Nonlinear semigroups and limit theorems for convex expectations

2022/10/25 by Blessing, Jonas, Kupper, Michael · 4 citations
#60F05 #60F10 #60G50 #FOS: Mathematics #Primary 47H20 #Probability (math.PR) #Secondary 47J25

paper · doi:10.48550/arxiv.2210.14096

Abstract

Based on the Chernoff approximation, we provide a general approximation result for convex monotone semigroups which are continuous w.r.t. the mixed topology on suitable spaces of continuous functions. Starting with a family (I(t))_{t≥ 0} of operators, the semigroup is constructed as the limit S(t)f:=lim_n→∞I((t)/(n))ⁿ f and is uniquely determined by the time derivative I'(0)f for smooth functions. We identify explicit conditions for the generating family (I(t))_{t≥ 0} that are transferred to the semigroup (S(t))_{t≥ 0} and can easily be verified in applications. Furthermore, there is a structural link between Chernoff type approximations for nonlinear semigroups and law of large numbers and central limit theorem type results for convex expectations. The framework also includes large deviation results.

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