$(10$ marks total$)($Analytical Solution of Special SDEs$)$

Apply Ito's Lemma to show that given $d{x}_t=$adt$+bd{Z}_t$ where a and $b$ are constant

$($a$).(5$ marks$)F=e^{x_t-\frac{1}{2}t}$ solves $dF=(-\frac{1}{2}+a+\frac{1}{2}{b}^2)Fdt+bFd{Z}_t$

$($b$).(5$ marks$)F=e^{2x_t-t}$ solves $dF=(2b^2-1+2a)Fdt+2bFd{Z}_t$