Convergence rates for Chernoff-type approximations of convex monotone semigroups

Abstract

We provide explicit convergence rates for Chernoff-type approximations of convex monotone semigroups which have the form $S\left(t\right)f={lim}_{n\to \infty }I{\left(\frac{t}{n}\right)}^{n}f$ for bounded continuous functions $f$. Under suitable conditions on the one-step operators $I\left(t\right)$ regarding the time regularity and consistency of the approximation scheme, we obtain ${\Vert S\left(t\right)f-I{\left(\frac{t}{n}\right)}^{n}f\Vert }_{\infty }\le c{n}^{-\gamma }$ for bounded Lipschitz continuous functions $f$, where $c\ge 0$ and $\gamma >0$ are determined explicitly. Moreover, the mapping $t\mapsto S\left(t\right)f$ is Hölder 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 $I\left(t\right)$, 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.