continuous-time stochastic process

Question

Give an example of a continuous-time stochastic process (Xt)t≥0 which is not a martingale, but such that E[Xt] = 0 for all t ≥ 0. Hint: consider f(Wt), for a well-chosen function f (where (Wt)t≥0 is, as usual, a standard Brownian Motion).

Answer 1

Take $f(x)=x^{3}$. To show that $f(W_t)$ is not a martingale let us compute $E(W_{t+s}^{3}|\mathcal F_t)$: We can write $W_{t+s}^{3}=(W_{t+s}-W_t)^{3}+3(W_{t+s}-W_t)^{2}W_t+3(W_{t+s}-W_t)W_t^{2}+W_t^{3}$. Hence $E(W_{t+s}^{3}|\mathcal F_t)=0+3sW_t+0+W_t^{3} \neq W_t^{3}$.