Local Martingales

Stop a process before it misbehaves and it is a martingale; let it run and the mean can leak away. The gap between local and true is a limit interchange — and a bubble.

The idea

A local martingale is a process that can be stopped into a martingale, and the definition exists because the Itô integral is not always one.

Betting a stake $\sigma(s)$ on the next Brownian increment cannot favour the bettor, since the stake is fixed before the increment arrives, so $\int_{0}^{t} \sigma\,dW$ ought to be a fair game with a frozen mean. For a general integrand that fails, and at a precise point: the integral is built path by path, requiring only $\int_{0}^{t} \sigma(s)^{2}\,ds < \infty$ along each path, and a pathwise condition controls no averages. A quantity can be finite on every path while its expectation is infinite, and once the expectation diverges the process is not a martingale.

Discarding such processes would discard most of the Itô integrals in actual use, so we weaken the claim instead.

Definition (Local martingale).

A process $M$ is a local martingale when there are stopping times $\tau_{1} \le \tau_{2} \le \cdots$ with $\tau_{n} \to \infty$ almost surely such that each stopped process $M(t \wedge \tau_{n})$ is a true martingale. Such a sequence is a localising sequence for $M$.

The process need not be fair outright, but stopping can cut it off before it misbehaves — and because the stopping times exhaust all of time, bad behaviour is never hidden permanently, only deferred.

Ways to work on it

• Walkthrough. Why the Itô integral forces the definition, and the hypotheses that make a local martingale a true one.
• Proof. Why a bounded local martingale is a true martingale, and an unbounded counterexample that escapes the argument.
• Practice. Test local martingales for the true martingale property, and read what a failure means.
• Hardest. Decide whether a strictly positive, unbounded local martingale is a true one.

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