Physical examples (and meaning) of the Ito process

Question

An Ito process $\{X\}_t$ is defined as

$$dX_t= K_tdt + H_tdW_t,$$

where $K_t, H_t$ are two progressive measurable process and $W_t$ is a standard Brownian motion.

My questions are:

(1) What (real-world) physical problems are usually modeled by Ito process (stochastic differential equation, SDE)?

Especially, (2) what is the physical meaning of the Brownian motion term? Is it a measurement error? or an accumulative mechanical error? or what?

Note: by "real-world physics", I mean something like pushing a box, lifting an object, collision of balls, ect.

(I found many papers claimed that the Ito process "is used in various physical applications", but having trouble finding examples.)

check out this wikipedia page to get started : https://en.wikipedia.org/wiki/Brownian_motion — Stratos supports the strike, Nov 08 '23 at 12:55
I know Brownian motion for particles movements. However, I want some "real-world physics" examples (please refer to my question). — Mingzhou Liu, Nov 08 '23 at 12:58
Langevin equation. You should do some of your own research and not ask MSE to start from scratch. — Kurt G., Nov 08 '23 at 13:05
In physics, it is often more convenient to use the Stratonovitch discrtization, so you'll find more examples from this angle (it's the mathematically rigorous way of treating Langevin equations). — LPZ, Nov 08 '23 at 13:58

score 2 · Accepted Answer · answered Nov 08 '23 at 18:44

(1) What (real-world) physical problems are usually modeled by Ito process (stochastic differential equation, SDE)?

Here are some links as mentioned here What practical applications do SDEs and SPDEs have?

In physics of turbulence see "Stochastic PDEs in Turbulence Theory" by E.Weinan.
In finance see Introduction to Stochastic Differential Equations (SDEs) for Finance
In biology see "Introduction to Stochastic Models in Biology"

Generally, the main difference with pdes is adding a degenerate process as a potential term that simulates random forcing from the environment. In statistics-ling it represents the sum total of all the confound variables that we can't identify but still affect the system.

Especially, (2) what is the physical meaning of the Brownian motion term? Is it a measurement error? or an accumulative mechanical error? or what?

As mentioned here Why is Brownian Motion so Big in the Theory of Stochastic Differential Equations? too,

any phenomenon where you have a lot of iid interactions eg. gas particles clashing, you get Gaussian universality ie. you have central limit theorem showing up for the various statistics involved. And since people care about the variation in time of those statistics, we also get random walks and in turn Brownian motion.

The most common occurrence of Brownian motion is in simulating random forcing.

So for example, as observed by Robert Brown and later Einstein in modelling the motion of pollen particles follows Brownian motion law as being moved by individual water molecules (these are the iid hits i.e. each hit by water molecule is assumed to be an iid occurence).

But it is also indeed a good model for error. Why? Because again there are many confounding variables that are assumed to act in an iid-manner. For example, the average height often follows Gaussian law (Bell curve) because there are so many different influences each of which adding/subtracting inches (eg. genes, diet, exercise etc).

Physical examples (and meaning) of the Ito process

1 Answers1