A Mean-Field Theory of $Θ$-Expectations

• URL: http://arxiv.org/abs/2507.22577v1
• Date: Wed, 30 Jul 2025 11:08:56 GMT
• Title: A Mean-Field Theory of $Θ$-Expectations
• Authors: Qian Qi,
• Abstract summary: We develop a new class of calculus for such non-linear models.<n>Theta-Expectation is shown to be consistent with the axiom of subaddivity.
• Score: 2.1756081703276
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The canonical theory of sublinear expectations, a foundation of stochastic calculus under ambiguity, is insensitive to the non-convex geometry of primitive uncertainty models. This paper develops a new stochastic calculus for a structured class of such non-convex models. We introduce a class of fully coupled Mean-Field Forward-Backward Stochastic Differential Equations where the BSDE driver is defined by a pointwise maximization over a law-dependent, non-convex set. Mathematical tractability is achieved via a uniform strong concavity assumption on the driver with respect to the control variable, which ensures the optimization admits a unique and stable solution. A central contribution is to establish the Lipschitz stability of this optimizer from primitive geometric and regularity conditions, which underpins the entire well-posedness theory. We prove local and global well-posedness theorems for the FBSDE system. The resulting valuation functional, the $\Theta$-Expectation, is shown to be dynamically consistent and, most critically, to violate the axiom of sub-additivity. This, along with its failure to be translation invariant, demonstrates its fundamental departure from the convex paradigm. This work provides a rigorous foundation for stochastic calculus under a class of non-convex, endogenous ambiguity.

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