I have a fundamental doubt regarding the memorylessness of a Markov model. I have searched everywhere, but couldn't find an answer The basic assumption of a Markov model is that the past state can only influence the present state and all information about the past is captured in the present state. However this past state can't influence the future state.
How is that possible? If the past state influences the present state which itself influences the future state, then doesn't that mean that the past state influences the future state?
 A: The Markov property is well summarized as: "The future is independent of the past, given the present".
The key is "given the present". Once you know the present state, it captures everything you need to know about the past. You can then make a prediction on the future state based on your knowledge of the present state alone.
Of course, the future and the past are not independent, but when you condition it on knowing the present, they become independent.
A: The idea here is that the probability of a random process $X$ at some future time $t$, namely $X_t$, conditioned to it known past (that is, conditioned to the previous known values $X_s$) is the same than just considering the last known value.
This means that just the last known state of the process gives you all possible information to predict it future. There are a lot of common and natural random processes that have this property, by example the prediction of many chemical processes cannot be improved knowing more than the last known state of the reaction because the reaction just depend on it actual state, not in it past.
By example a random walk is also a process of this kind: if you are at some position in a street and throw a dice to decide in what direction to walk the next step, then it doesn't care how you arrive to your actual place to know the probabilities to be in some place at the next step, it just depends on where you are now. That is: the information of the path you did to end in your actual position in the street doesn't give you more information to predict the next position in your walk as your actual position.
Similarly, the prediction of the climate can be seen as a Markov process, as the way the climate will evolve doesn't depends on what climate was three billion of years before, if not just on the present state of temperature, clouds, air pressure and many other actual values.
