Symmetric path integrals for stochastic equations with multiplicative noise

A Langevin equation with multiplicative noise is an equation schematically of the form dq/dt=-F(q)+e(q)ξ, where e(q)ξ is Gaussian white noise whose amplitude e(q) depends on q itself. I show how to convert such equations into path integrals. The definition of the path integral depends crucially on the convention used for discretizing time, and I specifically derive the correct path integral when the convention used is the natural, time-symmetric one whose time derivatives are (qₜ-q_t-Δt)/Δt and coordinates are (qₜ+q_t-Δt)/2. (This is the convention that permits standard manipulations of calculus on the action, like naive integration by parts.) It has sometimes been assumed in the literature that a Stratonovich Langevin equation can be quickly converted to a path integral by treating time as continuous but using the rule θ(t=0)=1/2. I show that this prescription fails when the amplitude e(q) is q dependent.