# Stochastic process with good sample paths

I am interested in a stochastic process on $[0, +\infty)$ which has a.s. Lipschitz continuous sample paths.

Does such a process with independent increments or markovian property exist? Are there any well known such examples?

• With independent increments I do not think it is possible under nontrivial circumstances. In this case the increments will follow some stationary distribution which makes them either deterministic (uninteresting), Gaussian (so locally like Brownian motion), or heavy-tailed (which will likely be worse than BM locally). With just Markov you will need to be more specific because again any deterministic process is Markov but this case is obviously not of interest here.
– Ian
Oct 21, 2016 at 15:09

There are a lot of nice ODEs of the shape $\dot x = f(x)$ which will have Lipschitz continuous paths. Now, since increments are non-probabilistic, they are independent. Furthermore, this process is Markovian. Similar construction applies to cadlag, piecewise constant paths.

If you would like to have non-trivial probability distribution of any increment, then for the latter case compound Poisson processes (which includes standard Poisson process) are Markov processes with independent increments.

At the same time, I do not think that there exist a process with Lipschitz continuous sample paths, whose increments are independent but follow non-Dirac distributions. Unfortunately, I am not sure whether this is true or whether I've seen a proof of this fact.

• Thanks! Yes, I need non-trivial examples. Could you suggest any references on compound Poisson processes? Oct 21, 2016 at 15:10
• regarding the Lipshcitz paths, I agree with yo, I suspect that a similar argument like the proof of roughness of brownian paths may work Oct 21, 2016 at 15:13
• @Pedro: the wikipedia article may be a good place to start, otherwise any book/lecture notes on Levy processes should work. The latter always have independent increments and cadlag paths, however they may have a Brownian motion component, so not every Levy process will have the paths that you like.
– SBF
Oct 21, 2016 at 15:13
• Thanks Ilya. I guess the primitive of a compound Poisson processes will be a process with Lipschitz paths :) Oct 21, 2016 at 20:34
• @Pedro I'm not sure it will have independent increments or markov property
– SBF
Oct 21, 2016 at 20:41