path integral expansion is provided that enables both
analytic and numerical temporal effect calculations for a range of stochastic
processes. Birth-death processes with linear rates are analysed via coherent
state Doi-Peliti techniques. The $\mathfrak{su}(1,1)$ Lie algebra is utilised
to capture quadratic rate birth-death processes. The techniques are also
adapted to diffusion processes. All methods rely on finding a suitable
reproducing kernel associated with the underlying algebra to perform the
expansion. The resulting series differ from those found in standard Dyson time
series field theory techniques.
