Stopped density-scaled jump versus chemical Langevin weak model error

Assume Countable Choice and Dependent Choice. Use the density-scaled generators in Density-scaled reaction jump and chemical Langevin generators. Let $K\subset U$ be compact and the closure of a bounded open stopping region with nonempty interior, let $x_0\in \mathrm{int}K$, and let ${\mathcal{L}}_V=x_0+\frac{1}{V}{\mathbb{Z}}^d$ be a containing lattice. For fixed $V$, the set $D=\mathrm{int}K\cap {\mathcal{L}}_V$ is finite: boundedness of $K$ bounds each integer coordinate index in the lattice. Let $X^V$ be the jump process stopped at ${\tau }_X=\inf \{t\ge 0:X_t^V\mathrm{\notin }\mathrm{int}K\}$ and let $Z^V$ be a chemical Langevin diffusion stopped at ${\tau }_Z=\inf \{t\ge 0:Z_t^V\mathrm{\notin }\mathrm{int}K\}$. Assume both stopped processes exist with $X_0^V=Z_0^V=x_0$, and write ${\bar{X}}_t=X_{t\wedge {\tau }_X}^V$ and ${\bar{Z}}_t=Z_{t\wedge {\tau }_Z}^V$.

Assume $a_j\in C^3(U)$, $0\le a_j\le A_j$ on $K$, and every segment $[x,x+{\nu }_j\mathrm{/}V]$ for $x\in D$ with $a_j(x)>0$ lies in $U$. For a bounded payoff $g$ with $\mathrm{\mid }g\mathrm{\mid }\le G$, suppose the stopped CLE backward solution $u(s,x)={\mathbb{E}}_x[g(Z_{(T-s)\wedge {\tau }_Z}^V)]$ has an extension to an open neighborhood of $K$ and all these jump segments that is $C^3$ in space and $C^1$ in time, with $u(T,y)=g(y)$ on all states reached by the stopped jump chain and $u(s,y)=g(y)$ on every exit state (including diffusion boundary points), and ${\mathrm{\partial }}_{s}u+{\mathcal{G}}_{V}u=0\text{on }[0,T]\times \mathrm{int}K,\mathrm{\parallel }{D}^{3}u(s,y)\mathrm{\parallel }\le M,$ and ${\mathrm{\partial }}_{s}u$ is continuous in time on the stated neighborhood. The third-derivative bound holds along every jump segment. The derivatives are spatial Fréchet derivatives with operator norm. Then $\mid \mathbb{E}g({\bar{X}}_T)-\mathbb{E}g({\bar{Z}}_T)\mid \le \frac{TM}{6V^2}{\sum }_{j=1}^R{A}_j\mathrm{\parallel }{\nu }_j{\mathrm{\parallel }}^3\mathrm{.}$ If the original local jump and diffusion models are defined through $T$ and are coupled to agree with their stopped versions up to their respective exits, then $\mid \mathbb{E}g(X_T^V)-\mathbb{E}g(Z_T^V)\mid \le \frac{TM}{6V^2}{\sum }_{j=1}^R{A}_j\mathrm{\parallel }{\nu }_j{\mathrm{\parallel }}^3+2G(\mathbb{P}({\tau }_X\le T)+\mathbb{P}({\tau }_Z\le T))\mathrm{.}$ The $V^{-2}$ term is density-model error from the generator expansion; the exit term is separate. Neither term is time-discretization error. The existence and regularity of the CLE backward solution are assumptions, not consequences of this estimate.