When $X_t = {W_t}^n - k\int^{t}_{0} {W_s}^{n-2} \, ds$ is a martingale?

Question

I know that $X_t = {W_t}^3 - 3\int^{t}_{0} W_s \, ds$ is a martingale, but my general question is: for what values of $k$, $X_t = {W_t}^n - k\int^{t}_{0} {W_s}^{n-2} \, ds$ is a martingale?

Itô formula allows you to answer this question. – Surb Mar 06 '21 at 22:19 — Surb, Mar 06 '21 at 22:19
@Surb, how can I use ito for this question? – Moh514 Mar 07 '21 at 15:02 — Moh514, Mar 07 '21 at 15:02

score 1 · Answer 1 · answered Mar 07 '21 at 15:37

Hint

You wish to find a function $f\in \mathcal C^2(\mathbb R)$ s.t. $$X_t=\int_0^t f(W_s)\,\mathrm d W_s=W_t^n-k\int_0^t W_s^{n-2}\,\mathrm d s.$$

According to Itô formula $$\int_0^tf(W_s)\,\mathrm d W_s=f(W_t)-\frac{1}{2}\int_0^t f''(W_s)\,\mathrm d s.$$ I let you conclude.

When $X_t = {W_t}^n - k\int^{t}_{0} {W_s}^{n-2} \, ds$ is a martingale?

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