Stochastic Processes Introduction (2012)

Before starting the introduction of Stochastic Processes, let us start with some important definitions related to statistics and stochastic processes.

Experiment: Any activity or situation having an uncertain outcome.

Sample Space:  The set of all possible outcomes is called sample space and every element $\omega$ of $\Omega$ is called sample point. In the Stochastic process, we will call it state space.

Event and Event Space:  An event is a subset of the sample space. The class of all events associated with a given experiment is defined to be the event space.

An event will always be a subset of the sample space, but for sufficiently large sample spaces, not all subsets will be events. Thus the class of all subsets of the sample space will not necessarily correspond to the event space.

Random Variable:
A random variable is a mapping function that assigns outcomes of a random experiment to real numbers. The occurrence of the outcome follows a certain probability distribution. Therefore, a random variable is completely characterized by its probability density function (PDF). Or

A random variable is a map $X:\Omega \rightarrow R$ such that $F\{X \le x\} = \{\omega \in \Omega:x(\omega)\le x\} \in F$ for all $x \in R$.

Probability Space:  A probability space consists of $(\Omega, \mathfrak{F}, P)$ of three parts, sample space, a collection of events, and a probability measure.

Cumulative Distribution Function (CDF): Probability distribution function for the random variable $X$ such that $F(a) = P\{X \le a\}$.

time line

Time: A point of time either discrete or continuous

State: It describes the attribute of a system at some point in time $S=(s_1, s_2, \cdots, s_t)$.

It is convenient to assign some unique non-negative integer as an index to each possible value of the state vector $S$.

Activity: Something that takes some amount of time (duration) to occur. The activity culminates in an event.

Transition (movement from one state to another) Stochastic Processes

Transition:  Transition is caused by an event and it results in some movement from one state to another state.

Probability Measure:  A probability measure intends to be a function defined for all subsets of $\Omega$.

What is a Stochastic Process?

The word stochastic is derived from the Greek word “stoΩ’kæstIk” meaning “to aim at a target”. Stochastic processes involve a state which changes randomly.
Given a probability space $(\Omega, \mathfrak{F}, P)$  stochastic process $\{X(t), t\in T\}$ is a family of random variables, where the index set $T may be discrete $(T=\{0,1,2,\cdots,\})$ or continuous $(T=[0, \infty))$. The set of possible values which random variables $\{X(t), t\in T\}$ may assume is called the state space of the process, and denoted by $S$.

A continuous time stochastic process $\{X(t), t \in T\}; (T=[0, \infty))$ is said to have an independent increment of for all choices of $\{t_1,t_2, \cdots, t_n\}$, the $n$ random variables $X(t_1) – X(t_0), X(t_2) – X(t_1), \cdots, X(t_n)-X(t_{n-1})$ are independent. Using discrete time the state of the process at time $n+1$ depends only on its state at time $n$.

It is often used to represent the evolution of some random value or system over time.

Examples of Stochastic Processes

Examples of processes modeled as stochastic time series include stock market and exchange rate fluctuations, signals such as speech, audio, and video, medical data such as a patient’s EKG, EEG, blood pressure or temperature, random movement such as Brownian motion or random walks, counting process, Renewal process, Poisson process and Markov process.

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