DES - Definition of 'Markov Process'

Markov Process:

For Markov process, we assume an independent stochastic process where the assumption of independence is partially relaxed:

Given the current value of the process, the future values do not depend on the past.

Markov Property:

$$\begin{array}{rl}& P(X(t+\tau )\le \bar{x}\mid X(s)=x(s)\forall s\le t)\\ & =P(X(t+\tau )\le \bar{x}\mid X(t)=x(t))\end{array}$$

Notice the difference with the Independent Stochastic Process Assumption where the process is also independent from the current value.

NOTE: When writing $X(s)=x(s)\forall s\in [0,t]$ it means to consider all the sample path up to time $t$, how and when the states evolved. The equivalent in discrete time, (which is more clear to understand) is given by: $X(n)=x(n),X(n-1)=x(n-1),\dots ,x(0)=x(0)$

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~Example:

If the processes are both Markov processes then the probabilities:

$$P(X_1(t+\tau )\le \bar{x}\mid X_1(t)=x(t))=P(X_2(t+\tau )\le \bar{x}\mid X_2(t)=x(t))$$

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