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Let $(\Omega, \mathcal F, P)$ be a probability space and $\{B_t\}_{t\ge0}$ a Brownian motion. Furthermore let $\{F_t\}_{t\ge0}$ be the natural filtration of $B$. Let

$$Y(t)=\sin(t+B_t), \ \ \ t\ge0$$

I want to determine $dY(t)$ using Ito's lemma.

As we didn't use this lemma yet I don't really know how to solve this problem. I managed to determine $dY(t)$ for $Y_t=\sin(B_t)$ and for $Y_t=\cos(B_t)$.So I thought about using $\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$ and applying these results somehow but I don't think that's the right way to solve this.

$dY(t)=\frac{\partial Y}{\partial t}dt+\frac{\partial Y}{\partial B_t}dB_t+\frac{1}{2}\frac{\partial^2Y}{\partial B_t^2}(dB_t)^2=\cos(t+B_t)dt+\cos(t+B_t)dB_t-\frac{1}{2}\sin(t+B_t)dt$

user671116
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  • Which version of Itô's formula do you know? You need the version for functions which are time-dependent, i.e. Itô's formula for functions of the form $f(t,B_t)$, see e.g. here (with $\mu := 0$ and $\sigma :=1$) – saz May 04 '19 at 18:54
  • @saz We didn't introduce it in lecture yet which does'nt make things easier but we can use wikipedia and anything we find online as long as it is correct – user671116 May 04 '19 at 19:03
  • I see. Feel free to ping me if you run into trouble when applying the Itô formula which I linked you in my previous comment. – saz May 04 '19 at 19:21
  • @saz I added something in my main post but I don't really know what to do – user671116 May 04 '19 at 20:05
  • It's correct (it should read $dB_t$ instead of $dW_t$ but that's obviously just a typo). – saz May 04 '19 at 20:08
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    @saz Good. I still need to read into it a lot to understand what these terms acutally mean. Two questions: Why is $\mu=0$ and $\sigma=1$? And why is $(dB_t)^2=dt$? – user671116 May 04 '19 at 20:42
  • See e.g. this for discussion on $(dB_t)^2= dt$: Wiener Process $dB^2=dt$. – Minus One-Twelfth May 04 '19 at 20:45

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