Test if a process (with no drift) is a martingale

Question

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)

Answer 1

the Lemma from Bjoerk's book is only valid for functions $g(u)$ not for function's $g(u,t)$.

What you can do instead is to use Ito's formula. First let's start with the process

$$ X_t = \int_{0}^t u^a \; dW_u. $$

This process can be written in differntial form as $$ dX_t = t^a dW_t. $$

Now we want to derive the differential form of $Z_t = \frac 1 {t^a}X_t$. Therefore we can use Ito's formula with $f(t,x) = \frac 1 {t^a}x$: $$ dZ_t = df(t,X_t) = f_t(t,X_t)dt + f_x(t,X_t)dX_t + \frac 12 f_{xx}(t,X_t) d[X]_t $$ Here $f_t$ resp. $f_x$ is the partial derivative of $f$ with respect to $t$ resp. $x$, and $f_{xx}$ is the second partial derivative of $f$ with respect to $x$. Note that we have $f_{xx}(t,x) = 0$, such that the differential form of $Z$ can be simplified to $$ dZ_t = - \frac a {t^{a+1}}X_t dt + \frac{1}{t^a} t^a dW_t = - \frac at Z_t dt + dW_t. $$ For a Ito process to be a (local) martingale the drift part has to be equal to zero. Therefore, $Z_t$ is a martingale if and only if $a = 0$ in which case $Z_t = W_t$.