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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Apply Itô's formula to the square of Brownian motion.
- Justify the conclusion "E[Bₜ²]=t and Bₜ²−t is a martingale" using the stated assumptions.
Before you start
Brownian motion and stochastic integrals.
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
Find the differential of Bₜ² and its expectation.
Why this math matters
Apply Itô's formula to the square of Brownian motion. 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:
- B is standard Brownian motion starting at zero.
- The integral is the Itô integral.
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.
See the extra term in stochastic differentiation
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the differential of Bₜ² and its expectation.
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
For f(x)=x², f′=2x and f″=2
Itô's formula contains a second-derivative correction.
Work through the mathematics
d(Bₜ²)=2Bₜ dBₜ+dt
The Brownian quadratic variation contributes the dt term absent in ordinary chain rules.
Check the conclusion
E[Bₜ²]=t and Bₜ²−t is a martingale
The stochastic integral has mean zero under the standard square-integrability conditions.
The result
E[Bₜ²]=t and Bₜ²−t is a martingale
The stochastic integral has mean zero under the standard square-integrability conditions.
Common mistakes to catch
- The notation (dB)²=dt is a mnemonic for quadratic variation, not ordinary algebra.
- An Itô integral is not a pathwise Riemann integral in general.
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
Apply the formula to (Bₜ+1)².
Show a hint
Differentiate the shifted square.
Reveal answer and explanation
d=2(Bₜ+1)dBₜ+dt
The second derivative remains two.
Practice 2
Why is ordinary differential algebra insufficient?
Show a hint
Brownian paths have nonzero quadratic variation.
Reveal answer and explanation
The second-order term survives
Squared increments accumulate instead of disappearing in the limiting stochastic sum.
Take the idea with you
Compare the evolution of a noisy signal's mean with that of its squared magnitude.
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: Accumulate squared increments instead of total displacement
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