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Applying Itô’s Formula to a Time-Scaled Brownian Motion

Article Quant Q&A · Author: Luca Camerani
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Summary

The document resolves a discrepancy in applying Itô’s formula to the process formed by multiplying standard Brownian motion by time. For Y_t = tB_t, the differential contains a drift term proportional to the current Brownian value and a stochastic term scaled by time. The proposed homework answer that leaves the Brownian differential unscaled is identified as an error.

The reasoning can be checked either by applying Itô’s lemma to the function f(t,x) = tx or by using the product rule for stochastic differentials. In the product-rule approach, the cross term between Brownian motion and time vanishes because the differential of time has zero quadratic variation with Brownian motion. The note is a focused calculus explanation; it does not discuss a trading model, market data, or applications beyond this example.

Key ideas

  • For Y_t = tB_t, the drift is B_t dt and the diffusion term is t dB_t.
  • Itô’s lemma applies to the function f(t,x) = tx.
  • The stochastic product rule gives the same differential.
  • The cross term between dt and dB_t is zero.

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# Ito formula for $Y_t=tB_t$


# Ito formula for $Y_t=tB_t$












someone can help me to solve this problem:

$B_t$ is a Standard Brownian Motion.

Let $Y_t=tB_t$. Using Ito formula, find drift and volatility of $Y_t$.

The result I found is $dY_t=B_tdt+t\cdot dB_t$ but in my homework paper the solution is $B_tdt+dB_t$. Which result is right?

Thank you

## Answer by Kevin (score 4, accepted)

https://quant.stackexchange.com/a/53430

I happen to agree with your solution $$ \mathrm{d}(tB_t)=B_t\mathrm{d}t+t\mathrm{d}B_t.$$ You can either apply Ito's Lemma to $f(t,x)=tx$ as you did or apply the product rule,$$ \mathrm{d}X_tY_t=X_t\mathrm{d}Y_t+Y_t\mathrm{d}X_t+\mathrm{d}X_t\mathrm{d}Y_t,$$ where, of course, $\mathrm{d}B_t\mathrm{d}t=0$.

There is presumably an error in your book/homework paper.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.