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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Verify the martingale property of a centered random walk.
- Justify the conclusion "E[Sₙ₊₁|Fₙ]=Sₙ" using the stated assumptions.
Before you start
Filtrations and conditional expectation.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Let ξₙ be independent ±1 variables with equal probabilities and Sₙ=Σⱼ₌₁ⁿξⱼ. Is Sₙ a martingale?
Why this math matters
Verify the martingale property of a centered random walk. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Fₙ is the natural filtration generated by the first n increments, with no future information.
- The increments are independent and integrable.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Check fairness through conditional expectation
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Let ξₙ be independent ±1 variables with equal probabilities and Sₙ=Σⱼ₌₁ⁿξⱼ. Is Sₙ a martingale?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Sₙ₊₁=Sₙ+ξₙ₊₁
The next increment is separated from the observed past.
Work through the mathematics
E[ξₙ₊₁|Fₙ]=0
Independence and symmetry make the next conditional increment mean zero.
Check the conclusion
E[Sₙ₊₁|Fₙ]=Sₙ
Adaptedness and finite expectations complete the martingale conditions.
The result
E[Sₙ₊₁|Fₙ]=Sₙ
Adaptedness and finite expectations complete the martingale conditions.
Common mistakes to catch
- Zero unconditional mean alone does not establish a martingale.
- A filtration describes information, not merely elapsed time.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
If P(ξ=1)=0.6, what process is centered?
Show a hint
The increment mean is 0.2.
Reveal answer and explanation
Sₙ−0.2n
Subtracting predictable mean growth restores zero conditional increments.
Practice 2
Does a martingale have constant sample paths?
Show a hint
Fairness concerns conditional means.
Reveal answer and explanation
No
Individual paths fluctuate even when the next conditional expectation equals the current value.
Take the idea with you
Identify a model's unpredictable residual after removing its predictable drift.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Use optional stopping only with its conditions
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