Write the stochastic differential of W_t^3 and of W_t^4 for standard Brownian motion, and use them to derive E[W_t^3] and E[W_t^4] without quoting the Gaussian moment formula.

Write the stochastic differential of W_t^3 and of W_t^4 for standard Brownian motion, and use them to derive E[W_t^3] and E[W_t^4] without quoting the Gaussian moment formula.

Approach: Apply Ito to f(x) = x^3 and to f(x) = x^4, take expectations so that the stochastic integral drops out, and solve the resulting ordinary differential equations for the moments.

d(W_t^3) = 3 W_t^2 dW_t + 3 W_t dt and d(W_t^4) = 4 W_t^3 dW_t + 6 W_t^2 dt, from which E[W_t^3] = 0 and E[W_t^4] = 3t^2. Ito's lemma gives df = f'(W) dW + (1/2) f''(W) dt, so the Ito correction term is (1/2)(6W) dt = 3 W dt for the cube and (1/2)(12 W^2) dt = 6 W^2 dt for the fourth power. The stochastic integrals are square integrable martingales on any finite horizon, so they vanish in expectation and the martingale property leaves ordinary differential equations for the moments of Brownian motion. For m_3(t) = E[W_t^3] this reads m_3'(t) = 3 E[W_t] = 0 with m_3(0) = 0, so m_3 = 0. For m_4 it reads m_4'(t) = 6 E[W_t^2] = 6t, so m_4(t) = 3t^2.

Follow-up: What does the same recursion give for E[W_t^{2n}], and why does the double factorial (2n-1)!! appear?

Key concepts: Ito's lemma, Ito correction term, moments of Brownian motion, martingale property.