Probability: Martingales Intro
Probability: Martingales Intro
💡 Key Takeaways
- Define a martingale and the fair-game condition E[X_{n+1} | F_n] = X_n with respect to a filtration.
- Identify natural examples (like a fair random walk) and determine when a process is a martingale.
- Differentiate martingales from submartingales and supermartingales and learn how to verify the martingale property.
- Get a basic intuition for stopping times and how martingales behave under stopping under suitable conditions.
❓ Frequently Asked Questions
What is a martingale?
A stochastic process {X_n} adapted to a filtration F_n with E|X_n|<∞ for all n, and E[X_{n+1} | F_n] = X_n almost surely. Intuition: given past information, the expected next value equals the current value (a fair game).
What is a filtration?
An increasing sequence of sigma-algebras F_0 ⊆ F_1 ⊆ ... that represents the information available up to each time n; X_n must be measurable with respect to F_n (the process is adapted to F_n).
What conditions must hold for a process to be a martingale?
The process must be adapted to the filtration (X_n is F_n-measurable), integrable (E|X_n|<∞ for all n), and satisfy E[X_{n+1} | F_n] = X_n almost surely.
How does a simple symmetric random walk relate to martingales?
Let S_n = sum of n i.i.d. steps ±1 with equal probability, and F_n be the natural filtration. Then E[S_{n+1} | F_n] = S_n, so {S_n} is a martingale; future increments have zero expected net change given the past.
What does the zero conditional mean property imply about forecasting?
It implies that, given past information, you cannot expect to gain or lose on average in the next step—the process has a fair game characteristic.