I am currently self-studying some financial mathematics and am stuck on a practice problem:
I have defined the below as hinted
\begin{align} Y_t &= x_0 e^{\alpha t} \\ Z_t &= \sigma e^{\alpha t} \\ R_t &= \int_{0}^{t} e^{-\alpha s} dWt \end{align}
and then computed Ito's rule on
\begin{align} F(y,z,r) &= y + zr \\ dX_t &= d(y + zr) \\ &= dY_t + rdZ_t + zdR_t + dZ_tdR_t \end{align}
which for me simplified to
$$dX_t = \alpha e^{\alpha t} x_0 + \alpha \sigma e^{\alpha t} \int_{0}^{t} e^{-\alpha s} \,dWs + \sigma e^{\alpha(t-s)}dWt + \alpha \sigma e^{\alpha(t-s)}dW_t$$
I'm not sure if I had applied multi-dimensional Ito's formula properly or if I had missed a term.
Any help is much appreciated.
Something is wrong in your last line.
Why you use lower case $r,z\,$? This should be
\begin{align}\require{cancel} dX_t &= dY_t + R_t\,dZ_t + Z_t\,dR_t + dZ_t\,dR_t\\[2mm] &=\alpha\, x_0\,e^{\alpha t}\,dt+\alpha\,\sigma\, e^{\alpha t}\Big(\int_0^te^{-\alpha s}\,dW_s\Big)\,\color{red}{dt}+\sigma\,\cancel{e^{\alpha t}\,e^{-\alpha t}}\,dW_t\,.\tag{1} \end{align} There is no $dZ_t\,dR_t$ term because $Z$ is of finite variation. Using $$\tag{2} X_t=e^{\alpha t}x_0+\sigma\,e^{\alpha t}\int_0^te^{-\alpha s}\,dW_s\,. $$
The equation (1) can be written as $$\tag{3} dX_t=\alpha\,X_t\,dt+\sigma\,dW_t\,. $$