Stochastic Processes

Introduction

A stochastic process is a mathematical object usually defined as a collection of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines such as physics, chemistry, biology, computer science, engineering, finance and economics.

Definition

Formally, a stochastic process is a function of two variables, one from the set indexing the collection of random variables (often representing time), and the other from the set of outcomes of the random variables (usually representing the states of a system). The state space can be discrete (as in a coin toss) or continuous (as in the growth of a population); the time variable often takes values in the nonnegative integers or the nonnegative real numbers.

Classification

Stochastic processes can be classified in various ways, for example, by their state space, their index set, or the statistical properties of the random variables.

By State Space

If the random variables take on real or complex values, the process is called a real-valued or complex-valued stochastic process; if the variables take on vector values, it is a vector stochastic process.

By Index Set

If the index set is the set of nonnegative integers, the stochastic process is a discrete-time stochastic process; if the index set is the set of nonnegative real numbers, it is a continuous-time stochastic process.

By Statistical Properties

If all the random variables in a stochastic process are statistically independent, the process is an independent stochastic process (or a process with independent increments); if they are not all independent but satisfy a weaker condition called Markov property, the process is a Markov process.

Examples

There are many examples of stochastic processes, and the study of these processes has become a major area of research. Some examples include:

- Random walks: This is a simple example of a stochastic process where the current position is dependent on the previous steps taken. It is often used to model stock market fluctuations and diffusion processes in physics.

- Markov chains: These are sequences of random variables where the probability distribution of each variable depends only on the value of the previous variable. They are used in many fields, including statistics, economics, and computer science.

- Poisson processes: These are processes where events occur continuously and independently at a constant average rate. They are used to model events such as radioactive decay and telephone calls arriving at a switchboard.

- Brownian motion: This is a continuous-time stochastic process used to model random movement, such as the motion of particles in a liquid or gas, or the fluctuating prices of financial instruments.

Properties

Stochastic processes have many interesting and important properties, some of which are outlined below.

- Stationarity: A stochastic process is said to be stationary if its statistical properties do not change with time. This means that the process is statistically the same at all points in time.

- Ergodicity: A stochastic process is said to be ergodic if its long-term average behavior is the same as the average behavior of a large number of independent realizations of the process.

- Markov property: A stochastic process has the Markov property if the conditional probability distribution of future states of the process depends only upon the present state, not on the sequence of events that preceded it.

Applications

Stochastic processes have many applications in various fields. For example, they are used in finance to model the behavior of asset prices, in physics to model the behavior of quantum systems, in biology to model the growth of populations, and in computer science to model the behavior of algorithms and networks.