Mean-Field BSVIEs: Memory and Mean-Field Effects

• Mean-field BSVIEs are stochastic integral equations incorporating memory effects and law-dependence to extend classical BSVIE frameworks.
• They involve adapted M-solutions, well-posedness under Lipschitz or quadratic conditions, and detailed existence and uniqueness analyses.
• Applications span dynamic risk measures, stochastic optimal control, and particle approximations showcasing propagation of chaos.

Mean-field backward stochastic Volterra integral equations (mean-field BSVIEs, sometimes abbreviated as MF-BSVIEs) constitute a class of stochastic integral equations used to model systems with both memory effects and nonlinear interactions involving the law (distribution) of the solution processes. These equations generalize classical backward stochastic Volterra equations by introducing mean-field (law-dependent) terms, and arise in a variety of applications including stochastic control with memory, dynamic risk measures, and systems with large populations or distributed parameters. Recent developments encompass analysis in both finite and infinite dimensions, equations with singular Volterra kernels, quadratic growth, and associated interacting particle systems.

1. General Formulation and Function Space Setting

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a complete probability space equipped with a $d$-dimensional Brownian motion $W=(W_t)_{t \in [0,T]}$ and the filtration $\mathbb{F} = (\mathcal{F}_t)_{0 \leq t \leq T}$. The classical mean-field BSVIE has the form

$$Y(t) = \psi(t) + \int_t^T g\bigl(t, s, Y(s), Z(t, s), \mathcal{L}(Y(s)), \mathcal{L}(Z(t, s))\bigr) ds - \int_t^T Z(t, s) dW_s, \quad t \in [0, T],$$

where

• $Y: [0,T] \times \Omega \to \mathbb{R}^n$ and $$Z: \Delta[0,T] \times \Omega \to \mathbb{R}^{n \times d}$$ are unknown adapted processes, with $\Delta[0,T] = \{ (t,s) : 0 \leq t \leq s \leq T \}$
• $\psi(t)$ is given, typically $\mathcal{F}_T$-measurable, and square-integrable
• $g$ is the generator, possibly exhibiting linear or quadratic growth, with explicit dependence on the law $\mathcal{L}(\cdot)$ of solution components

A mean-field BSVIE is said to admit an adapted M-solution if it holds in the Itô sense for almost every $t$, $Y\in L^2_\mathbb{F}([0,T])$, $Z\in L^2_\mathbb{F}(\Delta[0,T];\mathbb{R}^d)$ (or stronger spaces in the quadratic case), and the process $Y(t)$ admits the martingale representation

$Y(t) = \mathbb{E}[Y(t)] + \int_0^t Z(t, s) dW_s.$

In the infinite-dimensional setting, these equations are generalized for Hilbert-space-valued processes, with integration against cylindrical Brownian motions and operator-valued $Z$, and the function spaces generalized accordingly (Asadzade et al., 29 Nov 2024).

2. Main Assumptions: Generator Structure and Well-posedness

Regularity and growth conditions on the generator $g$ (or $P$ in infinite dimensions) are central for well-posedness:

• Measurability and Progressivity: $g$ must be progressively measurable in $s$ for fixed other parameters.
• Lipschitz/Kernel Condition:
  • Finite-dimensions: Uniform Lipschitz conditions in all arguments, possibly with constants $L$ for each variable or group of variables (see (Shi et al., 2011, Hao et al., 10 Nov 2025)).
  • Infinite-dimensions and Singular kernels: The kernel functions controlling the Lipschitz constants (e.g., $L_{x_1}(t,s)$) may belong to $L^2$ over the domain $\Delta$, hence possibly unbounded, which is termed "singular" (Asadzade et al., 29 Nov 2024).
• Growth: For linear growth, $$|g(\cdots)| \leq L(|y| + |z| + \mathcal{W}_2(\mu, \delta_0) + \mathcal{W}_2(\nu, \delta_0))$$; for quadratic growth, $|g(\cdots)| \leq C|z|^2 + \text{lower-order terms}$.

For generators of quadratic growth in $z$, $g$ may also admit law-dependence in $z$, under additional structural conditions for well-posedness and integrability (Hao et al., 10 Nov 2025).

