In the mathematics of finance, a stochastic process can be given by a stochastic differential equation: $$dX_t = a(X_t,t)dt + b(X_t,t)dB_t$$ where $dB_t$ is a Wiener process. What is the basic reason that the term $a$ is considered as the mean of process $dX_t$ and $b^2$ the variance?
Thanks.
For a diffusion process ${X_t,t\geq0}$ we have
$$ \lim_{h\downarrow 0} \frac{1}{h}\mathbb{E}[X(t+h)-X(t)|X(t)=x] = \mu(x,t) $$
so that $\mu(x,t)$ is an infinitesimal mean of the diffusion process (typically referred to as the drift parameter).
Similarly,
$$ \lim_{h\downarrow 0} \frac{1}{h} \mathbb{E}[(X(t+h)-X(t))^2|X(t)=x] = \sigma^2(x,t) $$
so that $\sigma^2(x,t)$ is the infinitesimal variance of the diffusion process (often referred to as the diffusion parameter).
For the SDE $dX_t=\mu(X_t,t)dt+\sigma(X_t,t)dB_t$ one can verify with some effort that the above conditions are true (but for the case where $\mu(X_t,t)=\mu$ (a constant) and $\sigma(X_t,t)=\sigma$ this is easy to see).