Local Martingales

A local martingale is a process that "looks like a martingale locally" — it becomes a martingale when stopped at appropriate times. Local martingales arise naturally as Itô integrals with unbounded integrands.

Definition

Definition5.1Local martingale

A process $(M_t)$ is a local martingale if there exists a sequence of stopping times $\tau_n \uparrow \infty$ a.s. such that the stopped processes $(M_{t \wedge \tau_n})$ are martingales for each $n$ .

Every martingale is a local martingale (take $\tau_n = n$ ), but the converse is false.

ExampleLocal martingale that is not a martingale

Let $M_t = \int_0^t B_s \, dB_s = \frac{1}{2}B_t^2 - \frac{1}{2}t$ . Then $(M_t)$ is a local martingale (stop at $\tau_n = \inf\{t : |B_t| \geq n\}$ ), but $\mathbb{E}[M_t] = -t/2 \neq 0 = M_0$ , so it is not a martingale. The issue: the integrand $B_s$ is not square-integrable uniformly over all paths.

Quadratic variation

Definition5.2Quadratic variation of local martingale

For a continuous local martingale $(M_t)$ , the quadratic variation $[M]_t$ is the unique increasing process such that $M_t^2 - [M]_t$ is a local martingale.

For an Itô integral $M_t = \int_0^t H_s \, dB_s$ , we have $[M]_t = \int_0^t H_s^2 \, ds$ .

Summary

Local martingales extend the martingale concept to processes that are martingales "up to stopping times." They are the natural outputs of stochastic integration.