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I am a beginner of SDE, and I am working on an exercise, the problem ask us to find the $d Y_{t}$ for different $Y_t$. The setting of problem is:

Suppose $B_{t}$ is a standard Brownian motion and $X_{t}$ satisfies $$ d X_{t}=X_{t}^{2} d t+X_{t} d B_{t} $$ Find $d Y_{t}$ for different $Y_t$

  1. $Y_{t}=B_{t}^{2}$
  2. $Y_{t}=X_{t}^{3}$
  3. $Y_{t}=\exp \left\{\int_{0}^{t}\left(X_{s}^{2}+1\right) d s\right\}$

I know how to use Ito's formula to find first and second answer, but don't how to proceed the third one, can anyone help or give some suggestion.

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  • $\begingroup$ intuitively, $dY_{t}=(X_{t}^{2}+1)\exp \left\{\int_{0}^{t}\left(X_{s}^{2}+1\right) d s\right\}dt$ $\endgroup$
    – PaulWH
    Commented Nov 16, 2020 at 23:02

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It sounds like the main difficulty is finding the differential of $\int_0^t (X_s^2+1)ds$, and for that I would give the hint: For a deterministic function $f$, what is $\frac{d}{dt} \int_0^t f(s)ds$? Did you need the chain rule for that, or even to know what $\frac{df}{dt}$ is?

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