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.


2 Answers 2


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).


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).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.