I've been stuck with the question how to find a measure to make a discounted price a martingale. I cannot use Girsanov because I am only given the SDE for which an unique strong solution exists but not known. So, I have $dX_t = (\frac{1}{X_t} - \frac{X_t}{T-t} ) dt + dW_t$, $0<t<T$, $X_0 = a$ and $W_t$ a standard Brownian motion. Now I want to find a measure to make $(e^{-rt} X_t)_t$ a martingale, at least a local martingale up to time T. Do I have to use itos lemma or any suggestions? If I use Itos lemma with $f(t,X_t) = e^{-rt} X_t$ then $df(t,X_t) = (-r + \frac{1}{X_t} - \frac{X_t}{T-t}) e^{-rt} dt + e^{-rt} dW_t$, hence, it is only a martingale if $\frac{1}{X_t} - \frac{X_t}{T-t} = r$, but under what measure und how do i change the measure to make it a martingale?
thanks in advance!