Why integrate against martingales rather than ordinary functions?

Martingale paths are too irregular to integrate in the ordinary sense, but their controlled fluctuation, measured by quadratic variation, allows a probabilistic integral that is itself a martingale and underlies stochastic calculus.

What is quadratic variation?

It is the limit of summed squared increments of a process over finer partitions; for martingale paths it is generally nonzero and acts as the natural variance clock for stochastic integration.

Martingales and Stochastic Integration

Continuous-time martingales, with their quadratic variation and decomposition into predictable and martingale parts, are the integrators against which stochastic integrals are built.

Definition

In continuous time a martingale is a process whose conditional expected increments vanish; its quadratic variation measures accumulated fluctuation, the Doob-Meyer decomposition splits submartingales into a predictable increasing part and a martingale, and these structures define stochastic integration against semimartingales.

Scope

This topic covers continuous-time martingales and local martingales, the Doob-Meyer decomposition of submartingales, the quadratic variation and bracket process, semimartingales as the largest natural class of integrators, the construction of the stochastic integral against a martingale, and the martingale representation theorem expressing Brownian martingales as stochastic integrals.

Core questions

How do continuous-time martingales and local martingales generalise the discrete case?
What is quadratic variation and why is it central to stochastic integration?
How does the Doob-Meyer decomposition identify the martingale part of a process?
Why are semimartingales the natural class of integrators, and what does martingale representation give?

Key theories

Doob-Meyer decomposition and quadratic variation: A submartingale decomposes uniquely into a local martingale plus a predictable increasing process, and the quadratic variation of a continuous local martingale is the predictable process whose subtraction makes its square a martingale, supplying the variance measure for stochastic integrals.
Stochastic integral and martingale representation: The stochastic integral of a predictable process against a square-integrable martingale is itself a martingale with computable quadratic variation, and the martingale representation theorem shows every Brownian martingale is such an integral, the basis of hedging in finance.

Clinical relevance

Martingale-based stochastic integration is the mathematical foundation of the Ito integral and stochastic differential equations, of filtering theory, and of arbitrage-free pricing and hedging in mathematical finance, where the martingale representation theorem yields replicating strategies for derivative securities.

History

Doob conjectured the decomposition that Meyer proved in 1962, the Strasbourg school led by Meyer developed the general theory of semimartingales and stochastic integration in the 1960s and 1970s, and Kunita and Watanabe's work on square-integrable martingales unified the integral against general martingale integrators.

Key figures

Joseph Doob
Paul-Andre Meyer
Kiyosi Ito
Hiroshi Kunita

Seminal works

karatzasShreve1991

Frequently asked questions

Why integrate against martingales rather than ordinary functions?: Martingale paths are too irregular to integrate in the ordinary sense, but their controlled fluctuation, measured by quadratic variation, allows a probabilistic integral that is itself a martingale and underlies stochastic calculus.
What is quadratic variation?: It is the limit of summed squared increments of a process over finer partitions; for martingale paths it is generally nonzero and acts as the natural variance clock for stochastic integration.