SYSTEMATIC MATHEMATICS

Stochastic Processes & Stochastic Calculus

How can a changing system be modeled when its next movement is genuinely uncertain? Across 41 developed chapters in eight content-sized units of lengths 4, 5, 5, 5, 4, 7, 4, and 7, begin by separating one observed path from the probability law of all possible paths, then read ensemble mean, covariance, and stationarity without treating a long-looking curve as proof. Build information through filtrations, predict with conditional expectation, and study step-by-step Markov chains before comparing Poisson arrivals with more general renewal processes and continuous-time chains. Martingales then formalize fair conditional change. Brownian motion supplies a continuous random driver, quadratic variation explains why ordinary calculus fails, and the Itô integral, Itô formula, Itô–Stratonovich conversion, and a proved constant-drift Girsanov change replace ordinary pathwise rules. The final units solve and approximate stochastic differential equations, distinguish forward density evolution from backward payoff equations, turn exit times into boundary-value problems through Dynkin’s formula, connect diffusions to PDEs, and report simulation error without treating one path as proof.

Before this course: Probability & Statistics; Measure & Lebesgue Integration through conditional expectation and L² convergence; Differential Equations & Dynamical Systems; Partial Differential Equations; Linear Algebra. Real Analysis is inherited through measure theory.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

A stochastic process is a family, not one noisy curve

Objective: What is the difference between a sample path and the random variable X_t?

A weather trace, queue length, share price, or particle position changes with time, but before observation its future is not one fixed curve. A stochastic process packages every possible time trace together with probabilities. Looking across time on one outcome gives a sample path; fixing a time and looking across all outcomes gives an ordinary random variable.

Write X_t(ω) with two inputs: t selects the time and ω selects the realized outcome. Finite-dimensional distributions describe the joint law at finitely many chosen times; they are stronger than separate one-time histograms because they retain dependence across time. Name the probability space, the information currently available, the time index, and every integrability or regularity assumption before manipulating a formula. A stochastic statement is about a family of random outcomes, so one simulated path can illustrate the statement but cannot prove its distribution, expectation, convergence, or error.

The first boundary to test is: Matching every one-time distribution does not determine temporal dependence: independent coin tosses and one coin copied forever can each be Bernoulli(1/2) at every time. This is not a footnote. It identifies the earliest assumption whose loss can change the answer, make a conditional quantity undefined, or turn a finite-time calculation into a false long-run claim.

The joint law of (X_t1,…,X_tn) determines the probability of every event involving only those selected times.

An event involving only t1,…,tn is the inverse image of a measurable set B under the random vector (X_t1,…,X_tn).

The joint law assigns that event probability P((X_t1,…,X_tn)∈B).

Therefore every finite-time probability question is answered by the corresponding finite-dimensional distribution.

Let X_n be independent fair coin indicators. Find P(X_1=1,X_2=0) and compare it with a process Y_n=Y_1 for all n where Y_1 is fair.

  1. Independence gives P(X_1=1,X_2=0)=P(X_1=1)P(X_2=0)=1/4.
  2. For Y, the event Y_1=1 forces Y_2=1 because the same value is copied.
  3. Hence P(Y_1=1,Y_2=0)=0 although X_1,X_2,Y_1,Y_2 all have the same one-time Bernoulli law.

Result: The independent process gives 1/4; the copied process gives 0, proving that marginal laws do not determine a process.