Martingales

Abstract

Let B be a Brownian motion. We shall study integrals with integrator B — say $$${X_t} = \int_0^t {{B_s}} d{B_s}$$$. This integral is a new stochastic process. Actually, Itô calculus implies $$${X_t} = \frac{1}{2}\left( {B_t^2 - t} \right)$$$. This process in turn might appear as an integrator — say in $$${Y_t} = \int_0^t {{B_s}d{X_s}} $$$ — and so on. Thus it is desirable to develop stochastic integration for a class of integrators rich enough to contain processes like X or Y. The appropriate class to start with is the class of (local) martingales.