IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A discrete-time stochastic process X_n on a finite state space satisfies P(X_{n+1}=j | X_n=i, X_{n-1}=i_{n-1}, ..., X_0=i_0) = P(X_{n+1}=j | X_n=i) for all n and all histories. Which property does this describe?
The condition describes the Markov property: given the present state, the next state is conditionally independent of the earlier history. Stationarity, independent increments, martingale behaviour and ergodicity are separate properties that this conditional-probability statement does not imply.
- AThe process is stationary in the strict sense
- BThe process has the Markov propertyCorrect
- CThe process has independent increments
- DThe process is a martingale
- The process is ergodic
Explanation
The stated equality says the future depends on the past only through the present state, which is the definition of the Markov property. Independent increments, stationarity and martingale behaviour are different properties and do not follow from it.
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