Daniel, have a look to Kunita book: "Stochastic Flows and Stochastic Differential Equations". where you can find formulations like this one:

Let X be a diffusion process, strong solution to the following SDE

[$] dX_t = a(\theta, X_t) dt + b(\theta, X_t) dW_t, ~~X_0 = x [$]

Then [$] Y_t = \frac{\partial X_t}{\partial \theta} [$] is defined as the solution of the following SDE:

[$] dY_t = [a'_{\theta}(\theta, X_t) + a'_{x}(\theta, X_t) Y_t ]dt + [b'_{\theta}(\theta, X_t) + b'_{x}(\theta, X_t)Y_t ]dW_t, [$]

[$] Y_0 = \frac{\partial X_0}{\partial \theta} [$]

where the primes denote standard derivatives.

Have a look also to Giles work on Vibrato.