The solution concept in these singular/infinite-dimensional settings is adapted as adapted M-solutions in the appropriate function spaces, often $L^2$ in time and Hilbert-space variables (Asadzade et al., 29 Nov 2024). For the finite-dimensional case, continuity in $L^2$ and integrability of $Z$ over $\Delta$ ensures the martingale representation.

3. Existence, Uniqueness, and Regularity

Existence and Uniqueness

• Linear Growth: Under the aforementioned regularity and growth conditions (e.g., (A1)-(A3) for $\zeta$ and $g$; or (A1) in (Shi et al., 2011, Hao et al., 10 Nov 2025)), there exists a unique adapted solution $(Y, Z)$ to the MF-BSVIE in $L^2$ spaces. The standard proof utilizes contraction mappings in weighted norms, parameterizing the BSVIE as a family of mean-field BSDEs on $[t,T]$ (Shi et al., 2011, Hao et al., 10 Nov 2025).
• Quadratic Growth: Local solvability (on small intervals $[T-\varepsilon, T]$) is established under bounded coefficients with further extension to global solutions by backward induction; this uses BMO-martingale and exponential transform techniques to control $Z$ (Hao et al., 10 Nov 2025).
• Singular/Infinite-Dimensional Case: For singular kernels, the mapping properties are preserved by partitioning the time interval and solving on each subinterval via contraction, then gluing the local solutions (Asadzade et al., 29 Nov 2024).

An a priori estimate of the form

$$\mathbb{E}\int_0^T |Y(t)|^2 dt + \mathbb{E} \iint_\Delta \|Z(t,s)\|^2 ds\,dt \leq C \mathbb{E}\int_0^T |\psi(t)|^2 dt$$

is typical, and similar stability estimates hold under perturbation of the input data.

Regularity and Malliavin Calculus

Under strengthened smoothness and Malliavin-differentiability of the generator and terminal data, the M-solution $(Y, Z)$ to a mean-field BSVIE/BDSVIE is Malliavin differentiable. The Malliavin derivatives $(D_r^iY(\cdot), D_r^iZ(\cdot, \cdot))$ satisfy a (linear) mean-field backward doubly stochastic Volterra equation, and in the region $t>r$: $$D_r^i Y(t) = Z_i(t, r) + \int_r^t D_r^i Z(t, s) dW_s, \quad Z_i(t, r) = \mathbb{E}[D_r^i Y(t)\mid \mathcal{F}_r].$$ This regularity result is established via Picard iteration and application of Malliavin calculus to the doubly stochastic framework (Yang et al., 2023).

4. Particle Approximations and Propagation of Chaos

Finite $N$-particle systems are constructed to approximate the solution to a mean-field BSVIE. Each particle $i$ solves

$$Y^{N,i}(t) = \psi^i(t) + \int_t^T g(t, s, Y^{N,i}(s), Z^{N,i,i}(t, s), \mu^N(s), \nu^N(t, s)) ds - \sum_{j=1}^N \int_t^T Z^{N,i,j}(t, s) dW^j_s,$$

with empirical measures

$$\mu^N(s) = \frac{1}{N}\sum_{k=1}^N \delta_{Y^{N,k}(s)},\qquad \nu^N(t, s) = \frac{1}{N} \sum_{k=1}^N \delta_{Z^{N,k,k}(t, s)}.$$

Under the linear growth regime, quantitative rates of convergence ($\mathscr{Q}(N)$) for the propagation of chaos are derived, where

$$\mathbb{E} \bigg[\int_0^T |Y^{N,i}(t)-Y^i(t)|^p\,dt + \int_0^T \Big(\int_t^T \sum_{j=1}^N |Z^{N,i,j}(t,s) - \delta_{ij} Z^i(t,s)|^2 ds\Big)^{p/2} dt\bigg] \leq C\, \mathscr{Q}(N),$$

with $\mathscr{Q}(N)$ scaling as $N^{-(2-p)/2}$ for $d=1,2,3$ (Hao et al., 10 Nov 2025). In quadratic-growth, law-independent-of-$Z$ cases, convergence is of order $\mathscr{O}(N^{-1/(2\lambda)})$, $\lambda>1$. The analysis uses empirical measure estimates and Grönwall-type arguments.

