What is Markov Property ?What Is Stationarity? Why Static Probability Models Fail in the Real World

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00:00 Why static models are not enough
01:00 Random variable vs stochastic process
03:00 Dynamic modeling intuition
05:00 Two-state system setup
07:00 Initial probabilities and interpretation
09:00 Transition probabilities explained
11:00 Homogeneous process intuition
13:00 Matrix multiplication logic
15:00 Long-run behavior intuition
17:00 Stationary distribution explained
19:00 Discrete vs continuous time
21:00 Discrete vs continuous state
23:00 Mortality and stock price intuition
25:00 What is the Markov property
27:00 Cricket match intuition for Markov models
30:00 Why only current state matters
33:00 State diagram interpretation
36:00 No-claim discount system example
40:00 Experience rating intuition
44:00 Multi-state transitions
48:00 Matrix powers and prediction
52:00 Why stochastic processes scale
55:00 Why this foundation matters long term

This session builds stochastic processes from first principles, without assuming prior intuition.

You start by separating two ways the world is modeled.
Static modeling uses a single random variable.
Dynamic modeling needs a collection of random variables indexed by time.
That shift is the core reason stochastic processes exist.

The class begins by explaining why most real systems cannot be captured by one distribution.
Markets change.Customers switch.Risk evolves.

A stochastic process is introduced as a sequence of random variables evolving over time.
Each time point has its own random variable.

You then build intuition using simple finite examples.
Two states are defined and time is discretized.
Probabilities are interpreted as proportions of a population, not abstract symbols.
Market share is used to ground the idea.
Initial probabilities represent how the system starts.

Transition probabilities are introduced next.
These quantify how the system moves from one state to another over a fixed time gap.
They are estimated from past data.
Once estimated, they are assumed constant in a homogeneous process.

All transition probabilities are organized into a transition matrix.
This matrix becomes the engine of prediction.
By multiplying the current distribution with the matrix, you obtain the distribution at the next time step.
Repeated multiplication pushes the system forward in time.

You then show how prediction replaces manual enumeration.
Instead of tracking individuals, the model tracks proportions.
This makes scaling possible when states grow from two to many.

A key idea is long-run behavior.
Even though individuals continue to move between states, the overall proportions may stabilize.
This stable vector is called the stationary distribution.

Stationarity is explained intuitively, not formally.

You then classify stochastic processes by time and state.
Discrete time, discrete state.
Continuous time, discrete state.
Discrete time, continuous state.
Continuous time, continuous state.

Mortality for continuous time with discrete states.
Stock prices for continuous time with continuous states.

The Markov property is introduced carefully.
It does not mean independence.
It means conditional sufficiency of the present.
Given the current state, past information adds no extra predictive power.

This idea is explained using a cricket match scenario.
At a late stage of the game, only the current score, balls remaining, and conditions matter.
The full scoring history is irrelevant for predicting the outcome.
This captures the essence of a Markov process.

You connect this idea to broader applications.
Time series models rely on stochastic processes.
Reinforcement learning is built on Markov decision processes.
Risk modeling, insurance pricing, and experience rating all depend on the same foundation.

A multi-state insurance no-claim discount system is analyzed in detail.
States represent discount levels.
Transitions represent claims or no-claims.
The initial distribution reflects new policyholders.
Matrix powers are used to predict future distributions over multiple years.

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