"Practical" use of time-continuous stochastic processes like Wiener process or Poisson (point) process?

Question

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?

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@ 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.

Answer 1

You answered your own question, I think. Many physics/economics models involve partial differential equations which often are studied using Feynman-Kac (among many other methods), which involves Brownian motion.

These have exact formulas compared to the finite-difference relations. So they are much easier to study and draw conclusions from compared to finite-difference relations. For example, in Durrett, I remember he proved a Random-Walk Laws of the Iterated Logarithm using the continuous one via the Skorokhod-embedding.

Another major reason is Universality (eg. KPZ universality). Here the discrete-interactions and specificities get averaged out. This is a general principle of physics saying that the discrete-details often don't matter, so we might as well reduce to studying only a few continuous models that are "Fixed Points" for many discrete models.

Answer 2

I could add a couple things to the already existing good answers.

1. Continuous time processes are not that bad/that much more difficult to define or to simulate numerically in practice. For a Markov chain in a discrete or even finite state space, going from discrete time to continuous time amounts to making the time intervals between when something happens random, e.g., exponentially distributed instead of fixed equal to some given time increment $\Delta t$. Moreover, some properties work nicely for this continuous time version but not so much for the discrete time version. See for example the correspondence with local times, related to the Ray-Knight theorem(s) or the Symanzik-Brydges-Fröhlich-Spencer-Dynkin isomorphisms. A good pedagogical review on this topic is "The geometry of random walk isomorphism theorems" by Bauerschmidt, Helmuth and Swan.

2. Although constructing the continuum random object requires some work, e.g., rigorously getting Brownian motion $B_t$ as a limit of (discrete time and space) simple random walk. This effort pays off in having beautiful exact symmetries properties hold for the continuum object. My favorite is the time inversion, i.e., equality in distribution of $tB_{1/t}$ with $B_t$. Good luck formulating that as a property of the simple random walk.

I like this example because this is the simplest instance of conformal invariance which is an important area in mathematical physics related to the work of W. Werner, S. Smirnov, M. Hairer, H. Duminil-Copin and many others.

Answer 3

As one example, the geometric Brownian motion is used in mathematical finance to model the price of the underlying asset, whence one derives the Black–Scholes formula for the price of the corresponding European call/put options.

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Quoted below is part of a response by ChatGPT (quite impressive, in my view) -- which I have read, understood, and approved. I have also added a detail at the end of the following quote:

Poisson point process: A simple example of a Poisson point process is the modeling of customer arrival times in a queuing system, such as customers arriving at a bank or a call center. The Poisson point process is often used to model the arrival process because it captures the randomness and independence of customer arrival times. Advantages over discrete analogues:

Simplicity: The Poisson process has very simple and intuitive properties, such as a constant arrival rate, which makes it easy to analyze and understand.

Analytical tractability: Many queuing models based on the Poisson process have closed-form solutions or can be analyzed using well-established techniques, such as Markov chains or the Poisson equation. This is often more difficult for discrete-time models, especially when arrival rates vary over time.

Time-scale invariance: The Poisson process is time-scale invariant, meaning that the statistical properties of the process do not change if the time scale is changed. This makes it suitable for modeling systems with various time scales, whereas discrete models may need to be adjusted for different time scales. [Added by me: Indeed, with the continuous-time Poisson point process, any variable arrival rate can be obtained from the constant rate by a simple, deterministic time change.]