Which of the following are martingales with respect to the Brownian filtration: W_t^2 - t, W_t^3 - 3 t W_t, exp(W_t - t/2), t^2 W_t, W_t^3? Justify each with Ito.

Which of the following are martingales with respect to the Brownian filtration: W_t^2 - t, W_t^3 - 3 t W_t, exp(W_t - t/2), t^2 W_t, W_t^3? Justify each with Ito.

Approach: Write the stochastic differential of each candidate and keep only those whose dt coefficient vanishes identically, checking integrability for the ones that pass.

W_t^2 - t, W_t^3 - 3 t W_t and exp(W_t - t/2) are martingales; t^2 W_t and W_t^3 are not. Applying Ito's lemma: d(W^2 - t) = 2W dW + dt - dt = 2W dW, so the drift term is zero. d(W^3 - 3tW) = 3W^2 dW + 3W dt - 3W dt - 3t dW = (3W^2 - 3t) dW, again driftless. For the exponential, d(exp(W - t/2)) = exp(W - t/2)(dW + dt/2 - dt/2) = exp(W - t/2) dW, the standard exponential martingale. For t^2 W_t, Ito gives 2t W dt + t^2 dW, whose drift is nonzero, and for W^3 it gives 3W^2 dW + 3W dt, also nonzero. Each surviving integrand is square integrable on finite horizons, so the local martingales are true martingales there.

Follow-up: Which polynomial p(t,x) of degree four in x makes p(t, W_t) a martingale, and what recursion do these Hermite polynomials satisfy?

Key concepts: Ito's lemma, drift term, martingale, exponential martingale.