Discrete analogy

In discrete time, a martingale transform has the form

$$ M_n=\sum_{k=0}^{n-1} Y_k(X_{k+1}-X_k), $$

where $(Y_k)$ is adapted.

Continuous version

For Brownian motion, the analogous object is

$$ M_t=\int_0^t Y_s\,dB_s. $$

Parallel properties

  • In both settings, the transform is a martingale.
  • In both settings, the second moment is controlled by the square of the integrand.

This is why the Ito integral can be viewed as the continuous-time version of a martingale transform.