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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.

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I think the notion of a Lévy process fits for your problem. Lévy processes with only positive increments are called Lévy subordinators. Poisson processes as lehalle proposes are a subclass of these. Compound Poisson processes are an easy generalization of Poisson processes, they have only positive increments if you assume that the "2nd" distribtion (jump size) is non-negative. I can provide details but you find good sources online too. EDIT: I just read positive integer. Then only compound poisson with integer valued jumps size distribution works ... (e.g. Negative Binomial).

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Are you sure that you do not need a Poisson Process (see for instance a course about them)?

Poisson processes are commonly used to model the sum of arrival times. They are very useful to model high frequency data (arrivale rates of buy / sell orders). You can couple several Poisson processes using Hawkes processes (see Modeling microstructure noise with mutually exciting point processes).

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