Why is Brownian Motion so Big in the Theory of Stochastic Differential Equations?

Question

I am reading some introductory material on stochastic differential equations at the moment. In almost all cases, the equations which are presented are of the form

$$ dX_t = \mu(t,X_t) dt + \sigma(t, X_t) dB_t, \quad (\dagger) $$

where $B_t$ usually is a standard Brownian motion. I am aware, that it is possible to generalize the above equaton by substituting $B_t$ with some semimartingale $H_t$; however, it seems that these cases are almost never studied in practise. In almost all books about applications of stochastic differential equations, the case where $H_t = B_t$ dominates completely.

Why is this the case? Are stochastic differential equations based on Brownian motion really that general? How is it possible, that these types of equations satisfy the need of so many practicioners? Are there any Theorems stating, that it is possible to model almost all stochastic processes of interest with equations like $(\dagger)$? I do not see how almot exclusively studying the special case of Brownian-motion based stochastic differential equations is not a massive loss of generality in the theory.

Answer 1

Actually it is a massive loss of generality. Here are two reasons why people use models based on Brownian motion:

• the central limit theorem. The CLT is saying that Gaussian distributions appear naturally in "many" situations as (limiting) distribution. Hence, if people want to model some phenomena and they don't have much information on the distribution, they like to use Gaussian distributions. Now, once you decided to settle for Gaussian distributions, Brownian motion is a quite natural choice (though, obviously, not the only one).
• computations. Stochastic differential equations driven by Brownian motion have many nice properties. They are relatively easy to compute numerically and there are quite a number of convergence results (e.g. Euler-Maruyama approximation). In contrast, if you use some general semimartingale to cook up a (very nice theoretical) model, you will most likely run into trouble when you try to perform numerical computations - which is clearly a huge disadvantage in applications.

Over the last decade there has been an increasing interest in using Lévy processes as driving processes in stochastic differential equations. The main point is that SDEs driven by Lévy processes allow for jumps whereas solutions to SDEs driven by Brownian motion are always continuous. Lévy-driven SDEs are, for instance, used in financial mathematics to model stocks, and there is also a number of applications in physics and biology. I'm not aware of applications where very "general" semimartingales are used.