I have a question on the calculation on Ito's Lemma:
${{Y}_{t}}={{t}^{{{W}_{t}}}}$
solve for $d{{Y}_{t}}$
the following is my solution
[\begin{align} & d{{Y}_{t}}=\frac{\partial Y}{\partial t}dt+\frac{\partial Y}{\partial {{W}_{t}}}d{{W}_{t}}+\frac{1}{2}\frac{{{\partial }^{2}}Y}{\partial {{W}^{2}}}{{\left( d{{W}_{t}} \right)}^{2}} \\ & ={{W}_{t}}{{t}^{{{W}_{t}}-1}}dt+{{t}^{{{W}_{t}}}}\ln td{{W}_{t}}+\frac{1}{2}{{t}^{{{W}_{t}}}}{{\left( \ln t \right)}^{2}}dt \\ & ={{t}^{{{W}_{t}}}}\ln td{{W}_{t}}+{{t}^{{{W}_{t}}}}(\frac{{{W}_{t}}}{t}+\frac{1}{2}{{\left( \ln t \right)}^{2}})dt \\ \end{align}]
is my solution correct? Thanks