Stochastic Processes

Mathematical Modeling · Axiom Academy

LESSON Stochastic Processes - Mathematical Modeling Unit 4: Probabilistic Models - Mathematical Modeling A stochastic process is a mathematical model for systems that evolve randomly over time. Unlike deterministic models where the future is completely determined by initial conditions, stochastic processes incorporate randomness at every step. "A collection of random variables indexed by time" where T is the index set (time) and X(t) is a random variable for each t Think of a stochastic process as a random function of time. At each moment t : X(t) is a random variable (not a fixed number) The process describes how randomness unfolds over time A single realization of the process is called a sample path or trajectory Sample Paths of a Stochastic Process Each run of the random experiment produces a different path: Path 1: 0 -> 1 -> 2 -> 1 -> 3 -> 2 -> ... Path 2: 0 -> 0 -> 1 -> 2 -> 2 -> 4 -> ... Path 3: 0 -> 1 -> 0 -> 1 -> 2 -> 1 -> ... Same stochastic process, different random outcomes The first major classification of stochastic processes concerns how time is represented. Time takes values in a countable set, typically or . Time takes values in an interval, typically or . Wiener process (Brownian motion) Discrete time: When observations or events occur at fixed intervals (daily, weekly, per generation). Continuous time: When events can occur at any moment (particle motion, arrivals at a queue, stock price changes). Discrete vs Continuous State Space

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