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

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.

• No-arbitrage pricing arguments for option pricing very specifically rely on continuous time formulations. They require that you dynamically hedge a contract continuously - such hedging arguments can't be used in discrete time. Mar 24 at 2:25
• @rubikscube09 Sorry, but I'm not sure that that's entirely correct. There definitely exist discrete time financial models that price using no-arbitrage arguments and hedging, e.g. the binomial model. These are analogous to the random walk idea mentioned in the question in that they do approach the continuous time solutions when the scale is sufficiently fine. Mar 24 at 18:34
• @user6247850 sure - agreed, but in those models there are only finitely many possible prices at each timestep. In the case of continuous prices we need continuous hedging. Mar 24 at 19:29
• For the updated question: It depends who you talk to. To speak more on the KPZ universality. There are people from many different fields eg. exciton-photon polariton, who have their own discrete models that they are interested to study and that happen to somewhat relate to KPZ. Mar 24 at 20:18
• They are not necessarily interested in studying other discrete models from other areas where KPZ showed up even if they are more computationally better eg. mRNA protein synthesis model, forest-fires and coffee percolation. But all of these people are interested to get nice formulas for KPZ that they can try to relate to their particular models (eg. via obtaining convergence rates). Mar 24 at 20:19

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.

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.

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.

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.]

• Why the downvote? Is anything wrong in this answer? Mar 23 at 19:28
• Some people might be offended by the very thought of an AI being able to answer Mathoverflow questions. If only there was someone (or some_thing_) to discuss this with... Mar 23 at 20:19
• -1 for the ChatGPT response, which (as is often the case with ChatGPT) sounds nice but is subtly incoherent. For example, it lists the constant arrival rate of the Poisson process as an advantage over discrete time analogs (is it?) but also claims the Poisson process is better equipped to deal with arrival rates that change over time. Mar 24 at 7:57
• Wow. That ChatGPT answer is quite impressive. What was the precise question used to generate it? Mar 24 at 15:13
• Lastly, how would you feel about a student who answered an assignment question (or a mathematician who wrote a paper!) but instead of pretending it was their own work, it was an openly acknowledged cut-and-paste from a chat bot? Since MO doesn't want (at least currently, as there's no guarantee of correctness) chatGPT answers (or similar) passed off as user contributions, I don't see how merely being open about it qualitatively changes the value of such an answer. Mar 25 at 6:23