Stochastic processes
Probability with time in it. A stochastic process is a family of random variables indexed by time, and the question changes accordingly: probability-theory asks what the distribution of an outcome is, this subject asks what a sequence of outcomes does — where it goes, whether it settles, when it first hits something, what it looks like in the limit.
The subject opened here on 2026-08-16 with three sources, at the curator's request, framed as "the main field of mathematical finance." The framing is right about the destination and understates the field: two of the three sources end at option pricing, and the third never goes near it.
The objects
- markov-chain — the process whose future depends on its present and not on how it got there. Finite and countable state spaces, stationary distributions, hitting times, mixing.
- martingale — the process whose expected next value, given everything so far, is its current value: a fair game. Carries optional stopping and the convergence theorems.
- brownian-motion — the continuous-time limit of a random walk, and the object the continuous theory is built on.
- stochastic-calculus — integration and differentiation along such a path, where the ordinary rules break and Itô's formula replaces them.
- Poisson and Lévy processes — the jump-driven side: events arriving at random times rather than a path that never leaves the page. wu-introduction-stochastic-processes reaches the Poisson process and continuous-time chains; lawler-stochastic-calculus devotes a chapter to Lévy processes, compound Poisson and the Lévy-Khinchin characterisation.
Three entrances, one destination
The most useful thing the three sources say together is about the shape of the subject rather than its content, because they teach it in three incompatible orders.
| Source | Starts from | Ends at |
|---|---|---|
| wu-introduction-stochastic-processes (MIT 18.445) | finite Markov chains | continuous-time chains, generators |
| lawler-stochastic-calculus (Lawler) | discrete-time martingales | Black-Scholes, Lévy processes, harmonic functions |
| gamarnik-advanced-stochastic-processes (MIT 15.070J) | metric spaces, large deviations | Itô calculus, financial economics |
Markov chains are the whole first half of one course and not a lecture topic in another. Brownian motion is the sixth lecture in one, chapter two of a book, and entirely absent from the third. What they agree on is the summit: two of the three end at Itô calculus and its application to pricing, and the third ends at the generator of a continuous-time chain, which is the same operator seen from the discrete side.
A fourth source arrived 2026-09-05 and it does not fit the table, because it is not a course. dynkin-yushkevich-markov-processes (Dynkin & Yushkevich, Moscow University, 1969) grew out of lectures and was then reorganised by problem area: recurrence, the probabilistic solution of equations, optimal stopping, boundary conditions. Its direction is the reverse of the three above. They build analytic machinery and point it at probability; this one proves the Dirichlet problem has a solution by probabilistic argument, and derives boundary conditions for differential equations from the paths of a birth-and-death process. There is no finance in it at all.
So the summit is not the only summit. Two of the four sources here climb toward Itô calculus and option pricing, one toward the generator of a continuous-time chain, and one leaves the subject entirely and lands in analysis — which is where the field started, decades before the general theory of Markov processes existed.
This is the spoke's second clear case of the pattern carroll-geometric-function-theory named for complex analysis — consensus about a subject's core, none about the route in.
Where the boundary sits
The maths is the subject here. ../quant-trading-wiki owns the claims about markets that this
machinery is used to make: black-scholes as a finance paper, option-pricing as a trading
concept, dealer-gamma-exposure as an empirical finding about hedging flow. What that spoke uses
and does not teach — geometric Brownian motion, the self-financing portfolio, the martingale
measure — is stochastic-calculus here.
The relationship with ../machine-learning-wiki is thinner than it looks: its
monte-carlo-methods page samples from distributions, and the Markov-chain machinery that usually
carries that name is in neither spoke (markov-chain records the gap).
../optimization-algorithms-wiki is the closer user — its metaheuristics are random walks with a
fitness function attached.
What is missing
No source here treats queueing or renewal theory, both named as applications by gamarnik-advanced-stochastic-processes and taught by neither text. Nothing covers stochastic control or filtering. And no source here is a measure theory text: lawler-stochastic-calculus indents that layer as optional reading, gamarnik-advanced-stochastic-processes folds measure-theoretic probability into its own lectures starting from metric spaces, and wu-introduction-stochastic-processes asks only for basic probability and matrix operations. The foundation is assumed, taught in passing, or avoided.
Related
markov-chain · martingale · brownian-motion · stochastic-calculus · probability-theory · mathematical-statistics · lawler-stochastic-calculus · gamarnik-advanced-stochastic-processes · wu-introduction-stochastic-processes · mathematics · synthesis