I want to write down an expression for $dX_t$ for both:
i. $X_t=t^2W_t^2-2\int_0^t(sW_s^2+s^2)ds$; and
ii. $X_t=W_t^2-tW_t$
What is the process I would use for differentiating these stochastic equations?
Also how can I tell which of these are $\mathcal{F} $ - martingales on [0,T] where ${\mathcal{F_t}} $ is the natural filtration generated by $W_t$ ?
I believe we can assume that $\int_o^T\mathrm{E_p}[|W_t|^n]dt<\infty$ for all $n>0$
Please let me know if you have any advice or solution, would be greatly appreciated, thanks.
Let's start with the first one; I leave the other one to you. First of all, it follows from the definition of $dX_t$ that
$$dX_t = d(t^2 W_t^2) -2 (t W_t^2+t^2) \, dt$$
i.e. we only have to calculate $d(t^2 W_t^2)$. By applying Itô's formula to $f(t,x) := t^2 \cdot x^2$ we see that
$$d(t^2 W_t^2) = 2 t^2 W_t \, dW_t + ( t^2 + 2t W_t^2) \, dt.$$
Hence,
$$dX_t = 2t^2 W_t \, dW_t - t^2 \, dt,$$
i.e.
$$X_t - X_0 = 2 \int_0^t s^2 W_s \, dW_s - \frac{t^3}{3}.$$
In order to decide whether $(X_t)_{t \geq 0}$ is a martingale we recall that any stochastic integral of the form $\int_0^t f(s) \, dW_s$ is a martingale. This means that
$$M_t := 2\int_0^t s^2 W_s \, dW_s$$
is a martingale. Since
$$X_t-X_0 - M_t = - \frac{t^3}{3}$$
is obviously not a martingale, this already implies that $(X_t)_{t \geq 0}$ is not a martingale.
Remark Note that the integrability condition $\int_0^T \mathbb{E}(|W_t|^n) \, dt<\infty$ is nothing "we can assume", we have to check it!