# 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

• Hi Luca! I happen to agree with your solution. You can either apply Ito's Lemma or the product rule, $\mathrm{d}X_tY_t=X_t\mathrm{d}Y_t+Y_t\mathrm{d}X_t+\mathrm{d}X_t\mathrm{d}Y_t$. Perhaps it's just a typo in your book? Apr 9 '20 at 0:26
• The result you find is correct. Which book is it? Apr 9 '20 at 8:14
• Must be a typo in the book. Please signal the error to the book writer, there must be an errata page in the book's website or something like this for these kind of situations. These things can happen sometimes :) Apr 9 '20 at 8:27
• Assuming the book title and pages are added to the post, I disagree with the close votes. The question, while basic, could be very useful to other students or readers of the book. Apr 9 '20 at 12:48
• @KeSchn Could you make your comment an answer so it's clear this has been resolved? Apr 19 '20 at 19:06

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, $$\mathrm{d}B_t\mathrm{d}t=0$$.