# ito formula apply to integral

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.

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

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?