Stochastic Calculus: Consequences of the Itô Formula

Proof. The stochastic differential equation (2) follows directly from the (multivariate) Itô theorem, using the fact that Zt has the form Zt = u(Mt , [M ]t ). Every Itô integral process V ·M is a local martingale provided that M is a local martingale and V 2 L§(M), so for each n <1 the stopped process Z øn is a local martingale, where øn is the first time t such that [M ]t = n, and so it follows (see Lemma 7 of the notes on Continuous and Local Martingales) that Z is a local martingale. Finally, if Mt 2 L2 and [M ]t is a bounded random variable then for any T <1 the stopped process Zt^T is the Itô integral of a bounded, progressively measurable process against a martingale in the class M2, and hence the process Zt is a martingale.