I've revised my question to hopefully be more concise:
Consider a simple Kalman filter that is designed to estimate a single state, the position of a car along a rail, $p$, and a time delay parameter, $t_d$, that is included in the state vector. The process model is driven by a velocity input (i.e., the derivative of position). Measurements of the position state occur at a time that either leads or lags the process model (as defined by $t_d$). Note that $t_d$ is a random variable.
Now, I wish to propagate the process model forward in time to next measurement update - this update is predicted to occur at $t + t_d$, but in reality we don't yet know $t_d$ precisely. Thus, we have a situation in which the upper limit of the integration is, in fact, a random variable. This means that the time interval over which we propagate the process model is a function of $t_d$ in the state vector.
I have not seen any work that deals with this situation - however, $t_d$ is unknown, and its value affects both the process and measurement model equations.
I am looking for any work on stochastic processes that properly treats the uncertainty associated with a state estimate when the integration bound is a random variable, and when one is trying to estimate the mean of that random variable.