Let $\zeta _s=(\xi^0_s,\xi^1_s)$ where $\xi^i_s$ are Markov process talking there value in $\mathbb Z^d$ and $P_s-$Measurable. The measure for $\zeta _s$ is $$\mathbb P_s^{(2)}((i,j),(k,l))=\mathbb P_s(i,k)\mathbb P_s(j,l).$$ How would you compute $$\int_0^t1_{\{\zeta _s\in \Delta\} }ds$$ where $\Delta =\{i,j\in\mathbb Z^d\mid i=j\}$ ?
To me it's $$\int_0^t1_{\{\zeta _s\in \Delta\} }ds=\mathbb P^{(2)}(\zeta _s\in \Delta \mid s\in [0,t]),$$ but how can I do better ?