Path-Dependent Multivalued MV-SDEs

Updated 23 August 2025

Path-dependent multivalued MV-SDEs are stochastic systems that combine history-dependent coefficients, mean-field interactions, and multivalued drifts via maximal monotone operators.
They employ rigorous fixed-point and Lipschitz approximation methods to establish well-posedness and stability in both Lipschitz and non-Lipschitz regimes.
Recent advances include generalized Itô formulas and particle-based numerical schemes, enabling effective modeling of complex phenomena in physics, finance, and control theory.

Path-dependent multivalued McKean–Vlasov stochastic differential equations (MV-SDEs) comprise a broad class of stochastic systems in which the evolution of the state variable depends simultaneously on its entire past trajectory (path dependence), the probability law of the solution (mean-field interaction), and potentially set-valued or multivalued drift mechanisms typically modeled by maximal monotone operators. These systems generalize classical SDEs, combining memory effects, interaction through population-wide statistics, and non-smooth constraint dynamics. Recent advances have established well-posedness, stability, and numerical methods for such equations under both Lipschitz and non-Lipschitz regimes, providing tools to analyze complex phenomena in statistical physics, control theory, neuroscience, and finance.

1. Structural Definition and Main Equation Forms

Path-dependent multivalued MV-SDEs extend the classical McKean–Vlasov SDE by allowing both path dependence in the coefficients and multivalued (set-valued) drift terms. A prototypical form is:

$\mathrm{d}X(t) \in -A(X(t))\, \mathrm{d}t + b(t, X_{t}, \mu_{t})\, \mathrm{d}t + \sigma(t, X_{t}, \mu_{t}) \, \mathrm{d}W(t), \quad X_{0} = \xi$

where:

$A$ is a maximal monotone operator (possibly multivalued), encoding constraints, reflections, or singularities.
$X_{t}$ denotes the memory segment, $X_{t}(\theta) = X(t+\theta)$ , $\theta \in [-r_{0}, 0]$ for some history length $r_{0} > 0$ .
$b$ and $\sigma$ depend on the entire segment $X_{t}$ and the law $\mu_{t} = \mathsf{Law}(X_{t})$ .
$W$ is a standard Brownian motion.

The coefficients may also exhibit non-Lipschitz continuity and polynomial growth, and the inclusion of $A$ models phenomena like reflection (as in variational inequalities) or hysteresis.

Table: Variants of the MV-SDE with Path and Measure Dependence

Feature	Classical MV-SDE	Path-dependent MV-SDE	Multivalued MV-SDE
Drift $b$	$b(t, x, \mu_{t})$	$b(t, X_{t}, \mu_{t})$	$-A(X) + b(t, X_{t}, \mu_{t})$
Diffusion $\sigma$	$\sigma(t, x, \mu_{t})$	$\sigma(t, X_{t}, \mu_{t})$	$\sigma(t, X_{t}, \mu_{t})$
Multivalued Set	None	None	$A$ : maximal monotone operator

This formalism encompasses equations with history-dependent coefficients, measure-driven feedback, and set-valued nonlinearities (Ma et al., 21 Aug 2025, Ning et al., 2023).

2. Well-posedness: Existence and Uniqueness in Lipschitz and Non-Lipschitz Regimes

Initial results establish global existence and uniqueness via Picard-type iterations under global Lipschitz conditions for the path-dependent coefficients and uniform monotonicity for the multivalued operator $A$ . The fundamental scheme iterates over the space of continuous trajectories and laws, leveraging maximal monotone operator theory for multivalued drift handling (Ma et al., 21 Aug 2025):

For Lipschitz $f$ , $g$ (in path $X_t$ ), and $A$ maximal monotone, the sequence of solutions defined by successive substitution converges in $L^2(\Omega; C([0,T]))$ , granting strong well-posedness.
Non-Lipschitz cases use Lipschitz approximations: coefficients $f,g$ are approximated by regularized functions (e.g., by mollification) and the sequence of solutions for the regularized systems is shown to converge tightly to a solution for the original non-Lipschitz equation.

For the full path-dependent multivalued MV-SDE with measure dependence (i.e., McKean–Vlasov type), the authors implement an iterative scheme in distribution: the law $\mu_t$ is updated at each step, and convergence is shown in the space of measure-valued path-functions. Assumptions may require concavity, integrability (e.g., $\int_0^+ \frac{1}{\kappa(x)} dx = \infty$ for the control function $\kappa$ ), and continuity for non-Lipschitz settings (Ma et al., 21 Aug 2025, Qiao et al., 2021, Ning et al., 2023).

