Martingale Convergence and Square-Integrable Martingales

Risk-neutral pricing rests on the assumption that discounted price processes are martingales. But for a Monte Carlo estimator to converge, or for the continuous-time theory to produce a well-defined limit, we need more than a martingale — we need it to converge. The martingale convergence theorem says that any martingale whose absolute values are uniformly bounded in expectation has an almost-sure limit. This result is what allows us to pass from a sequence of conditional expectations $V_t = \mathbb{E}^{\mathbb{Q}}[V_T \mid \mathcal{F}_t]$ to a well-defined terminal random variable as the time grid refines to zero.

But a.s. convergence is only the first half of the story. For integration theory and approximations, we need convergence in the $L^2$ sense — and this requires a stricter $L^2$ bound on the martingale. The theory of square-integrable martingales reveals an elegant structure: their increments are orthogonal in $L^2$, giving rise to a Pythagorean identity for the second moment. The quantity that tracks this growth — the predictable quadratic variation $\langle M \rangle_n$ — is the discrete version of the bracket process $[W, W]_t = t$ that drives Itô's formula.

Learning objectives. By the end of this note you should be able to: