Give an explicit strictly positive local martingale that fails to be a true martingale, prove the failure by computing its expectation, and say what this implies for a price process modelled that way.

Give an explicit strictly positive local martingale that fails to be a true martingale, prove the failure by computing its expectation, and say what this implies for a price process modelled that way.

Approach: Take the reciprocal of the radial part of a three dimensional Brownian motion, apply Ito to see the drift vanish, then evaluate its expectation directly from the known radial density.

X_t = 1/|B_t| for a three dimensional Brownian motion B started at a point of norm 1 is a strictly positive local martingale with E[X_t] = 2 N(1/sqrt t) - 1 < 1 = X_0 for every t > 0. The radial part R_t = |B_t| is a Bessel process of dimension 3 with dR = d beta + dt/R, and applying Ito's lemma to f(r) = 1/r gives dX = -R^{-2} dR + R^{-3} (dR)^2 = -X^2 d beta, so the drift cancels exactly and X is a local martingale. A positive local martingale is a supermartingale, so its expectation can only fall, and here the direct computation from the radial density shows a strict decrease that starts immediately and sends E[X_t] to 0. A discounted price modelled by such a process admits no equivalent martingale measure making it a true martingale, and the gap X_0 - E[X_t] is the standard quantitative description of a bubble in that model.

Follow-up: Which measure change has this X as its density process, and where does the missing mass sit?

Key concepts: local martingale, Bessel process, supermartingale, Ito's lemma.