Global Structure and Local Specifications in Sublinear Valuation
Abstract
To what extent does a local operator on smooth test functions determine a sub-Markov semigroup and its underlying structure of laws? Many existing approaches work with Feller semigroups on the space Cb of bounded continuous functions and rely on uniqueness, often established through a comparison principle for the associated evolution equation. However, even when the local operator is linear, uniqueness may fail and a Cb-Feller restriction may exclude the associated canonical semigroup. We discover that the space USCb of bounded upper semicontinuous functions offers a natural setting for this study. Within this framework, a complete description of the relationships among sublinear local operators, sublinear sub-Markov semigroups, and their underlying structures of laws is achieved without assuming uniqueness. Under suitable regularity and Lyapunov conditions, we construct the greatest semigroup compatible with a prescribed sublinear local operator and obtain a stochastic representation in terms of a Markov process under sublinear expectations. We characterize all sublinear sub-Markov semigroups dominated by this greatest semigroup in two ways: through a one-to-one correspondence with stable subfamilies of its representing laws, and through an order relation between their infinitesimal generators and the prescribed local operator. We further characterize the greatest semigroup via the greatest viscosity subsolution of the associated Hamilton--Jacobi--Bellman equation. These results clarify what information about global stochastic evolution can be recovered from local infinitesimal descriptions and what freedom remains in the absence of uniqueness. The resulting characterizations are new even when the local operator is linear.
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BibTeX
@misc{park2026global,
title = {{Global Structure and Local Specifications in Sublinear Valuation}},
author = {Jongjin Park and David Criens and Hyungbin Park},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37881},
eprint = {2609.37881},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37881},
}
RIS
TY - UNPB
TI - Global Structure and Local Specifications in Sublinear Valuation
AU - Jongjin Park
AU - David Criens
AU - Hyungbin Park
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37881
UR - https://arxiv.org/abs/2609.37881
AB - To what extent does a local operator on smooth test functions determine a sub-Markov semigroup and its underlying structure of laws? Many existing approaches work with Feller semigroups on the space C_b of bounded continuous functions and rely on uniqueness, often established through a comparison principle for the associated evolution equation. However, even when the local operator is linear, uniqueness may fail and a C_b-Feller restriction may exclude the associated canonical semigroup. We discover that the space USC_b of bounded upper semicontinuous functions offers a natural setting for this study. Within this framework, a complete description of the relationships among sublinear local operators, sublinear sub-Markov semigroups, and their underlying structures of laws is achieved without assuming uniqueness. Under suitable regularity and Lyapunov conditions, we construct the greatest semigroup compatible with a prescribed sublinear local operator and obtain a stochastic representation in terms of a Markov process under sublinear expectations. We characterize all sublinear sub-Markov semigroups dominated by this greatest semigroup in two ways: through a one-to-one correspondence with stable subfamilies of its representing laws, and through an order relation between their infinitesimal generators and the prescribed local operator. We further characterize the greatest semigroup via the greatest viscosity subsolution of the associated Hamilton--Jacobi--Bellman equation. These results clarify what information about global stochastic evolution can be recovered from local infinitesimal descriptions and what freedom remains in the absence of uniqueness. The resulting characterizations are new even when the local operator is linear.
ER -