If one uses the Wiener process as an ingredient to model something, then for practical purposes one could just as well take a simple discrete random walk (with sufficiently fine scale). If one uses a homogenous Poisson point process, one could just as well partition space and/or time into equal cells and take a sequence of independent 0-1 random variables describing whether the cells are occupied or empty.

"Practical" questions never force you to use the continuum. Similarly, instead of using integrals one could always use finite sums. However, sometimes there is a very simple expression for an integral, but not for the corresponding sum, for the solution of a differential equation, but not for the difference equation, or certain manipulations may be easier in the continuous than in the discrete case. In this sense, the continuum also has a purely practical use.

My question is: What would be the simplest examples for "practical" applications of Wiener process and Poisson (point) processes and practical advantages over the discrete analogues? Putting it slightly differently: If one is not interested in the continuum for fundamental reasons or as something that has its value in itself, what can be reasons to study these processes?

@ Thomas Kojar and also others:

OK, one can reduce everything to a few continuous models, but one could also replace each of these by a computationally particularly convenient discrete model - because "details do not matter" anyway. I am looking for "practical" examples where it is better not to do the last step.

In case of the corresponding analysis question "Why not always use sums instead of integrals?", a simple example would be $\sum_{k=1}^nk^p$ and $\int_l^u x^p\text dx$ and an everyday application of this would be to compute positions from accelerations. Some examples like this would satisfy me. Stochastic processes are a more advanced topic, so the simplest possible answers might be more complicated than this, but I am looking, as far as possible, for simple, maybe unspectacular examples where all details can be checked with a reasonable amount of effort and some general mathematical knowledge. Can you come up with something like this? "Studying PDE via Feynman-Kac formula" does not satisfy this...

And maybe, since I wrote it in the title but then maybe obscured it in the text, I should empasize: I am particularly wondering about discrete vs. continuous time. I am not worried whether to use a random walk with +/- 1 steps or with Gaussian steps, but about situations when it is good to add an evolution for all rational (or real, but that will hardly matter) times to the Gaussian random walk.

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