A Mean-Field Theory of $\Theta$-Expectations

TL;DR

This paper develops a new stochastic calculus for non-convex models of uncertainty, introducing a class of mean-field FBSDEs with a maximization-driven driver, and reveals a valuation functional that violates classical axioms.

Contribution

It introduces a novel mean-field FBSDE framework for non-convex uncertainty models, establishing well-posedness and properties of the resulting valuation functional.

Findings

The $ heta$-Expectation is dynamically consistent.

The $ heta$-Expectation violates sub-additivity.

The framework ensures unique and stable solutions under concavity assumptions.

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.