Part VIII — Stochastic Processes

Randomness indexed by time: arrival processes (Bernoulli, Poisson, renewal), Markov chains and their hidden-state extension, and the continuous-time processes — Brownian motion and its geometric variant — that underlie price models.

Topics

Topic	Focus
Random Processes	Collections of random variables indexed by time
Bernoulli Processes	Arrival counts and waiting times in discrete time
Poisson Processes	Continuous-time arrivals with exponential interarrival times
Renewal Processes	Arrival processes with general interarrival distributions
Markov Chains	Transition matrices, state classification, and steady-state behavior
Continuous-Time Markov Chains	Continuous-time Markov dynamics with exponential holding times
Hidden Markov Models	Filtering, smoothing, decoding, and learning for hidden-state chains
Brownian Motion	The continuous-time limit of random walks
Geometric Brownian Motion	The standard model of multiplicative price dynamics
Martingales	Fair-game processes and the optional stopping theorem
Random Walks	Sums of i.i.d. steps and their long-run behavior