Stochastic Process- Definition, Specifications and Classification

The probabilistic phenomena depending on time are called Stochastic process. For Randomly Evolving processes, Associating some probability laws by using probabilities to possible future states (But do not depend on the previous states).

For example, the Number of telephone calls in a telephone service center during a given time interval can be solved using Stochastic Process technique.

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Definition of Stochastic Process or Random Process:

Stochastic Process is the family of Random Variables [X(t) | t ∈T] defined on a given probability space, indexed by an index parameter t ∈T where T = set of parameters = Index Set.
The value assumed by X(t) is called states, set of all possible values form State-Space 'E' of Stochastic process. The Stochastic Process is also called Random Process.

Specifications of Stochastic Process

1. If the index set 'T' is Discrete, then the process is called Discrete-parameter (Discrete Time) process. 
2. If the index set 'T' is Continuous, then the process is called Continuous-Parameter (Continuous-Time) Process. 
3. If the State Space 'E' is Discrete, then the process is called Discrete-State process. 
4. If the State Space 'E' is continuous, then the process is called Continuous-State process.

Classification of Stochastic Process

Some of the Classifications of Stochastic Process are as follows:

1. Stationary Process:

A Random Process  [X(t) | t ∈T] is said to be Stationary or strict-sense stationary if, for all 'n' and every set of time instances t_i ∈T, i = 1,2,3,4....
F_x(x1,x2,x3,....,x_n;t1,t2,t3,t4.....,tn) = F_x(x1,x2,x3,....,x_n;t1+τ,t2+τ,t3+τ.....,tn+τ)
For any Tau(τ) , Hence the distribution of the stationary process will not be affected by whifting the time origin by τ. and X(t) and X(t+τ) will have the same distribution for any τ.

2. Markov Process:

Let [X(t) | t ∈T] be a Random/Stochastic Process then, 

P(X(t_n) ≤ x_n+1 / X(t₀)= x₀, X(t₁)= x₁,..., X(t_n)= x_n )
= P(X(t_n+1) ≤ x_n+1 /  X(t_n)= x_n)
Then Random Process is called Markov process.



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