A plausible implication is that such results pave the way for rigorous numerical simulation of BSVIE/MF-BSVIEs via particle schemes and Monte Carlo methods.

5. Comparison Theorems and Duality

Mean-field BSVIEs admit comparison principles in the scalar case (and certain special multi-dimensional settings), provided the driver is monotone and non-decreasing in the mean-field arguments. Specifically, if two sets of data $(\zeta_1, f_1)$ and $(\zeta_2, f_2)$ satisfy $\zeta_1 \leq \zeta_2$ and $f_1 \leq f_2$ pointwise, then the corresponding solutions satisfy $Y_1(t) \leq Y_2(t)$ almost surely for all $t$ (Yang et al., 2023).

Duality relationships exist between linear mean-field forward SVIEs and their adjoint MF-BSVIEs, ensuring key structural properties and enabling variational analysis in stochastic control (Shi et al., 2011). The duality identities have the form

$$\mathbb{E} \int_0^T \langle X(t), \Psi(t) \rangle dt = \mathbb{E} \int_0^T \langle Y(t), \Phi(t) \rangle dt,$$

where $X$ solves an MF-FSVIE and $Y$ its adjoint MF-BSVIE.

6. Applications: Dynamic Risk Measures and Stochastic Optimal Control

MF-BSVIEs and their extensions (such as MF-BDSVIEs and singular/infinite-dimensional instances) are closely linked to dynamic risk measures and control problems involving memory and non-Markovianity.

Dynamic risk measures $\rho(t; \zeta(\cdot)) := Y^{-\zeta}(t)$, where $(Y^{-\zeta}, Z^{-\zeta})$ solves an MF-BSVIE or MF-BDSVIE with terminal data $-\zeta$ and an appropriate generator, inherit properties such as past-independence, monotonicity, translation invariance, and convexity from the structure of the generator and the comparison theorem (Yang et al., 2023).

In stochastic optimal control settings, stochastic maximum principles are established where the adjoint equation is an MF-BSVIE. Variational inequalities are derived via spike variation, with the adjoint processes $(Y,Z)$ appearing in the optimality conditions (Shi et al., 2011, Asadzade et al., 29 Nov 2024).

7. Singular and Infinite-dimensional Mean-field BSVIEs

Singular mean-field BSVIEs extend the theory to cases where the Volterra kernel may be unbounded or only square-integrable, and the unknowns, data, and generators are operator-valued over separable Hilbert spaces. This advances the analysis towards infinite-dimensional stochastic systems, such as those arising in distributed parameter systems, stochastic PDEs with memory, and models of viscoelasticity or population dynamics (Asadzade et al., 29 Nov 2024).

Analytical techniques include partitioning the time interval and solving by contraction on each piece, reduction to parametric MF-BSDEs or Fredholm integral equations, and glueing together the local solutions. The extension to associated forward equations, and the use of a resolvent family, are essential in this infinite-dimensional setting.

A plausible implication is that the mean-field BSVIE formalism is sufficiently robust to accommodate both path-dependent interactions (memory) and large-system effects (mean-field), across finite and infinite-dimensional state spaces.

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Summary Table: Key MF-BSVIE Types and Results

Setting	Growth in $Z$	Dimension	Existence/Uniqueness (Main Hypotheses)
Finite-dim., linear	Linear	$\mathbb{R}^n$	Lipschitz generator, $L^2$ data
Finite-dim., quadratic	Quadratic	$\mathbb{R}^n$	BMO/quadratic techniques, $\|\psi\|_{L^\infty}$ bounded
Singular, inf.-dim.	Linear	Hilbert spaces	$L^2$-integrable kernels, adaptation, martingale representation
Doubly stochastic	(Linear)	$\mathbb{R}^n$	Additional backward Itô terms; comparison/regularity via Malliavin

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Mean-field backward stochastic Volterra integral equations provide a comprehensive mathematical framework for systems where both the law of the process and memory effects impact evolution. Recent advances establish well-posedness and convergence in highly general cases, extend comparison and duality results, and connect their analytical structure to practical applications in risk, control, and large population systems (Shi et al., 2011, Yang et al., 2023, Asadzade et al., 29 Nov 2024, Hao et al., 10 Nov 2025).