Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 27K (Applied Probability)

Condition on the first short time interval. A jump contributes the generator term $Qf$, while local time grows at unit speed in the current coordinate and contributes ${\partial }_{{\ell }_i}f$. The Markov property and division by the interval length therefore give ${\partial }_t{f}_t=M{f}_t$.

Symmetry of $Q$ gives ${\partial }_{y_i}{e}^{y^{T}Qy/2}=e^{y^{T}Qy/2}{\sum }_j{q}_{ij}{y}_j$. Also ${\partial }_{y_i}f(i,y^2/2)=y_i{\partial }_{{\ell }_i}f$. Substitute these identities into the left side and integrate the resulting total $y_i$ derivative by parts. Compact support removes the boundary term and leaves exactly the right side.

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 $y_i{y}_j$.