Modeling Positive Cumulative Series with Lévy Subordinators
Summary
The question concerns a cumulative series of positive integer values across independent entities. Its cross-sectional distributions appear lognormal, suggesting geometric Brownian motion, but the usual GBM specification has normally distributed random shocks with zero mean. That assumption conflicts with increments that can only be positive or zero, with a positive mean.
The replies suggest modeling the process with a Lévy subordinator, a process with nonnegative increments. Poisson processes are one example, while compound Poisson processes allow random jump sizes; for integer-valued data, the jump sizes must also be integer-valued, with a negative binomial distribution offered as an example. A second reply points to Poisson processes for modeling accumulated arrivals and notes that Hawkes processes can couple several arrival streams. These are candidate model families, not a fitted solution: the document gives no estimation procedure, empirical comparison, or evidence that any particular process fits the original data.
Key ideas
- Standard GBM uses zero-mean normal shocks, which do not match strictly nonnegative increments.
- Lévy subordinators provide a framework for processes with nonnegative increments.
- Compound Poisson models can represent positive jumps with random sizes.
- Integer-valued cumulative data requires integer-valued jump sizes.
- Poisson and Hawkes processes are suggested for modeling arrival streams.
Tags
Full text
# Geometric Brownian Motion with non-negative random increments # 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. ## Answer by Richi Wa (score 2) https://quant.stackexchange.com/a/7427 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). ## Answer by lehalle (score 1) https://quant.stackexchange.com/a/7426 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).
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.