Derive the $N-$ factor Merton $(1973)$ model in the HJM framework. Therefore, derive:

the solution for $r(t)={\displaystyle \underset{i=1}{\overset{N}{}}}{x}_i(t),$ where $d{x}_i(t)={}_{i}dt+{}_{i}d{W}_i(t),$ for iin$\{1,2,$dots,$N\},$ and where $$ is

the $(NN)$ covariance matrix for the $N$ factor innovations with ${}_{i,j}dt=E(d{W}_i(t)d{W}_j(t))={}_i{}_j{}_{i,j}dt$

being the instantaneous covariance between innovations in factors i and $j$ and ${}_{i,j}$ is the correlation coefficient

between the $i\text{'}$th and $j\text{'}$th factors,

the zero$-$coupon bond price $P(t,T),$

the instantaneous forward rate $f(t,T),$

the ${}_i(t)$ and ${}_i(T)$ terms such that ${}_i(t,T)={}_i(t){}_i(T)$ for the iin$\{1,2,$dots,$N\}$ factors.