Finite-state Gillespie and next-reaction methods have the specified CTMC law

For one channel clock sequence with independent unit exponential gaps, the joint density of exactly $m$ ordered thresholds $0<s_1<\cdots <s_m\le u$ and no next threshold by $u$ is ${\prod }_{i=1}^m{e}^{-(s_i-s_{i-1})}{e}^{-(u-s_m)}=e^{-u},s_0=0.$ This agrees with [F4] at unit rate, so each $Y_j$ in the next-reaction construction is a unit-rate Poisson process on finite windows. The streams for distinct channels are independent by [F5], and after an event the unused sequence coordinates remain independent: partition on the finite index of the last used coordinate and use the finite-coordinate product law.

At state $x$ with $\lambda (x)>0$, the direct method's waiting time $E\mathrm{/}\lambda (x)$ has survival $e^{-\lambda (x)t}$ by [F2]. The disjoint cumulative intervals for $U$ have lengths $a_j(x)\mathrm{/}\lambda (x)$, so channel $j$ is selected with that probability; because $U$ and $E$ are independent, the channel is independent of the waiting time. Thus the direct method uses exactly the reference event kernel. If $\lambda (x)=0$, it returns the absorbing path specified in the Statement.

At a state $x$, the next-reaction clock of channel $j$ has residual internal threshold distributed as an independent $\mathrm{Exp}(1)$ variable. By [F2] their minimum has the same event-time and channel law as in step 1.2. After a reaction at elapsed time $\delta$, every nonwinning channel $k$ is conditioned only on its threshold exceeding $a_k(x)\delta$; memorylessness follows exactly from $\mathbb{P}(E_k>a_k(x)\delta +u\mid E_k>a_k(x)\delta )=e^{-u}\mathrm{.}$ The residual thresholds remain independent because the conditioning events factor across independent clocks, and the fired channel receives a fresh independent exponential increment. Induction over the reaction count proves that every successive event kernel agrees with step 1.2, hence the full marked path law agrees. The threshold construction also gives $X_t=X_0+{\sum }_j{\nu }_j{Y}_j({\int }_0^t{a}_j(X_{s-})ds)$ by its definition.

Apply the bounded-envelope thinning theorem with the time-constant rates $a_j(t,x)=a_j(x)$ and envelope $\mathrm{\Lambda }={\max }_{x\in E}\lambda (x)$. It gives finitely many events almost surely and $\mathbb{E}N(T)\le \mathrm{\Lambda }T$. For bounded $f$ and small $h$, the probability of no reaction is $e^{-\lambda (x)h}=1-\lambda (x)h+O(h^2)$; the probability of exactly one channel-$j$ reaction is $a_j(x)h+O(h^2)$ by its first-event density; and the probability of two or more is $O(h^2)$ by the candidate Poisson bound in [F6]. Therefore ${\lim }_{h\downarrow 0}\frac{{\mathbb{E}}_x[f(X_h)]-f(x)}{h}={\sum }_{j:a_j(x)>0}{a}_j(x)(f(x+{\nu }_j)-f(x)),$ which is the operator in [F1]. A zero-increment channel contributes zero to this expression even though its marked event is counted.

Steps 1.2 and 2.1 identify both algorithmic event kernels, starting at the fixed $x_0$, with the reference kernel; the finite event bound in step 3.1 makes the unlimited-budget comparison a full-horizon path-law equality. A firing at a fixed deterministic horizon has probability zero, since each successive event time has a density conditional on the preceding history; the countable union over event indices is still null. Thus the algorithms' endpoint branches do not alter the path-law conclusion. With a finite budget, the same event-kernel comparison holds up to the algorithm's stopping rule and gives exactly the stated prefix law. Step 3.1 computes the generator and expected event-count bound.