Martingales On A Euclidean Manifold With A Boundary And Reflected BSDES In Non-Convex Domains

Abstract: The purpose of this paper is twofold. First, we introduce the notion of a $Γ$-martingale on a Euclidean manifold with a boundary (i.e., the closure of an open connected domain in R d ), we provide its equivalent characterization through the $Γ$-convex functions, and we establish its connection with the reflected backward stochastic differential equations (BSDEs) in the associated domain. Second, we show how the tools of stochastic geometry can be used to develop a new method for proving existence and uniqueness of solutions to reflected BSDEs. We implement this method and obtain a well-posedness result for reflected BSDEs in any bounded, two-dimensional, simply-connected domain that is locally C2 -diffeomorphic to a convex set. This work extends the results of [6] and [16].

• The paper introduces a novel framework for T-martingales on manifolds with boundary and links them to reflected BSDEs using coordinate-based T-convex functions.
• The paper employs stochastic geometric methods and CAT(0) space properties to prove existence and uniqueness of solutions without standard penalization schemes.
• The paper demonstrates stability and well-posedness of reflected BSDEs, paving the way for advances in stochastic control and financial mathematics in nonconvex settings.

Martingales on Euclidean Manifolds with Boundary and Reflected BSDEs in Non-Convex Domains

Introduction and Motivation

This work develops a rigorous framework for martingales (specifically, "𝕋-martingales") on Euclidean manifolds with boundary and uses this framework to analyze reflected backward stochastic differential equations (BSDEs) in non-convex domains. The primary motivation stems from the challenge that, in multidimensional settings, reflected BSDEs are well-posed only under restrictive geometric conditions—most notably, domain convexity. The zero-generator case in non-convex domains is especially intricate due to the failure of standard comparison principles and the delicate role of the martingale and reflection terms in maintaining domain constraints.

The paper achieves two main objectives:

1. The introduction and robust characterization of 𝕋-martingales in manifolds with boundary, connecting their structure to reflected BSDEs.
2. The development of new analytic tools, grounded in stochastic geometry, to establish existence and uniqueness of solutions for reflected BSDEs in a class of bounded, simply connected, locally convex-like two-dimensional domains.

These results broaden the scope of reflected BSDE well-posedness beyond convex and star-shaped domains, and provide a precise analytic and geometric understanding of martingale behavior constrained by non-convex boundaries.

Notion of 𝕋-Martingale in Manifolds with Boundary

The paper defines a 𝕋-martingale on a manifold with boundary (formally, the closure of a domain $D \subset \mathbb{R}^d$) as a continuous Euclidean semimartingale constrained by a reflection process $K$ whose increments always point in the exterior normal cone $n(x)$ at the boundary, and a local martingale component. The explicit coordinate-based construction leverages the ambient Euclidean structure, permitting characterization in terms of so-called 𝕋-convex functions—test functions convex along geodesics (Definition 2.1-2.4).

This approach is advantageous in settings where intrinsic differential geometry is technically prohibitive due to boundary irregularity. The connection between reflected BSDEs and 𝕋-martingales is made explicit: the solution process $Y$ of a reflected BSDE is always a 𝕋-martingale with drift given by the driver $f$.

A critical contribution is the equivalence between the 𝕋-martingale property and a monotonicity condition for finite variation processes applied to 𝕋-convex functions (Proposition 2.2, Corollary 2.1). This analytic characterization paves the way for both existence and uniqueness results.

Geometric Properties of Two-Dimensional Domains

The analysis restricts to two-dimensional, bounded, simply connected domains ($d=2$) possessing a local boundary regularity property (C2-diffeomorphic to convex). Under these conditions, the closed domain endowed with its geodesic metric is a $\mathrm{CAT}(0)$ space (Theorem 3.1), ensuring:

• Existence and uniqueness of minimizing geodesics,
• Geodesic distance being 𝕋-convex,
• Global convexity properties necessary for constructing Lyapunov functions and controlling variational processes.

These geometric properties, particularly strong convexity of the squared geodesic distance, underpin the adaptability of stochastic calculus tools (Itô's formula extensions) to the setting (Corollary 3.2, Proposition 3.7).

Existence and Uniqueness Results for Reflected BSDEs

Existence

The existence result departs from previous methods heavily reliant on penalization schemes and strict domain assumptions. Instead, it adapts Kendall's construction for martingales in manifolds to the reflected BSDE setting. Key steps involve:

• Iterative construction of finite-step approximations (via Fréchet averages in CAT(0) spaces) for terminal and drift data,
• Backward recursion and dynamic programming based on geodesic convexity,
• Inclusion of a transport step accounting for the generator in the BSDE,
• Passage to the limit in a suitable function space, leveraging compactness and regularity.

This construction is shown to produce continuous adapted solutions in $\mathcal{D}$ (Proposition 4.5, Theorem 4.2).

Uniqueness and Stability

Uniqueness is shown using a Lyapunov-type argument with the squared geodesic distance as the controlling functional, exploiting its strong convexity (Theorem 3.2, 4.1). Estimates are performed in the quadratic mean, with further generalizations employing BMO arguments for the martingale components.

The methods also yield stability with respect to initial conditions, terminal values, and generators, facilitating approximation and regularization arguments essential to the existence proofs.

Characterization via Fréchet Means and Jensen Inequalities

A technical highlight is the extension of properties of the Fréchet mean (minimizer of the expected squared geodesic distance) to the present setting. This enables dynamic programming-style iterations and transfer of classical Jensen inequalities and submartingale conditions to the nonlinear, constrained domain (Appendix, Propositions 5.1-5.3).

Implications and Future Directions

The formulated theory advances the martingale and BSDE toolkit for domains exhibiting significant non-convexity but sufficient local regularity. Practically, reflected BSDEs in non-convex domains appear in constrained stochastic control and mathematical finance, where dynamic constraints rarely conform to convexity.

The results indicate that stochastic geometric methods—particularly those exploiting non-positively curved (CAT(0)) spaces—are powerful for addressing constraint problems not susceptible to comparison principles or coordinate tricks. Geodesic convexity and metric structure replace much of the lost linearity.

The paper conjectures extension of existence/uniqueness results to two-dimensional Cartan-Hadamard manifolds (complete, simply connected, non-positive curvature), conditioned on analogous regularity and convexity properties for the squared distance and boundary projections.

A significant open challenge is the extension to higher dimensions, where CAT(0) properties are much more restrictive and where minimal geodesic uniqueness may fail even in topologically simple domains. Analysis of reflected BSDEs in such settings will likely require novel geometric and analytic techniques.

Conclusion

This paper provides a comprehensive framework for the study of martingales and reflected BSDEs on Euclidean manifolds with boundary, establishing well-posedness for a broad class of non-convex domains in dimension two. The main contributions are the coordinate-based definition and characterization of 𝕋-martingales, the adaptation of probabilistic geometric methods for existence, and the demonstration of uniqueness and stability rooted in CAT(0) geometry. These advances set a foundation for further investigations into stochastic analysis on general manifolds with constraints, and the extension of BSDE theory to settings of practical and theoretical importance.