Expected directed jump counts for a finite-state rate process

On the canonical path space, write a path with $n$ jumps as its state sequence and ordered jump times. Its jump count is $N_{xy}(T)={\sum }_{j=1}^n{\mathbf{1}}_{\{x_{j-1}=x,x_j=y\}}\mathrm{.}$ Using the nonnegative path densities [F2], the expectation is the sum over $n\ge 1$, state sequences, and ordered jump-time simplices of this count times the path density. The state-sequence sums are finite at each $n$, and their union over $n$ is countable; the increasing partial sums and [F3] justify the countable sum. For each fixed $n$ and jump index $j$, write the ordered-time simplex integral as an integral over $[0,T{]}^n$ with its simplex indicator. This is a nonnegative measurable function: the factors are continuous rates and survival functions, and the simplex is Borel. Lebesgue measure on the cube is finite and counting measure on the finite state sequences is finite, so [F4] permits integrating first in the selected jump time $s=t_j$ and then summing the other times. Summing over $n$ and $j$ is again an increasing nonnegative sum, so [A1] and [F3] justify its rearrangement.

For each selected jump at time $s$, the Markov property factors its path density into the prefix density through state $x$, the transition factor ${\kappa }_{xy}(s)ds$, and the suffix density starting in $y$. At fixed $s$, all finite prefixes ending in $x$ partition $\{X_s=x\}$ up to the null event of a jump at the fixed time $s$, so their densities sum to $p_x(s)$. All finite suffix densities starting in $y$ sum to $1$ by the normalization and nonexplosion conclusion of [F2]. The nonnegative sum/integral interchange justified in step 1.1 therefore gives the Lebesgue integral ${\int }_{[0,T]}{p}_x(s){\kappa }_{xy}(s)d{\lambda }_1(s)$.

Conditional on $X_s=x$, the probability of at least one jump by time $t>s$ is $1-H_x(s,t)=1-e^{-u}$ with $u:={\int }_s^t{\lambda }_x(r)dr\ge 0$. The inequality $1-e^{-u}\le u$ and the rate bound give this probability at most $M(t-s)$, uniformly in $x$. Hence $\mathrm{\mid }{p}_x(t)-p_x(s)\mathrm{\mid }\le \mathbb{P}(X_t\mathrm{\ne }{X}_s)\le M\mathrm{\mid }t-s\mathrm{\mid }$, so each $p_x$ is continuous. The product $p_x(t){\kappa }_{xy}(t)$ is consequently continuous and bounded on $[0,T]$, so its Lebesgue integral is finite. If $T>0$, [F5] identifies that integral with the Riemann integral in the Statement. If $T=0$, both the count and integral are zero by [F6]. This proves the identity for every $T\ge 0$.