Stochastic Processes

Randomness evolving in time — random walks and Markov-chain behavior through Poisson processes, martingales, Brownian motion, and Itô calculus.

Core Stochastic Processes

• Random Walks — Coin-flip steps summed over time: mean 0, spread √n, exact odds by counting paths.
• Gambler's Ruin — A bettor between two absorbing barriers: the fair game wins with probability k/N.
• Poisson Processes — Random arrivals at a steady rate: Poisson counts, exponential gaps.
• Stationary Distributions — Where a Markov chain settles: the distribution one step leaves unchanged.
• Martingales — Fair games: given the history, tomorrow's expected value is exactly today's.
• Optional Stopping Theorem — A fair game stays fair at an honest stopping time — gambler's ruin in two lines.

Stochastic Calculus

• Brownian Motion — The random walk run in continuous time: normal increments whose variance is the elapsed time.
• Quadratic Variation — Squared Brownian wiggles add up to the elapsed time: [W]_T = T, the theorem behind (dW)^2 = dt.
• Itô Integral — Integrate against Brownian wiggle: bet before the kick, and pick up a - t2 that ordinary calculus never sees.
• Itô's Lemma — The chain rule for Brownian motion: the second-order term survives, so df = f' dW + 12 f'' dt.
• Stochastic Differential Equations — Drift plus noise, dX = (X) dt + (X) dW — and solving GBM with Itô's lemma.

Martingale Theory & Change of Measure

• Martingale Convergence — Bound a fair game's whole path, and watch it settle down with probability 1.
• Reflection Principle — Mirror a path after it first touches a level: the running maximum's tail is twice the endpoint's.
• Feynman-Kac Formula — A PDE becomes an average over random paths: u(t,x) = E [e^-r(T-t)g(X_T) | X_t = x ].
• Girsanov's Theorem — Reweight the scenarios and a drift appears — but the volatility cannot be touched.
• Martingale Representation — Every fair game on a Brownian filtration is a bet on that Brownian motion — and the bet size is the hedge.

Jumps, Limits & Boundaries

• Compound Poisson Processes — Random arrivals carrying random sizes: the mean, the variance, and the jump measure.
• The Lévy–Khinchin Formula — Every Lévy process is a drift, a diffusion coefficient, and a jump measure.
• Donsker's Invariance Principle — The Central Limit Theorem for the whole path: every mean-zero walk rescales to the same Brownian motion.
• Skorokhod Embedding — Every mean-zero law hides inside a Brownian path — you only have to know when to stop watching.
• Bessel Processes — The distance |B(t)| from home: an outward drift d-1/2R made of pure geometry, and the dimension that decides whether the wanderer returns.
• Fractional Brownian Motion — One dial H turns Brownian motion's independent increments into memory: persistent above 12, antipersistent below.
• The Dirichlet Problem — Steady-state temperature from a random walker: u(x) is the average boundary value where Brownian motion exits.
• Lévy Processes — The family built from independent identical pieces of time, with Brownian motion and the Poisson process as its two poles.
• Multidimensional Brownian Motion — Independent coordinates, invariant under every rotation, coupled through one matrix: dW^i dW^j = _ij dt.
• The Infinitesimal Generator — One operator summarizes a process: (Af)(x) is the instantaneous rate of change of E[f(X_t)], and for Brownian motion it is 12 d^2dx^2.
• Local Martingales — Stop a process before it misbehaves and it is a martingale; let it run and the mean can leak away. The gap between local and true is a limit interchange — and…

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