# Generalize WienerProcess

Together the WhiteNoiseProcess, which includes the option of specifying a specific distribution for the noise, and the TransformProcess, which allows functions of processes, provide a great deal of flexibility in creating discrete-time processes.

There does not appear to be a similar approach to creating a custom continuous-time process. For instance, it would be nice it the WienerProcess function included an option to specify a distribution other than the Gaussian.

I see from other questions that the discrete-time processes can be converted into continuous time processes by using Interpolation; however, it's not clear to me how to do this conversion in a way that the resulting process can be use with RandomFunction. To be more specific, while a Interpolation can be used to specify a specific conversion given {t_min, t_max, dt} it's not clear how to define a general continuous time process that RandomFunction will recognize.

• You can do Accumulate[RandomVariate[yourdistribution,numsamples]] but a Wiener-like process with a non-normal finite-variance errors is redundant. There is some discussion in this question but surely all random walks of finite variance error term ultimately converge to the WienerProcess in the limit because of the CLT? ListPlot[Accumulate@RandomVariate[LaplaceDistribution[0,1],10^5]] is practically indistinguishable from ListPlot[Accumulate@RandomVariate[NormalDistribution[0, Sqrt[2]], 10^5]] – flinty Aug 14 at 20:33
• ^ in other words - if your process had a finite-variance non-gaussian underlying errors, then in continuous time it would be indistinguishable from the WienerProcess. If the errors were from a fat-tailed distribution like Cauchy, then it wouldn't exist in continuous time at all because it would blow up to infinity on any arbitrarily small time window. – flinty Aug 14 at 20:40
• @flinty Thanks for the suggestions. Indeed my interest is in modeling fat-tailed distributions, but you bring up an interesting point regarding whether a process with infinity variance can be properly defined. This is what I'd like to explore further. I understand the weight of the process will extend to infinity, but I'm not as sure that a formal definition is impossible. – Kenric Sep 6 at 14:54
• There are families of stable-stochastic-processes from stable distributions even with infinite variance. For example en.wikipedia.org/wiki/Cauchy_process but in this case they're not like the usual processes, and I don't think they're continuous like the Wiener Process (see en.wikipedia.org/wiki/Lévy_process) . In the case of the Cauchy Process, the time is subordinated too. The number of time steps between jumps is itself random from a Lévy process. It appears Mathematica has no way to model this at the moment. – flinty Sep 6 at 15:45
• Thanks, this is also helpful – Kenric Sep 6 at 17:41