Tanaka's formula
/ tah-NAH-kah /
Ito's formula requires the function to be twice continuously differentiable, but the most natural function on a Brownian path — its absolute value, or distance to a level — has a corner. What is the stochastic differential of |B_t|? Tanaka's formula answers this, extending Ito's formula to f(x) = |x - a| at the cost of a new ingredient: a local-time term that exactly captures the contribution of the kink.
The formula reads: |B_t - a| = |B_0 - a| + integral over [0,t] of sgn(B_s - a) dB_s + L_t^a, where sgn(x) = +1 for x > 0 and -1 for x < 0 (the value at 0 is immaterial since B spends no time there), and L_t^a is local time at level a. Compare ordinary Ito: if f were C^2 we would get f'(B_s) dB_s plus (1/2) f''(B_s) ds. Here f' = sgn is the right first derivative, but f'' is the Dirac delta 2 delta_a in the distributional sense, and the second-order term (1/2) integral 2 delta_a(B_s) ds is precisely the local time L_t^a — Tanaka's formula is the rigorous reading of '(1/2) f''(B) ds = L^a' when f'' is a point mass. The general Ito-Tanaka formula does the same for any convex (or difference-of-convex) f: f(B_t) = f(B_0) + integral f'_-(B_s) dB_s + (1/2) integral over R of L_t^a f''(da), with f'' the (signed) measure of second derivatives.
Why it matters: Tanaka's formula is the standard CONSTRUCTION of local time (as the difference |B_t - a| - |B_0 - a| - stochastic integral, it is automatically continuous and increasing), it shows |B - a| is a semimartingale with the local time as its bounded-variation part, and it underlies skew Brownian motion, the Skorokhod reflection problem, and pricing of payoffs with kinks (the local-time term is the 'gamma at the strike'). The key honesty: you CANNOT apply naive Ito calculus to |B| — doing so silently drops the local-time term and gives a wrong answer. The corner contributes a genuinely new, path-dependent increasing process, and the sign function inside the integral makes integral sgn(B_s) dB_s itself a Brownian motion (Levy's characterisation), so |B| = (a Brownian motion) + L, which is Levy's theorem in disguise.
Apply Tanaka at a = 0: |B_t| = integral over [0,t] of sgn(B_s) dB_s + L_t^0. The stochastic integral has quadratic variation t, so by Levy's characterisation it is itself a Brownian motion; hence |B| is that Brownian motion plus its local time, which recovers Levy's theorem |B| =d M.
The corner of |x| forces an extra local-time term that ordinary Ito calculus would miss.
Never apply C^2 Ito to |B|: the kink contributes the local-time term L_t^a, which the smooth formula omits and which is genuinely path-dependent.