Question about Ito Process. Stochastic Processes

Question

How to prove, that $W_{t/(1-t)}$ at $[0,1)$ is Ito Process ? (Have stohastic differential)

Answer 1

You justo have to solve $W_{\frac{t}{1-t}}=\int_0^t b_sdB_s$, if i'm not mistaken with the calculations, you first must get with Ito's isometry that $b_s=1/(1-s)$, then you get $B_t=\int_0^t b_s^{-1}dW_{\frac{s}{1-s}}=\int_0^{\frac{t}{1-t}}\frac{dW_s}{(1+s)^2}$, this is a Brownian Motion because Levy's Characterization Criterion apply. So you can ensure that $$W_{\frac{t}{1-t}}=\int_0^t \frac{dB_s}{1-s},\qquad t\in[0,1).$$