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?

• 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, ...? – LocalVolatility Oct 29 '17 at 19:15
• 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? – Zhiyuan Wang Oct 30 '17 at 13:31