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Markov Process and Markov Chain

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The Markov process is one of the part of stochastic process in which it is necessary for us to satisfy a Markovian equation:

P {Xt + 1Â =Â jÂ | X0Â =Â k0,Â X1Â =Â k1… , Xt-1Â =Â kÂ t-1Â ,XÂ t= i}

= P {XÂ t+1Â Â =Â j |Â XtÂ = i} forÂ tÂ = 0, 1, …

This equation shows that the environment that is going to occur in future depends directly and only on the environment of the next day.

There are two of transition probability:

• First one is known as one step transition probability which is given by the equationP { Xt+1Â  =Â jÂ | XtÂ = i} which shows the process between time period t and t+1.

Further a one-step conditional probability is divided into stationary and non-stationary probabilities. The stationary one-step conditional probability is shown by the equation:

P {XÂ t+1Â  =Â j |Â XtÂ = i} = P {X1Â =jÂ | X0Â = i} for allÂ tÂ = 0, 1, …

Where transition remain fixed with time. This equation is denoted by Pij.Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

• Another type is known as n-step transition probability which is shown by the equation:

P {Xt+n=jÂ | X, = i} = PÂ {XnÂ =Â jÂ | X0= i} for allÂ tÂ = 0, 1, ..,

And is denoted by pij(n) for all i and j.

Both the equation will further satisfy two important properties:

• pij(n )â‰¥Â 0 for all iÂ and j Â andÂ nÂ = 0,Â 1,Â 2, …
• âˆ‘ pij(n )=1 for all i andnÂ = 0, 1, 2, … , and Â M =No. of states.

Matrix form of transition probabilities:

Transition of conditional probabilities can also be represented in matrix form.

1. N-step transition probabilities:
1. One-step transition probabilities:

Thus, we see from the above matrix forms that P denotes one-step transition probabilities and P(n) Â is called n-step transition probabilities.

Finite state Markov chain:

A countable number of forms, the Markovian property, stationary transition probabilities and the set of probabilities P {X0 = i} together satisfy the finite state of Markov theorem of a stochastic process {X1} (t=0,1,2,â€¦).

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