Let $(B_t)$ be a one-dimensional Brownian motion and $y \in \mathbb{R}$. Show that the solution to the SDE $$dX_t^y=dB_t + \frac{y-X_t^y}{1-t}dt$$ with initial value $X_0^y = 0$ on $[0,1)$ is given by $$X_t^y = yt+(1-t)\int_0^t \frac{1}{1-s}dB_s.$$
Ok, so I am new to the SDE theory and I do not know yet, how I need to proceed here. I need the Ito formula for sure, but applied to which function?!
I would appreciate any help!
Thanks in advance!
Best Answer
Hint: use integrating factor technique after shifting the X from the right to the left. Then you will get a full differential on the left (after applying the integrating factor technique). Integrate both sides from $0$ to $t$ afterwards