FX forward brainteaser, part $1$

Let $x_t$ be the euro dollar exchange rate $($in dollars per euro$)$ at time $t,$ and suppose

$x_T|x_{t}lognormal(log{x}_t+(T-t),{}^2(T-t)).$

$($a$)$ Calculate $E(x_T|x_t).$ Find in terms of $r_{\text{\$}},r_{}$ and $$ the value of $$ such that

$E(x_T|x_t)=F(t,T),$ where $F(t,T)$ is the forward euro dollar FX rate $($see

Question $3).$

$($b$)$ Let $Y_t$ be the dollar euro exchange rate $($in euro per dollar$).$ Find the distribution

of $Y_T|Y_t$ and calculate $E(Y_T|Y_t).($Hint$.$ Lengthy calculations are not required.$)$

For what value of $$ does $E(Y_T|Y_t)=$tilde$(F)(t,T),$ where tilde$(F)(t,T)$ is the forward dollar

euro FX rate?

$($c$)$ What condition has to be satisfied such that both $E(x_T|x_t)=F(t,T)$ and

$E(Y_T|Y_t)=$tilde$(F)(t,T)?$ What does this imply in practice?

$($d$)$ Suppose that $x_t=1.$ Show that for the value of $$ found in $($a$),$

$E(x_T|x_t=1)=E(Y_T|Y_t=1)$ if and only if $=\sqrt[\phantom{2}]{2(r_{\text{\$}}-r_{})}$ and $r_{\text{\$}}{r}_{}.$

Note Does the lack of symmetry in results $($c$)$ and $($d$)$ trouble you?