Consider the process $$Z(t)=\int_{0}^{t} \frac{u^a}{t^a}dW_u$$ for some real constant $a$ and $W_t$ is a wiener process. I want to check whether this process is a $F_t^W$-martingale. I noticed Lemma 4.9 in Bjork's book which states the following:
for $g\in L^2$ and $X_t$ defined as $$X_t=\int_{0}^{t} g(u) dW_u$$ is a $F_t^W$-martingale.
If we let $g(z,t)=z^at^{-a}$; Can we then use the lemma to conclude that $Z(t)=\int_{0}^{t} g(u,t)dW_u$ is $F_t^W$-martingale?
And that leads me another question: How do I test if a function $g \in L^2$
With Bjork I referto the textbook Bjork, Arbitrage Theory in Continous Time Finance (3rd Edition)