Explain why Ito's lemma cannot be applied directly to |W_t|, state Tanaka's formula, and use it to compute E[L_t^0] where L^0 is local time at zero.

Explain why Ito's lemma cannot be applied directly to |W_t|, state Tanaka's formula, and use it to compute E[L_t^0] where L^0 is local time at zero.

Approach: Note the smoothness requirement of Ito's lemma at the origin, then take expectations in Tanaka's identity so that the stochastic integral vanishes and only the absolute value remains.

Tanaka's formula reads |W_t| = ∫_0^t sgn(W_s) dW_s + L_t^0, and taking expectations gives E[L_t^0] = E|W_t| = sqrt(2t/pi). Ito's lemma requires the function to be twice continuously differentiable, and the absolute value has no second derivative at zero: formally f'' is a point mass there, which is not a function that can be integrated against ds along a path. Local time is exactly the object that replaces it, and it is also the occupation time density, the limit of (1/(2 eps)) times the Lebesgue measure of {s ≤ t : |W_s| ≤ eps}. Since sgn(W) is bounded and adapted, the stochastic integral is a square integrable martingale with mean zero, so E[L_t^0] = E|W_t|, and for a centred Gaussian with variance t that expectation is sqrt(2t/pi), which is 0.7979 sqrt(t). Local time is continuous and non-decreasing in t and grows only on the zero set, which has Lebesgue measure zero.

Follow-up: What static option position replicates a contract paying the local time accumulated at a strike?

Key concepts: Tanaka's formula, local time, occupation time, second derivative.