Math 2.0

Probability theory and stochastic processes Definition

Markov property

Builds on Stochastic processConditional expectationConditional independence

Definition

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space, let T⊆[0,∞)T \subseteq [0, \infty), and let (ℱt)t∈T(\mathcal{F}_t)_{t \in T} be a filtration: σ-algebras with ℱs⊆ℱt⊆ℱ\mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} for s≤ts \le t. Let (Xt)t∈T(X_t)_{t \in T} be a stochastic process with values in (S,𝒮)(S, \mathcal{S}) such that each XtX_t is ℱt\mathcal{F}_t-measurable. The process has the Markov property with respect to (ℱt)(\mathcal{F}_t) if, for all s≤ts \le t in TT and each A∈𝒮A \in \mathcal{S}, P(Xt∈A∣ℱs)=P(Xt∈A∣Xs)almost surely.P(X_t \in A \mid \mathcal{F}_s) = P(X_t \in A \mid X_s) \quad \text{almost surely}.

Remarks

Here P(B∣𝒢)=E[1B∣𝒢]P(B \mid \mathcal{G}) = E[1_B \mid \mathcal{G}] is a conditional expectation. For a discrete-time process with a countable state space and the natural filtration, the property becomes P(Xn+1=j∣X0,…,Xn)=P(Xn+1=j∣Xn)P(X_{n+1} = j \mid X_0, \dots, X_n) = P(X_{n+1} = j \mid X_n). This defines a Markov chain. The property says that the future is conditionally independent of the past, given the present. The strong Markov property is the same statement with a stopping time in place of the fixed time ss.

Reading path

To understand this page from the ground up, read these 49 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Real numbers object
  9. Set definition
  10. Ring of integers statement
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Ordered pair definition
  15. Union definition
  16. Sample space definition
  17. The rationals form a field statement
  18. Existential quantifier definition
  19. Universal quantifier definition
  20. Cartesian product definition
  21. Subset definition
  22. The rationals form an ordered field statement
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Universal set definition
  26. Order on the reals definition
  27. Power set definition
  28. Complement definition
  29. Function definition
  30. Interval definition
  31. Injective function definition
  32. Preimage of a set definition
  33. Sigma-algebra definition
  34. Countable set definition
  35. Event definition
  36. Borel sigma-algebra definition
  37. Probability axioms concept
  38. Probability measure definition
  39. Probability space definition
  40. Conditional probability definition
  41. Random variable definition
  42. Independence of two events definition
  43. Discrete random variable definition
  44. Stochastic process concept
  45. Random variable with a density definition
  46. Conditional independence definition
  47. Expected value definition
  48. Conditional expectation definition
  49. Markov property definition

Where this leads

17 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Markov_property, Markov property in general
About this page
Tag0HB8SXHPFA
URIm2://0HB8SXHPFA
Kinddefinition
WritingW0 stub
MSC60J05
DefinesMarkov property
UsesStochastic process, Conditional expectation, Conditional independence

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