Set-Valued Stochastic Analysis

• Set-Valued Stochastic Analysis is the study of stochastic processes whose states are sets, addressing ambiguity and nonsmooth dynamics.
• It extends classical calculus via set-valued integrals, differential inclusions, and martingale representations using convex analysis and measurable multifunctions.
• Applications include nonlinear filtering, mathematical finance, robust control, and mean-field games to manage uncertainty and risk.

Set-valued stochastic analysis is the rigorous paper of random dynamical systems, stochastic processes, integration, and evolution equations where states, trajectories, or observables take values in families of sets (typically convex, closed, sometimes unbounded) rather than in vector spaces. Driven by challenges in modeling nondeterminism, ambiguity, nonsmoothness, multi-objective optimization, risk under uncertainty, and mean-field limits, the field develops and analyzes set-valued analogs of stochastic integrals, differential inclusions and equations, sub/supermartingales, and dynamic programming, using the tools of measurable multifunctions, convex analysis, and geometric measure theory. Recent advances document precise path regularity, martingale representation, propagation of chaos, superhedging, robust control, and infinite-dimensional dynamics, with applications spanning nonlinear filtering, mathematical finance, control under Knightian uncertainty, mean-field games, and more.

1. Set-Valued Integration: Definitions and Core Constructions

A fundamental object in set-valued stochastic analysis is the set-valued stochastic integral, which generalizes the classical Itô or Young integrals by integrating set-valued (multivalued) predictable processes with respect to Brownian motion, Poisson random measures, Lévy processes, or more general semimartingales.

Given a filtered probability space $(\Omega, \mathcal F, \mathbb F, P)$ and a Banach (commonly Hilbert) space $X$:

• Aumann–Itô integral: For a set-valued process $\Gamma_t$ with closed convex nonempty values, the Aumann–Itô integral is defined as the closed convex hull of all Itô integrals of measurable selectors:

$$\int_0^T \Gamma_s\,dB_s := \overline{\mathrm{conv}}\left\{ \int_0^T \gamma_s\,dB_s : \gamma \in L^2_{\text{prog}},\ \gamma_s \in \Gamma_s\ \text{a.s.} \right\}$$

(Ararat et al., 2020, Ararat et al., 2023, Tuyen et al., 2023)

• Closed decomposable hulls: Since pointwise set-valued images of stochastic integrals are typically not decomposable or measurable, the closed decomposable hull (with respect to the filtration and the $L^p$ norm) is imposed to produce random sets whose selectors are stable under gluing along partitions (Tuyen et al., 2023, Ararat et al., 2023).
• Volterra-type (convoluted) set-valued integrals: For integration against memory kernels $K(t,s)$, for a predictable $h$ and kernel $K$ (square-integrable in $(t,s)$), the integral

$I_h(t) = \int_0^t K(t,s)\,dL(s)$

is extended to set-valued integrands via the closed decomposable hull of all such functionals (Xia, 2023).

• Multifunctional Young integrals: When the integrator is a Hölder path (e.g., for fBm with Hurst $H > 1/2$), the set-valued Young integral is defined via a compact set of Hölder-continuous selections, leading to closed convex-valued images with controlled regularity (Coutin et al., 2021, Michta et al., 2020).
• Generalization to UMD Banach spaces: In UMD (Unconditional Martingale Differences) Banach spaces, set-valued stochastic integrals are developed via y-radonifying operator-valued theory, with compatible martingale representation (Essaky et al., 9 Dec 2024).

2. Set-Valued Stochastic Differential Equations and Inclusions

Set-valued SDEs and stochastic differential inclusions (SDIs) generalize standard SDEs by allowing drift, diffusion, and jump coefficients to take values in families of sets, yielding equations of the form:

$$X_t \in \xi \oplus \int_0^t F(s, X_s)\,ds \oplus \int_0^t G(s, X_s)\,dW_s$$

where $X_t$ evolves in a set-space (e.g., all convex closed sets, or "upper sets" generated by a solvency cone), and $F, G$ are set-valued (Almuzaini et al., 23 Mar 2024).

• Existence and uniqueness: Under Carathéodory or measurable-selection hypotheses, Lipschitz-type conditions (in Hausdorff or other appropriate metrics), and appropriate initial data, strong existence/uniqueness can be established via set-valued Picard iterations in complete metric (Hausdorff or Lusin) spaces (Almuzaini et al., 23 Mar 2024, Ararat et al., 2020, Zhang et al., 2021).
• Unbounded coefficients: By working with "LC-spaces" of convex closed sets generated by a cone, one recovers cancellation properties and closedness necessary for analysis in financial applications and for risk measures that are unbounded above (Almuzaini et al., 23 Mar 2024).
• Fractional and rough noise: For analysis under fractional Brownian motion, the pathwise theory of Young and Zähle integration with multifunctional selections is critical. When jumps are present and kernels are singular at the diagonal, set-valued stochastic convoluted integrals (Volterra type) can become unbounded or even "explode" (take extended vector values) due to the interplay of singularity and jump structure (Xia, 2023).
• Backward SDEs and set-valued BSDEs: Both Minkowski addition and the Hukuhara difference are employed to generalize backward SDEs to the set-valued domain. Well-posedness (existence and uniqueness) holds under appropriate Lipschitz and measurability criteria, provided the generator satisfies a two-argument (in $Y,Z$) Hukuhara-Lipschitz condition (Ararat et al., 2020, Zhang et al., 2021, Essaky et al., 9 Dec 2024).

3. Martingales, Submartingales, and Path Regularity in the Set-Valued Context

Classical martingale theory is extended to random sets valued in closed convex subsets of Banach or Hilbert spaces.

• Set-valued submartingales/martingales: A family $\{F_t\}_{t \ge 0}$, with $F_t: \Omega \to K_{bc}(X)$ (bounded, closed convex sets), is a (sub)martingale if $$E[F_t | \mathcal F_s] \supseteq (\text{resp. } =) F_s$$ for $s \leq t$ (Tuyen et al., 2023, Ararat et al., 2023, Zhang et al., 2020).
• Martingale representation: Every $L^2$-integrably bounded set-valued martingale can be represented as a revised set-valued stochastic integral—explicitly, for Banach/M-type 2 or UMD spaces, selectors can be written as sums of an initial value and a stochastic integral, with the full process characterized as the closed decomposable hull of all possible such representations (Essaky et al., 9 Dec 2024, Tuyen et al., 2023).
• Path regularity: Set-valued (sub)martingales admit modifications which are right-continuous with left limits in the Hausdorff topology, and under Brownian filtrations, set-valued martingales are even Hausdorff-continuous (Ararat et al., 2023).
• Necessary and sufficient conditions for martingality: The set-valued stochastic integral of a family of integrands yields a true martingale if and only if the $L^2$ closure and decomposability conditions hold for the set of terminal values (Ararat et al., 2023, Tuyen et al., 2023).

4. Boundedness, Explosion, and Regularity of Set-Valued Integrals

• Boundedness dichotomy: For set-valued stochastic integrals with respect to Poisson random measures or Lévy processes, integrable boundedness for bounded integrands holds if and only if the Poisson measure is of finite variation. In the infinite variation case, the integral can become unbounded in $L^2$ unless integrands are restricted to singletons (Xia, 2023).
• Explosion due to jump/singularity interaction: For indefinite integrals with Volterra-type (e.g., fractional) kernels that are singular at the diagonal and Poisson random measures with jumps, the