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Stochastic Process

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Stochastic process is a concept that forms the base of a lot of statistical methods used in different fields of study. This is a simple yet very powerful idea that can help us capture the random nature of problems and formulate them in mathematical notation and hence use them in different applications.

This process has many applications and is readily used in the finance, physics and mathematics. Did you know that stochastic probabilities are used in many asset pricing models and also in the heat diffusion equation? The stochastic process is the way of representing and analyzing the random behavior in the stock prices or in heat diffusion.


The word stochastic means 'random'. You can think of 'stochastic process' as a family of random functions. A few examples of stochastic processes are: brownian motion, poisson process, time series etc. In essence, each one of the random functions are stochastic but there are peculiarities that make them different from each other. Please look at the following examples to get an idea of 'stochastic processes'.

Examples:
- Brownian motion is a stochastic process and its randomness is defined by random movement of particles suspended in a fluid.
- Poisson process is a stochastic process and its randomness is defined by the number of event occurring in a length of time.


Mathematical Representation

A stochastic process is a process of random process that yields a value x from a set of random values X with respect to time (or indexed by) t.

 \{X_t : t \in T\} [1]
 

Discreet Stochastic Process

If the time (t) in from [1] belongs to the integer number set then the stochastic process is a discreet stochastic process. This means that there is a fixed number of values t can take in any range of values in the set T, e.g. we have the set T(1, 2, 3, 4, 5,...), we can easily say that the number of elements between 2 and 4 is 1, similarly the number of elements between 2 and 3 is 0.


Continuous Stochastic Process

If the time (t) in from [1] belongs to the real number set then the stochastic process is a continuous stochastic process. This means that there is an infinite number of values t can take in any range of values in the set T, e.g. we have the set T(1.0,..., 2.0,..., 3.0,...), we can say that the number of elements between 1.0 and 3.0 is infinite and a few of the numbers are: 1.5001, 1.67, 1.93401, 2.7623,....