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?

Answer 1

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.