Exotic one-parameter semigroups of endomorphisms of a symmetric cone

TL;DR

This paper constructs a novel one-parameter semigroup of endomorphisms on a symmetric cone, revealing a generator not decomposable into a Lie group generator and an endomorphism, with implications for affine processes in finance.

Contribution

It introduces an exotic semigroup with a generator that defies traditional decomposition, addressing a long-standing theoretical open problem.

Findings

Constructed an exotic semigroup of endomorphisms

Identified a generator not expressible as sum of a Lie group generator and an endomorphism

Implications for affine processes on symmetric cones

Abstract

We construct an exotic one-parameter semigroup of endomorphims of a symmetric cone $C$, whose generator is not the sum of a Lie group generator and an endomorphism of $C$. The question is motivated by the theory of affine processes on symmetric cones, which plays an important role in mathematical finance. On the other hand, theoretical question that we solve in this paper seems to have been implicitly open even much longer then this motivation suggests.