stokastik

Stokastik is a term used in mathematics, statistics, and related disciplines to describe processes or systems that involve randomness. In probability theory, a stochastic process is a collection of random variables indexed by time or space, used to model how a quantity evolves under uncertainty. Formally, X_t denotes the state of the process at time t, defined on a probability space (Ω, F, P). The joint behavior of the variables, their distributions, and dependencies determine the process’s properties.

Key concepts include independence, stationarity, and the Markov property, among others. A process may be discrete-time or continuous-time, and its state space may be finite, countable, or continuous. Common examples are discrete-time Markov chains, continuous-time Markov chains, Poisson processes, Brownian motion (Wiener process), and Gaussian processes.

Stochastic modeling often uses stochastic differential equations to describe continuous-time dynamics influenced by random noise, with Itô calculus and martingale techniques providing analytical tools. Probability distributions, expectations, variances, and covariances summarize behavior, while ergodicity and mixing describe long-run tendencies.

Applications span finance (stock prices modeled as stochastic processes and option pricing), physics (diffusion), queueing theory, biology (population dynamics, gene frequencies), and computer science (random walks, Monte Carlo methods). Stochastic methods underpin simulations, risk assessment, and data assimilation across disciplines.

Stochastic models contrast with deterministic ones: given the same initial conditions, deterministic models yield a single outcome, while stochastic models produce a range of possible trajectories with associated probabilities. The field has a long history in probability theory, with foundational work by early 20th-century mathematicians and continued development in theory and applications.