Skip to content
All library documents

Modeling Monotone Cumulative Data with Lévy Subordinators

Article Quant Q&A · Author: Zhiyuan Wang
Read the original

Summary

The question asks how to model cumulative mobile data use, which cannot decrease, using a process such as geometric Brownian motion. The response suggests Lévy subordinators, a class of processes with nonnegative increments, as a suitable alternative. A gamma subordinator is one example when increments follow a gamma distribution, while compound Poisson processes are another possible member of the class.

The note points toward a family of established stochastic models rather than modifying geometric Brownian motion directly. The best choice depends on the modeling objective and desired behavior of increments; the brief response does not compare model fit, estimation, or forecasting performance.

Key ideas

  • Lévy subordinators model processes that do not decrease over time.
  • A gamma subordinator is an option when increments are gamma-distributed.
  • Compound Poisson processes are another nondecreasing Lévy-process model.
  • Model selection depends on the intended use and the desired behavior of increments.

Tags

Statistics
Full text
# How to adjust Geometric Brownian Motion to be monotone?


# How to adjust Geometric Brownian Motion to be monotone?












I want to use stochastic process to model subscriber's mobile data consumption as time going in a month. So I think about Geometric Brownian Motion.

However, people's cumulative data consumption will never decrease. Thus, how can I adjust the formulation of Geometric Brownian Motion to make it monotone? Or is there any other stochastic process more suitable?

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/36652

You could model this as a Lévy-process. The class of subordinators can be used to model processes that never decrease. If the increments are Gamma-distributed then this is a gamma subordinator. Depending on what you want to do with the model the class of Lévy processes is studied in detail. You can look for more details here. Compound Poisson processes fall in this class as well but I think they don't come that natural in your case.

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.