Brownian motion

The continuous-time limit of a random walk, and the object almost everything continuous in stochastic-processes is built on. Take a walk with smaller and smaller steps taken faster and faster, and what survives in the limit is a path that is continuous everywhere, moves at every scale, and has no derivative anywhere.

Built, not assumed

lawler-stochastic-calculus reaches it that way on purpose. Chapter 2 goes limits of sums of independent variables → the multivariate normal → limits of random walks → Brownian motion → a construction of it, because a limit object needs to be shown to exist before it can be integrated against.

He then looks at the same process four ways, and the list is worth keeping because each view is a different subject's handle on it:

• a continuous martingale — the fair-game property, which is what makes stochastic integrals against it behave;
• a Markov process — the future depends on where it is now, linking it to markov-chains;
• a Gaussian process — every finite collection of its values is multivariate normal;
• a self-similar process — rescale time by t and space by √t and it is the same process, the reason √Δt appears everywhere in simulation.

Quadratic variation, and why calculus breaks

The property that separates it from anything in vector-calculus or real-analysis: over [0,t] the sum of squared increments does not vanish as the partition refines. It converges to t. An ordinary differentiable path has quadratic variation zero, which is exactly why the second-order term in a Taylor expansion can be dropped for it and cannot be dropped here. That single fact is the whole content of the extra term in Itô's formula (stochastic-calculus).

Its other lives

The heat equation. lawler-stochastic-calculus follows Brownian motion straight into the heat equation and its generator: the expected value of a function of the process at a future time solves a PDE. The probabilistic object and the analytic one are two descriptions of the same thing, which is what makes the Feynman-Kac formula possible and what puts the option-pricing PDE within reach.

In the plane. His last chapter takes complex Brownian motion into the Dirichlet problem and harmonic functions — the same conformal-mapping territory complex-analysis holds here from the analytic side.

In markets. ../quant-trading-wiki's black-scholes assumes the stock price follows a geometric Brownian motion, and option-pricing there records the lognormal random walk as borrowed machinery. This page is where the machinery lives. stirzaker-elementary-probability arrived in this spoke already making the connection: its second edition added the Wiener process because the option formula rests on it.

Related

stochastic-processes · stochastic-calculus · martingale · markov-chain · probability-theory · lawler-stochastic-calculus · gamarnik-advanced-stochastic-processes · complex-analysis · mathematics · synthesis