State the Dambis-Dubins-Schwarz theorem. Use it to give P(sup_t M_t ≥ a) for a continuous local martingale with M_0 = 0 and infinite terminal quadratic variation, and say what changes when that quadratic variation is finite.

State the Dambis-Dubins-Schwarz theorem. Use it to give P(sup_t M_t ≥ a) for a continuous local martingale with M_0 = 0 and infinite terminal quadratic variation, and say what changes when that quadratic variation is finite.

Approach: Represent the martingale as a Brownian motion run on its own quadratic variation clock, then apply the reflection principle to the resulting Brownian path over the total clock time.

The Dambis-Dubins-Schwarz theorem states that a continuous local martingale M with M_0 = 0 and quadratic variation growing to infinity satisfies M_t = B_{[M]_t} for a Brownian motion B, so P(sup_t M_t ≥ a) = 1 for every a when [M]_∞ = ∞. The time change runs B on the quadratic variation clock [M]t, and since that clock reaches every level, the supremum of M equals the supremum of B over all of [0,∞), which is infinite almost surely because Brownian motion is recurrent. When [M]∞ = c is finite the clock stops, sup M is the running maximum of B over [0,c], which is finite, and the reflection principle gives the conditional probability 2(1 - N(a/sqrt c)). Unconditionally the answer is the expectation of that expression over the law of the clock, which requires knowing the joint behaviour of the clock and the path when the two are dependent.

Follow-up: Where does the argument use that the clock is continuous, and what replaces the theorem for a purely discontinuous martingale?

Key concepts: time change, Dambis-Dubins-Schwarz, quadratic variation clock, reflection principle.