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

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.

• I think its motivated from physics (Einstein). – Wuestenfux Apr 14 at 15:36