We provide explicit convergence rates for Chernoff-type approximations of convex monotone semigroups which have the form $S(t)f=\lim_{n\to\infty}I(\frac{t}{n})^n f$ for bounded continuous functions $f$. Under suitable conditions on the one-step operators $I(t)$ regarding the time regularity and consistency of the approximation scheme, we obtain $\|S(t)f-I(\frac{t}{n})^n f\|_\infty\leq cn^{-\gamma}$ for bounded Lipschitz continuous functions $f$, where $c\geq 0$ and $\gamma>0$ are determined explicitly. Moreover, the mapping $t\mapsto S(t)f$ is H\"older continuous. These results are closely related to monotone approximation schemes for viscosity solutions but are obtained independently by following a recently developed semigroup approach to Hamilton-Jacobi-Bellman equations which uniquely characterizes semigroups via their $\Gamma$-generators. The different approach allows to consider convex rather than sublinear equations and the results can be extended to unbounded functions by modifying the norm with a suitable weight function. Furthermore, up to possibly different consistency errors for the operators $I(t)$, the upper and lower bound for the error between the semigroup and the iterated operators are symmetric. The abstract results are applied to Nisio semigroups and limit theorems for convex expectations.

翻译：本文给出凸单调半群的Chernoff型逼近的显式收敛速率，该半群对一致连续有界函数$f$具有形式$S(t)f=\lim_{n\to\infty}I(\frac{t}{n})^n f$。在单步算子$I(t)$满足关于时间正则性和逼近方案相容性的适当条件下，对Lipschitz连续有界函数$f$，我们得到$\|S(t)f-I(\frac{t}{n})^n f\|_\infty\leq cn^{-\gamma}$，其中$c\geq 0$和$\gamma>0$被显式确定。此外，映射$t\mapsto S(t)f$是Hölder连续的。这些结果与粘性解的单调逼近方案密切相关，但通过最近发展的Hamilton-Jacobi-Bellman方程的半群方法独立得到，该方法利用$\Gamma$-生成子唯一刻画半群。这种不同方法允许考虑凸方程而非次线性方程，且通过引入适当权函数修改范数可将结果推广至无界函数。进一步地，在算子$I(t)$的相容性误差可能不同的情形下，半群与迭代算子之间误差的上界和下界具有对称性。这些抽象结果被应用于Nisio半群和凸期望的极限定理。