 Solution

For $f_t(i,\ell )={\mathbb{E}}_i[g(\ell +L(t))1_{X(t)=j}]$, part (a) gives $M{f}_t={\partial }_t{f}_t$. Apply part (b), integrate in $t$ from zero to infinity, and integrate by parts in time. Compact support makes the terminal term vanish, while $f_0(i,\ell )=g(\ell )1_{i=j}$. Moving the resulting Gaussian derivatives once more by parts yields the stated identity with $y_i{y}_j$.