[Math] Proving that a process is a Brownian motion

Question

Let $B$ be a Brownian motion with natural filtration $(\mathcal{F}_t)_{t\geq 0}$ and let $\mathcal{H}_t$ be the $\sigma$-algebra generated by $\mathcal{F}_t$ and $B_1$.

Define

$$A_t = B_t-\int_0^{\min(t,1)} \frac{B_1-B_s}{1-s}ds$$

I'm trying to show that $A_t$ is a Brownian motion with respect to $(\mathcal{H_t})_{t\geq0}$.

As a first step, I'm attempting to show that $A_t$ is a martingale, but haven't made much progress.

Thank you.

Answer 1

Look at the quadratic variation of the process.

Regards.