For example, $$ dS(t) = \alpha S(t) dt + \sigma S(t) dW(t) $$ $$ dR(t) = (\alpha -\beta R(t))dt + \sigma dW(t) $$ $$ dR(t) = (\alpha -\beta R(t))dt + \sigma \sqrt{R(t)} dW(t) $$ There is always a Brownian motion / Wiener process ,"W(t)".
It looks like the standard Brownian motion "W(t)" is the atom of all kinds of stochastic process that Stochastic-Analysis studied. Why?
Are there any Theorems saying "All kinds of stochastic process could convert to a function of standard Brownian motion ,a.s., existing and unique."?
I know in the "Jump" circumstance, Poisson process is included. However, the kinds of "atom" process are still very rare.
Any hints or recommendation of papers or book chapters or web sites is of great grateful.Thanks.