I have to show that for $0 \leq s <t \leq 1$ the random variable $X_t = \int_0^t (B_t - B_s)/(1-s) ds$ is well defined ($B$ is Brownian motion, though I'm not sure if that's important).
I have been given that for all $\alpha \in (0,1/2)$ there is a random variable $H_{\alpha}$ with $E[|H_{\alpha}|]< \infty$ such that $|B_t - B_s | \leq H_{\alpha}|s-t|^{\alpha}$.
I'm not really sure what I'm meant to show here, I don't think its asking me to show that a strong solution exists, but what else should I show? Am I meant to show something about $E[\int_0^t (B_t - B_s)/(1-s) ds]$? If so, how does this relate to the integral being well-defined?
Once I've done this, I need to show $B_t - X_t$ is a martingale with respect to the filtration $\mathcal{F}_t = \sigma(\sigma (B_t), \sigma(B_1))$. I have shown that for $0 \leq s <t <1$, $E[B_t | \mathcal{F}_S] = \frac{t-s}{1-s}(B_1 - B_s) + B_s$, and also that $E[W_{s} | W_t] = (s/t)W_t$, but this hasn't gotten me anywhere.