# Geometric Brownian Motion with non-negative random increments

I am attempting to model a cumulative time-series of a positive integer variable across independent entities. The cumulative series appears to follow a process of Geometric Brownian Motion (GBM) based on lognormal distributions seen cross-sectionally at each time point.

The standard treatments and estimation methods for GBM drift ($m$) and diffusion ($s$) coefficients are based upon a specification where the random variation at each time point comes from a Wiener process $W(t)$ with normally distributed increments of zero mean:

$$dX(t) = m X(t) dt + s X(t) dW(t)$$

In my problem, this cannot apply since X(t) is a cumulative sum of positive numbers. Random increments can be positive or zero only, and the mean will be non-zero and positive. A normal distribution truncated below 0 appears to be appropriate.

Can anyone point me to a treatment and estimation approach for this type of problem? I believe the standard estimation methods do not apply here.

-