Math 2.0

Probability theory and stochastic processes Definition

Martingale

Builds on Conditional expectationStochastic processExpected value

Definition

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space with a filtration (ℱn)n≥0(\mathcal{F}_n)_{n \ge 0}: σ-algebras with ℱ0⊆ℱ1⊆⋯⊆ℱ\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \dots \subseteq \mathcal{F}. A martingale with respect to (ℱn)(\mathcal{F}_n) is a sequence (Mn)n≥0(M_n)_{n \ge 0} of real random variables with these three properties:

  1. MnM_n is ℱn\mathcal{F}_n-measurable for each nn.
  2. E[∣Mn∣]<∞E[|M_n|] < \infty for each nn.
  3. E[Mn+1∣ℱn]=MnE[M_{n+1} \mid \mathcal{F}_n] = M_n almost surely for each nn.

Remarks

If "==" in property 3 is replaced by "≥\ge", the sequence is a submartingale. If it is replaced by "≤\le", it is a supermartingale. By the law of total expectation, E[Mn]=E[M0]E[M_n] = E[M_0] for each nn. Examples: the simple random walk SnS_n and Sn2−nS_n^2 - n are martingales for the filtration of the steps. Doob's optional stopping theorem and Doob's martingale convergence theorem are the main tools of the theory.

Reading path

To understand this page from the ground up, read these 47 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. Discrete random variable definition
  43. Stochastic process concept
  44. Random variable with a density definition
  45. Expected value definition
  46. Conditional expectation definition
  47. Martingale definition

Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Martingale_(probability_theory), General definition
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Kinddefinition
WritingW1 stated
MSC60G42
DefinesMartingale
UsesConditional expectation, Stochastic process, Expected value

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