Stochastic calculus based on non-Brownian motion, or other-than-normal distributions

Question

Based on the book "Financial Calculus" by Baxter and Rennie, I understand that stochastic calculus is based on the equation $$dx(t)=\mu(t) dt+\sigma(t) dW(t),$$ in the same way that standard calculus has the standard derivative: $$\frac{dx(t)}{dt}=\mu(t).$$

In the stochastic equation, $W(t)$ is a Wiener process (Brownian motion), which forms the "building block" for stochastic processes, in the same way that the "straight line" forms the "building block" for derivatives.

My question is, does this "stochastic building block" necessarily have to be based on the normal distribution, as the Wiener process is? It seems arbitrary to restrict the stochastic element to only normal distributions, or is there some reason for it relating to the central limit theorem or something similar?

Answer 1

In (stochastic) calculus one tends to think in terms of integrals rather than derivatives. Even when one writes something like $dX = \mu dt + \sigma dW$ one means an integral, $$ X_t = X_0 + \int_0^t \mu(s) ds +\sigma(s)dW(s).$$

And as long as the integral on the right-hand side is well defined one can replace $W$ with pretty much anything, like a Poisson process, Levy process, or, if you don't like jumps, some continuous process whose quadratic variation is not absolutely continuous with respect to time. Equally, one can replace $ds$ (which signifies the Lebesgue measure on the time axis) with some other measure, like the Cantor measure, which is not absolutely continuous with respect to time.

So, you do not have to base your stochastic modelling on Brownian motion and your process does not have to run in line with the Lebesgue measure on the time axis.

Answer 2

To give a specific example of something that people care about other than Brownian motion is fractional Brownian motion (fBm), a slight generalization of regular Brownian motion. Here is a paper: https://arxiv.org/abs/0801.4963

There's an associated parameter associated to fBm, called Hurst parameter, $H$. When $H=1/2$ we get regular Brownian motion.

Some very interesting and technical things happen when $H<1/2$. You need something called rough paths theory which has some pretty cool uses.