existence/uniqueness of solution and Ito's formula Given the Ito SDE 
$$
  dX_t=a(X_t,t)dt + b(X_t,t) dB_t
$$
where $a(X_t,t)$ and $ b(X_t,t)$ satisfy the Lipschitz condition for existence and uniqueness of solutions.
Given a function $f(X_t,t) ∈ C^2$ 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.
 A: You shall distinguish between the equation and the 'direct' definition of something. For example, whenever you have an expression of the form 
$$
  x = g(x) \tag{1}
$$ 
where $g$ is a certain function/operator that is given to you, you may be asked to find $x$ which satisfies such expression. You can never be sure whether such $x$ exists, or whether there is only one such $x$. Indeed, if you change the $x$ in the RHS, the LHS changes as well since it depends on $x$. Anyway, suppose we found such $x$ and it is unique.
Now, imagine that you also have an expression
$$
  y = h(x) \tag{2}.
$$
This is not an equation, but rather a definition of $y$. Indeed, to find the value of $y$ the only thing we need to do is to substitute $x$ (found on the previous step) as an argument of $h$ - that's it.
What do you have in your original post (OP) is that $x$ is the process $(X_t)_{t\geq 0}$ which satisfies
$$
  X_T = X_0 + \int_0^Ta(X_t,t)\mathrm dt + \int_0^Tb(X_t,t)\mathrm dB_t \tag{$1^\prime$}
$$
which can be compactly written through differentials as in your case. Here the operator $g$ as in $(1)$ is just these integrals and functions $a,b$ applied to $X$. Again, the LHS and the RHS both contain $X$, so that it's not obvious whether there exists such $X$ which makes both sides be equal. Thus, we need conditions on $a$ and $b$ to assure such existence, or uniqueness. 
Now, let $Y_t = f(X_t,t)$ be another process. For $f\in C^2$ we can use an alternative definition of $Y$:
$$
  Y_T = Y_0 + \int_0^T\frac{\partial f}{\partial t}(X_t,t)\mathrm dt + \int_0^T \frac{\partial f}{\partial X}(X_t,t)\mathrm dX_t + \int_0^T \frac{\partial^2 f}{\partial X^2}(X_t,t)\mathrm dX_t^2 \tag{$2^\prime$}
$$
which yet again can be written in a compact symbolic form as via differentials as you did. Here you can think of the RHS as a function of the process $X$: $h((X_t)_{t\geq 0})$. It is only used to define $Y$. The only thing that you need to take care of is that $h$ is defined for a particular value of the argument, that is all the integrals in the RHS of $(2')$ are well-defined. Actually, here the only Ito integral is the middle one (a part of $\mathrm dX_t$) and you indeed shall check that 
$$
  Z_t:=b(X_t,t)\frac{\partial f}{\partial X}(X_t,t)
$$
satisfies for example $(iii)$ in Definition 3.1.4 or $(iii)'$ in Definition 3.3.2 of Oksendal: "Stochastic Differential Equations".
As a result, you actually shall not talk about solutions of $(2')$ as much as you don't talk about the solutions of $y = 3$. Instead, you talk about solutions of $x = x^2+1$.
