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Stochastic processes
Lesson 39 of 75
Challenge

Martingales

Objective. Use conditional expectation and stopping ideas.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A martingale is a stochastic process whose expected future value, conditional on all information up to the present, equals its current value, capturing the idea of a 'fair game' with no predictable drift. Formally, M is a martingale with respect to a filtration if E[Mₙ₊₁ | information up to n] = Mₙ for all n. Stopping times, random times τ determined without looking into the future, interact with martingales through the optional stopping theorem, which under suitable conditions gives E[M_τ] = E[M₀]. Example: for a symmetric random walk Sₙ starting at 0, Sₙ is a martingale since each step has mean zero, so E[Sₙ] = 0 for all n.

Worked example

Show that the simple symmetric random walk Sₙ = X₁ + ... + Xₙ, with each Xᵢ = ±1 equally likely, is a martingale with respect to its own history.

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Illustrated card showing 1
  1. 1By definition, a martingale requires E[Sₙ₊₁ | S₁,...,Sₙ] = Sₙ.
  2. 2Write Sₙ₊₁ = Sₙ + Xₙ₊₁, where Xₙ₊₁ is independent of the past with E[Xₙ₊₁] = (1)(0.5) + (−1)(0.5) = 0.
  3. 3Taking conditional expectation: E[Sₙ₊₁ | past] = Sₙ + E[Xₙ₊₁] = Sₙ + 0.
  4. 4Since this equals Sₙ exactly, the martingale property E[Sₙ₊₁ | past] = Sₙ holds.

Answer: E[Sₙ₊₁ | past] = Sₙ, so Sₙ is a martingale

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Common mistake. Students often apply the optional stopping theorem without checking its conditions, such as bounded stopping times or uniform integrability; for an unbounded symmetric random walk stopped at the first time it hits 1, E[M_τ] ≠ E[M₀] in general.

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