A Mean-Field Theory of $Θ$-Expectations

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.

• The paper introduces a framework using mean-field FBSDEs to systematically address non-convex, law-dependent uncertainty models.
• It proves local and global well-posedness via uniform Lipschitz stability and strong concavity, ensuring solution uniqueness and stability.
• The study reveals unique Θ-expectation properties that bridge stochastic calculus with non-local PDEs and have practical implications in finance and control.

Mean-Field Theory of $\Theta$-Expectations: A Rigorous Framework for Non-Convex Ambiguity

Introduction and Motivation

The paper develops a rigorous stochastic calculus for a class of non-convex, law-dependent uncertainty models, addressing a fundamental limitation in the canonical theory of sublinear expectations. In the convex paradigm, valuation functionals are insensitive to the non-convex geometry of the underlying model set, depending only on its convex hull. This "identifiability impasse" is problematic in applications where the fine structure of uncertainty—such as multimodal or disconnected model sets—matters. The work introduces a new class of mean-field forward-backward stochastic differential equations (FBSDEs) where the backward driver is defined by a pointwise maximization over a non-convex, law-dependent set, and the ambiguity is endogenous, shaped by the law of the value process itself.

Mathematical Framework

Function Spaces and Wasserstein Geometry

The analysis is set on a filtered probability space supporting a $d$-dimensional Brownian motion. The solution spaces for the FBSDEs are standard Banach spaces of adapted processes: $\mathcal{S}^p$ for continuous processes, $\mathcal{H}^p$ for square-integrable predictable processes, and $\mathcal{L}^p$ for control processes. The mean-field structure requires a metric on the space of probability measures, for which the $p$-Wasserstein distance is used, endowing $P_p(\mathbb{R}^k)$ with a Polish space structure. Differentiability on Wasserstein space is formalized via the Lions derivative, which is essential for the analysis of master equations.

Mean-Field $\Theta$-FBSDE System

The central object is a fully coupled mean-field FBSDE system:

• The forward SDE evolves under coefficients $b$ and $\sigma$ that depend on the state, control, and the law of the backward component.
• The backward SDE has a driver $F$ that is maximized pointwise over a non-convex, law-dependent set $U_{g(\mu_t)}$.
• The control process $A_t$ is the unique maximizer of $F$ at each time, with the uncertainty set itself depending on the law of the value process.

The system is characterized by a multi-layered coupling: standard FBSDE coupling, mean-field (law) dependence, and endogenous ambiguity through the law-dependent uncertainty set.

Well-Posedness Theory

Structural Assumptions

Mathematical tractability is achieved by imposing:

• Uniform strong concavity of the driver $F$ in the control variable, ensuring uniqueness and stability of the optimizer.
• Lipschitz continuity of all coefficients and the set-valued map $\theta \mapsto U_\theta$ (in the Hausdorff metric).
• Regularity and non-degeneracy of the boundary of the uncertainty sets (LICQ and strict complementarity).

Lipschitz Stability of the Optimizer

A central technical result is the uniform Lipschitz continuity of the optimizer map $a^*$ with respect to the state, backward variables, and the law. This is established via sensitivity analysis of parametric nonlinear programs, leveraging the implicit function theorem applied to the KKT system. The uniformity of the structural assumptions is critical for global stability.

Local and Global Well-Posedness

• Local Existence and Uniqueness: A contraction mapping argument on a product space of processes, using a weighted norm, yields local well-posedness for small time horizons.
• Global Existence and Uniqueness: Under additional strong monotonicity assumptions on the coefficients, a global well-posedness result is established via the method of continuity. The uniqueness argument is based on an energy estimate and a Gronwall-type argument, while existence is shown by constructing a homotopy of FBSDEs and demonstrating closedness and openness of the solution set.

Properties of the $\Theta$-Expectation

The unique solution to the BSDE component defines a non-linear valuation functional, the $\Theta$-Expectation, with the following properties:

• Dynamic Consistency: The operator is time-consistent, satisfying the tower property.
• Monotonicity: The operator preserves order.
• Failure of Sub-additivity: The operator is not sub-additive; explicit counterexamples are constructed where $E[\xi_1 + \xi_2] > E[\xi_1] + E[\xi_2]$.
• Failure of Translation Invariance: The operator is not translation invariant unless the driver is independent of the value variable.

These properties demonstrate a fundamental departure from the convex paradigm and confirm the operator's sensitivity to the non-convex geometry of the uncertainty set.

Stochastic Calculus and PDE Connections

$\Theta$-Martingales and Semimartingale Representation

A process is a $\Theta$-martingale if and only if it is the backward component of a solution to the FBSDE with zero driver. The value process admits a semimartingale decomposition, with the driver acting as the negative drift rate. The associated process, which accumulates the drift, is a classical martingale.

Nonlinear Feynman-Kac Formula and PDEs

In the Markovian setting, the value function $u(t,x)$ defined by the FBSDE is shown to be a viscosity solution to a non-local, non-linear Hamilton-Jacobi-Bellman (HJB) equation of McKean-Vlasov type, where the law of the value process enters as a parameter. The PDE is highly non-local and involves a supremum over a law-dependent, non-convex set.

Master Equation on Wasserstein Space

A formal derivation yields a master equation on the Wasserstein space, encoding the full dynamics of the endogenous, non-convex ambiguity model. The equation involves the Lions derivative and its Jacobian, and is both non-linear and non-local. The analysis of this equation is identified as a major open problem, due to the lack of a robust solution theory for such infinite-dimensional, non-smooth PDEs.

Application: Ambiguous Dynamical System

A concrete example is provided: a mean-field FBSDE with quadratic penalty on the control and a non-convex uncertainty set composed of disjoint intervals. The optimal control is the projection of a reference point onto the non-convex set, and the resulting dynamics are shown to be fundamentally different from those obtained by convexification (as in sublinear expectation theory). This illustrates the theory's ability to resolve the identifiability impasse and to model systems where the geometry of uncertainty is essential.

Implications and Future Directions

The framework provides a rigorous foundation for stochastic calculus under non-convex, endogenous ambiguity, with immediate applications in finance, economics, and control where model uncertainty is structured and systemic. The strong concavity assumption, while restrictive, is natural in settings with quadratic regularization or risk penalties. The theory opens new avenues for modeling and analysis, but also delineates significant open problems, particularly the direct analysis of the associated master equation and the development of a robust PDE theory on Wasserstein space for non-convex, law-dependent drivers.

Conclusion

This work establishes a mathematically rigorous and structurally novel theory for stochastic calculus under non-convex, endogenous ambiguity. By leveraging strong concavity and geometric regularity, it achieves well-posedness and dynamic consistency while decisively breaking from the convex paradigm. The framework is sensitive to the fine structure of uncertainty, resolves the identifiability impasse, and provides a foundation for further developments in the analysis of non-convex mean-field systems and their associated PDEs.