3. Analytical Techniques and Generalized Itô Formula

Analytical treatment relies on advanced stochastic calculus and functional analysis:

Itô Formula Generalizations: For multivalued MV-SDEs, the extended Itô formula encompasses derivatives not only in the state variable $x$ but also in the law $\mu$ and incorporates terms from the multivalued operator $A$ (Qiao et al., 2021). For path-dependent cases, functional Itô formulas are derived in the spirit of Dupire and Lions, treating measure derivatives and pathwise perturbations (Cosso et al., 2020).
Fixed Point Arguments: Well-posedness is proven using contraction mappings in probability spaces equipped with Wasserstein metrics.
Lipschitz Approximation and Truncation: Non-Lipschitz coefficients are regularized via cutoff functions or convolution, allowing usage of the Lipschitz existence theory followed by tightness/convergence.

Representative formula for well-posedness estimate (Ma et al., 21 Aug 2025):

$\sup_{t\in [0,T]} \mathbb{E}|X^{(n+1)}(t) - X^{(n)}(t)|^2 \leq C \int_0^t [\kappa_1(\|X^{(n)}_s - X^{(n-1)}_s\|_\infty^2) + \kappa_2(W_2^2(\mu^{(n)}_s, \mu^{(n-1)}_s)) ] ds$

Key technical ingredients from monotone operator theory, Wasserstein metric analysis, and concatenation of path-dependent control functions are central.

4. Regularity Properties, Stability, and Gradient Estimates

Regularity and stability are ensured under sharp conditions:

Stability and Exponential Estimates: Under polynomial growth and monotonicity, solutions exhibit stability in moments, with explicit rates for mean-square decay (Reis et al., 2017, Kalinin et al., 2021, Kalinin et al., 2022).
Gradient and Harnack Inequalities: For equations with singular coefficients and Dini continuity, gradient estimates of the transition semigroup are established, including dimension-free Harnack inequalities (Huang, 2019). This extends ergodic and regularity theory to settings with weak regularity in coefficients.
Lyapunov Functions: Construction of suitable Lyapunov functions enables proofs of asymptotic stability for the second moment and almost sure stability under additional negative drift conditions.

The stability and regularity framework enables large deviations analysis, sensitivity to initial conditions, and robustness under model perturbations.

5. Large Deviations Principles and Functional Laws

Advanced probabilistic properties are characterized using LDPs and functional limit theorems:

Freidlin–Wentzell LDPs: Path-dependent MV-SDEs (including those with super-linear drift growth) admit two types of large deviations principles—in uniform and Hölder topologies—with rate functions determined by skeleton equations incorporating measure dependence (Reis et al., 2017).
Functional Strassen Laws: Law of iterated logarithms is extended to the path-dependent mean-field setting, with compactness and characterization of oscillation amplitude for rescaled solution trajectories (Reis et al., 2017).

These results are instrumental for understanding rare events and the limiting behavior of interacting mean-field systems.

6. Numerical Methods and Computational Approaches

Numerical simulation of path-dependent multivalued MV-SDEs requires carefully designed particle methods and temporal discretization schemes:

Particle System Approximation: Interacting particle systems whose coefficients depend on empirical measure and trajectory segments approximate the MV-SDE solution. Convergence is proven in Wasserstein distance with explicit rates depending on number of particles and time-steps (Bernou et al., 2022, Liu et al., 2020).
Euler–Maruyama Schemes and Interpolated Paths: Temporal discretization handles the past-dependence via piecewise interpolation, enabling algorithms in high-dimensional state/path spaces.
Error Estimates: The total error is split into sampling error (controlled by the number of particles $N$ ) and discretization error (controlled by time step $h$ ), which together enable precise quantification of numerical approximation quality in practical implementations (Bernou et al., 2022, Liu et al., 2020).

These computational strategies are validated in applications such as mean-field neural population models and delayed Ornstein–Uhlenbeck processes with path-dependent drift.

7. Applications and Generalizations

Applications span statistical physics (mean-field particle systems with memory), finance (market models with collective feedback and path-dependent risk), biology (epidemiological systems with history effects), and control theory (mean-field games with constraints and reflections):

Path-dependent multivalued MV-SDEs are foundational for modeling systems with constraint dynamics (via maximal monotone operators), interaction through mean-field variables, and memory, addressing practical problems where conventional Markovian or single-valued SDEs are insufficient (Ma et al., 21 Aug 2025, Ning et al., 2023).
Generalization to control problems in infinite-dimensional settings and optimal control on Wasserstein spaces links analysis to master equations in non-Markovian mean-field games (Cosso et al., 2020).
The methodology readily accommodates non-Lipschitz (e.g., singular) coefficients, rough noise inputs (via rough integration and pathwise SPDE analysis (Bugini et al., 23 Jul 2025)), and variational inequalities (reflected processes, coupled forward-backward equations (Ning et al., 2023)).

Ongoing research is directed at path-dependent systems with rough/common noise, higher-order memory kernels, and coupling to nonlinear Fokker–Planck equations.

In sum, recent developments have furnished a comprehensive theory for path-dependent multivalued McKean–Vlasov stochastic differential equations, encompassing existence, uniqueness, stability, and algorithmic simulation under minimal regularity, and extending the reach of stochastic analysis in mean-field, memory-dependent, and constrained dynamical systems (Reis et al., 2017, Ma et al., 21 Aug 2025).