"A stochastic process is a mathematical object usually defined as a sequence of random variables, where the index of the sequence has the interpretation of time."
A collection of random variables that change over time.
Probability Theory: An introduction to the basic concepts of probability theory, including random variables, probability distributions, and the law of large numbers.
Markov Chains: A type of stochastic process that describes the evolution of a system over time in a discrete state space.
Poisson Processes: A type of stochastic process that models the occurrence of events over time.
Brownian Motion: A continuous-time stochastic process in which the random variable representing the system's state changes continuously over time.
Martingales: A type of stochastic process that models the growth or evolution of a system whose values depend on future outcomes of random events.
Queuing Theory: A branch of applied probability theory that deals with the mathematical modeling and analysis of waiting lines.
Renewal Processes: A class of stochastic processes used to model the occurrence of events in a system where the times between events are not necessarily constant.
Stochastic Calculus: A mathematical framework used to model and analyze stochastic processes, including Brownian motion and other types of continuous-time stochastic processes.
Time Series Analysis: A statistical method used to analyze and model time-dependent data, including forecasting future values and identifying patterns and trends.
Monte Carlo Methods: A numerical technique used to solve complex mathematical problems that involve probabilistic simulation, including computer simulation of stochastic processes.
Markov Chain: A Markov chain is a stochastic process that follows the Markov property, which states that the future state of the process depends only on the present state, and not on any past states.
Brownian Motion: Brownian motion is a stochastic process that describes the random movement of particles in a fluid, such as the movement of pollen on water.
Poisson Process: A Poisson process is a stochastic process that models the occurrence of random events over time, such as the arrival of customers at a store or the occurrence of radioactive decay.
Gaussian Process: A Gaussian process is a stochastic process that follows a Gaussian distribution, such as the noise in a signal or the variation in a stock price.
Wiener Process: A Wiener process is a stochastic process that is used to model the changes in a stock price or other financial asset over time.
Levy Process: A Levy process is a stochastic process that is used to model the behavior of a wide variety of systems, from the movement of particles in a fluid to the dynamics of financial markets.
Ornstein-Uhlenbeck Process: An Ornstein-Uhlenbeck process is a stochastic process that models the relaxation of a system to equilibrium, such as the return of a stock price to its mean value.
Martingale Process: A martingale process is a stochastic process that has the property that the expected value of the next state, given the current state, is equal to the current state itself.
Hidden Markov Model: A hidden Markov model is a stochastic process that is used in many applications, such as speech recognition and machine translation, to model a sequence of observations that are generated by an underlying sequence of hidden states.
Renewal Process: A renewal process is a stochastic process that models the occurrence of random events that happen independently of each other and that have a finite lifespan, such as the failure of a machine or the arrival of a new customer at a store.
"Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner."
"Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule."
"Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, and telecommunications."
"Seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance."
"Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time."
"The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space."
"The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables."
"If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead."
"The values of a stochastic process are not always numbers and can be vectors or other mathematical objects."
"Based on their mathematical properties, stochastic processes can be grouped into various categories, which include random walks, martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes."
"The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis."
"The theory of stochastic processes is considered to be an important contribution to mathematics."
"It continues to be an active topic of research for both theoretical reasons and applications."
"The terms stochastic process and random process are used interchangeably."
"In probability theory and related fields, a stochastic (or random) process is a mathematical object usually defined as a sequence of random variables."
"The index of the sequence has the interpretation of time."
"Examples include biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, and telecommunications."
"Seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance."
"The Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time."