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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?

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    $\begingroup$ Instead of modelling your total consumption to follow a GBM, you could e.g. model the instantaneous rate of data usage as a GBM $X_t$ then your total consumption is $Y_t = \int_0^t X_u \mathrm{d}u$. However, you should question if the a GBM is a reasonable model for the instantaneous rate of consumption - e.g. should it have a drift but no intra-day/week/month seasonality, ...? $\endgroup$ – LocalVolatility Oct 29 '17 at 19:15
  • $\begingroup$ Thanks a lot! Could you please explain how to compute Y_t? There is a Wiener process within X_t, how to deal with it in the time intergral? $\endgroup$ – Zhiyuan Wang Oct 30 '17 at 13:31
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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.

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