Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2008 / A / 2 / b

Itô's formula identifies the infinitesimal generator on twice continuously differentiable functions:Subtract the drift coefficient integral from that formula to obtainStopping when $|X|$ leaves a bounded interval makes the integrand bounded on each finite time interval. Therefore $M^f$ is a continuous local martingale. In fact $f^{\prime }\in L^{\infty }$ and the lognormal distribution solution has finite second moments on finite intervals, so here it is square integrable on each such interval as well.

The quadratic covariation of two stochastic integrals against the same Brownian motion is the integral of the product of their integrands. ConsequentlyThis is the quadratic covariation, which agrees with the pathwise quadratic covariation for these continuous local martingales.