For $W_t$ a BM, I want to show that $X_t=t\cdot W_{1/t}$ is a martingale using Ito.
As shown in
$\quad$ https://math.stackexchange.com/questions/182107
$\quad$ https://math.stackexchange.com/questions/1358542
we know $X_t$ is a BM, and in particular a martingale.
However, I was wondering if it was possible to show the martingale feature more "directly" by showing that the resulting SDE of $X_t$ from using Ito has a zero drift term, which tells us that we have a local martingale. Ensuring appropriate integrability assumptions we arrive at a proper martingale. Similarly as in
$\quad$ https://math.stackexchange.com/questions/525390
$\quad$ https://quant.stackexchange.com/questions/2435
Having $f(t,Y_t)$ Ito tells us that
$$df(t,Y_t) = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial y}dY_t + \frac12\frac{\partial^2 f}{\partial y^2}d\langle Y\rangle_t$$
But now I am unsure how to populate $Y_t$ with $W_{1/t}$. If we had $X_t=t\cdot W_t$ instead we could easily set $f(t,x) = t\cdot x$ and $Y_t=W_t$. But I am unsure how to treat $W_{1/t}$ in this context.
Any help would be appreciated.
