Existence/uniqueness of solution and Ito's formula

1. Jun 6, 2013

mbp

Hi everybody,

I have an Ito's stochastic differential equation

$dX_t = a(X_t,t) dt + b(X_t,t) dB_t$

where $a(X_t,t)$ and $b(X_t,t)$ satisfy the Lipschitz condition for existence and uniqueness of solutions.
Given a function $f(X_t,t) \in C^2$ using Ito's formula I can derive the SDE

$df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dX_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} dX_t^2$

where $dX_t^2$ is computed using Ito's lemma.

The question is: are there any requirements that $f(X_t,t)$ must satisfy to guaranty
the existence and uniqueness of solutions (I would say yes).

Any reference is welcome. Thanks in advance.