Let $W_t$ be a Wiener process. Let $X_t = W_t^2 − t$. Show that {${X_t
;t ≥ 0}$} is a martingale with respect to the natural filtration.
To prove that it is in fact a Martingale I must prove 2 properties:
1) $\mathbb E [|X_t|] < ∞ $
2) $\mathbb E [X_t | F_s] = X_s$ for $ 0 ≤ s ≤ t$
For the first part I have: $$\mathbb E [|X_t|] = \mathbb E [|W_t^2 -t|] ≥ t \mathbb E [W_t^2] +t =2t $$
Although I am not sure if this is correct?
For the second part I am unsure what the answer is, if anyone can help, I'd be very grateful.
Best Answer
For the first part, $X_t$ is just a sum of a constant and an integrable random variable, so it is integrable. For the second:
$E[ W_t ^2 - W_s ^2 | F_s ] = E[(W_t - W_s)(W_t + W_s) | F_s]= E[(W_t - W_s)W_t | F_s] + E[(W_t - W_s)W_s | F_s] $
The right summand equals zero, since $W_s$ is a martingale, and since $W_s$ goes outside of the expectation, because it is $F_s$-measurable. For the other summand: $=E[(W_t - W_s)(W_t - W_s + W_s) | F_s]= E[(W_t - W_s)^2 | F_s] +0= E[(W_t - W_s)^2]= \text{Var}(W_t - W_s)= t -s $, where we passed from conditional to simple expectation, because $W_t - W_s$ is independent from $F_s$.
Hence $E[ W_t ^2 - W_s ^2 | F_s] = t-s$ , as desired.