Let $Y_t=\left(\int_0^t e^{t-s}dB_s\right)^2$. Find the stochastic differential equation that is solved by $e^{Y_t}$. The answer must be given in differential notation.
What I did was rewrite $Y_t$ as
$$Y_t=e^{2t}X_t^2$$
where $X_t=\int_0^te^{-s}dB_s$ is an Itô process. Then,
$$e^{Y_t}=e^{e^{2t}X_t^2},$$
but that seems to make the problem more complicated. Is there a more efficient way of solving it?
So indeed by Itô formula we find
$$e^{Y_{t}}=1+\int e^{Y_{s}}dY_{s}+\frac{1}{2}\int e^{Y_{s}}d[Y]_{s}.$$
Here we need to apply Itô formula again $X_{t}:=e^{t}Z_{t}=e^{t}\int^{t} e^{-s}dB_{s}$
$$Y_{t}=\int 2X_{s}dX_{s}+\int d[X]_{s}$$
$$=\int 2X_{s}e^{s}Z_{s}ds+\int (2X_{s}+1)dB_{s}.$$
So
$$e^{Y_{t}}=1+\int e^{Y_{s}}2X_{s}e^{s}Z_{s}ds+\int e^{Y_{s}}(2X_{s}+1)dB_{s}+\frac{1}{2}\int e^{Y_{s}}(2X_{s}+1)^{2}ds.$$