What phenomena are better modelled by SDE instead of ODE?

Question

Both stochastic differential equations (SDE) and ordinary differential equations (ODE) can be used to model a variety of different phenomena, whether physical or otherwise. Most deterministic ODE models can be turned into stochastic models via adding a noise term in the dynamics, which changes the qualitative features of the model. However, it is not clear that the stochastic model is “more accurate” than the deterministic one.

I am interested in when an SDE model would be more appropriate than a deterministic one. As I am interested in the use of SDE in particular, I wish to restrict to continuous time examples.

One instance I am aware of is mathematical finance - most things in finance need to be stochastic to be of interest, indeed a deterministic model of stock prices or buy orders does not make too much sense, except in rare cases. What other phenomena/naturally occurring processes require the use of SDE over ODE? I am open to examples from machine learning/AI as well.

Answer 1

This is a really broad question, but in general noise terms become important if there are few degrees of freedom; for example, chemical reaction kinetics can be accurately described by coupled ODE's, until the concentration of the reactants becomes too low; then you need SDE's to account for the stochasticity in reaction rates and molecular collisions.

In the context of AI there is the topic of "probabilistic machine learning", see this text book. Instead of using a fixed-form parametric model for regression, one postulates a Gaussian process prior over the model functions. The Gaussian process regression is described by linear stochastic differential equations.

Answer 2

Brownian motion is an obvious example. Brownian motion described particles dispersed in a liquid that are large enough that the random jossling of the water molecules becomes important. Being one of the first models that were studied this way, it is an often quoted example. Einstein also used Brownian motion to provide evidence for the existence of atoms.

Stochastic differential equations nicely describe the behaviour of such particles. For example, it correctly predicts the linear growth of the mean squared displacement of a particle $$\langle R^2(t)\rangle\propto t.$$ When modelled using ODE, we get the very boring result

$$\langle R^2(t)\rangle= 0.$$

Answer 3

Many biological phenomena can be modelled using SDEs or other stochastic models. For example, disease transmission models which keep track of the numbers of infected $I$ and susceptible $S$ individuals in a population. Stochastic models can capture the observed phenomenon that $S$ and $I$ are typically negatively correlated (because, when an individual becomes infected, $I$ increases by $1$ and $S$ decreases by $1$). Not something that can easily be modelled deterministically.

More generally, compartmental models for biological populations, with processes such as disease transmission, births, deaths, etc. can typically be modelled using something like a discrete state space continuous time process. Such populations are typically subject to both demographic and environmental stochasticity. As the population size increases, demographic stochasticity decreases but environmental stochasticity doesn't - it's for larger populations that applications for SDEs are typically found. For example this article handles large populations using a SDE model which is constructed as a limit of a discrete state space continuous time Markov process.

For a general comparison of deterministic and stochastic models in the context of disease transmission, see this book, Mathematical Tools for Understanding Infectious Disease Dynamics by Odo Diekmann, Hans Heesterbeek, and Tom Britton.

Also, this book, An Introduction to Stochastic Processes with Applications to Biology by Linda Allen, has a chapter on biological applications of SDEs.

Answer 4

I am not sure what you mean by ''more accurate'' but I will just point out that many physical processes involve some type of inherent randomness which can be modelled using a stochastic differential equation.

One example is that of a particle which is in a quickly evolving random medium. This system is governed by a stochastic differential equation of the form

$d x_t = A(x_t, t/ \epsilon) d t + \sigma ( x_t, t / \epsilon ) d B$,

for some small parameter $\epsilon > 0$.

Some other motivating examples can be found in these lecture notes of Martin Hairer. The examples include a random string (or polymer), stochastic Navier-Stokes equations for fluid flows with random fluctuations, and stochastic heat flow.

Edit: As stated in some of the above answers, another obvious example is Brownian motion (see, for example

• M. Hairer and X.-M. Li, Generating Diffusions with Fractional Brownian Motion, Comm. Math. Phys. 396, 91-141 (2022) Link

Answer 5

Since the Earth is not at zero degrees Kelvin, thermal noise is a reality that cannot be ignored. Much of electrical engineering is designing analog filters to remove as much noise as possible from noisy signals. Such filters can be modeled using SDEs, though electrical engineers might prefer to work with spectral densities instead. The interested reader might want to take a look at Wiener–Khinchin.