I think my statement is wrong, but cannot find exactly in which point I am confused. Please forgive my sloppy notations.
There is a SDE : $ dX_{t}=\mu(X_{t})dt+\sigma(X_{t})dW_{t}$
where $\mu$ and $\sigma$ are both uniformly Lipschitz continuous and an initial condition $x_{0}$
Then according to Ito’s uniqueness/existence theorem, there is a unique strong solution to the SDE given by:
$X_{t}=x_{0}+ \int_{0}^{t}\mu(X_{s})ds+\int_{0}^{t}\sigma(X_{s})ds$ almost surely.
Suppose I consider two stochastic processes following the same SDE and filtration with only difference in their initial condition. Then according to the theorem, is the following statement true?
For any given $t>0$,
$X_{A,t}-X_{B,t}=x_{A,0}+ \int_{0}^{t}\mu(X_{s})ds+\int_{0}^{t}\sigma(X_{s})ds-(x_{B,0}+ \int_{0}^{t}\mu(X_{s})ds+\int_{0}^{t}\sigma(X_{s})ds)$ $X_{A,t}-X_{B,t} =x_{A,0}-x_{b,0}$ almost surely.
I think this must be wrong, as this statement can be easily rejected in a simple ODE.
Thanks