# How to deal with differential in Itô

Suppose I have two Brownian Motion $W$ and $B$ which are connected through Girsanov, i.e. $W_t=B_t-\int_0^t v(u,T)du$. Furthermore I have the following expression

$$\exp{(\int_0^tv(u,T)-v(u,S) dB_u)}$$

If I want to write this with respect to dynamics of $W$, I get $dB_u=dW_u+v(u,T)$. But how should I plug this in?

$$\exp{(\int_0^tv(u,T)-v(u,S) dB_u)}=\exp{(\int_0^tv(u,T)-v(u,S)dW_u+v(u,T))}$$

seems not be correct. Thanks for your help.

-

I think, the only thing which is needed here is to remember that you get $$\mathrm dB_u=\mathrm dW_u+v(u,T)\color{red}{\mathrm du}$$