Convergence rates for Chernoff-type approximations of convex monotone semigroups

TL;DR

This paper establishes explicit convergence rates for Chernoff-type approximations of convex monotone semigroups, extending previous methods to convex equations and unbounded functions, with applications to Nisio semigroups and convex expectations.

Contribution

It introduces a new semigroup approach to derive convergence rates for convex monotone semigroups, broadening the scope beyond sublinear equations and bounded functions.

Findings

Derived explicit convergence rates with error bounds for semigroup approximations.

Proved Hölder continuity of the semigroup mappings in time.

Extended results to unbounded functions using weighted norms.

Abstract

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.