Conditional McKean–Vlasov Jump Diffusions

• Conditional McKean–Vlasov jump diffusions are stochastic systems with coefficients driven by conditional laws from common noise, integrating explicit and emergent jump mechanisms.
• They employ diverse formulations such as regime-switching, common-noise control, and random-environment FBSDEs, leading to stochastic Fokker–Planck equations, SPIDEs, and measure-valued processes.
• Particle system approximations demonstrate conditional propagation of chaos with explicit convergence rates, underscoring strong well-posedness, ergodicity, and control applications.

Conditional McKean–Vlasov jump diffusions are distribution-dependent stochastic systems in which the coefficients depend on a conditional law—typically the law of the state given a common-noise filtration—and the dynamics include either explicit jump terms or jump-like behavior generated by the interaction mechanism itself. In the modern literature, the conditional law is a random measure-valued process, the state equation is coupled to that random law, and the resulting law flow is often governed by a stochastic Fokker–Planck equation, a stochastic partial integro-differential equation, or a measure-valued SPIDE. This framework includes Brownian common noise, jump common noise, Markovian regime-switching, random environments, absorbing boundaries, and endogenous blow-up mechanisms (Shao et al., 2023).

1. Conditional structure and model classes

The defining feature is the replacement of the unconditional law $\mathcal L(X_t)$ by a conditional law such as

$$\mu_t=\mathcal L(X_t\mid \mathcal F_t^0) \quad\text{or}\quad \mu_t=\mathcal L(X(t)\mid \mathcal F_t^{1}),$$

where the conditioning filtration is generated by a common source of randomness. In one formulation, the common environment is a continuous-time Markov chain $A_t$, and the coefficients depend on $\mathcal L(Y_t\mid\mathcal F_t^0)$ together with the current regime. In another, the common noise is the first Brownian component $B^1$, so the coefficients depend on $\mathcal L(X(t)\mid\mathcal F_t^1)$. In a jump-common-noise formulation, the conditional law is taken with respect to the filtration generated by a common Lévy process $Z^0$. A related FBSDE framework treats the conditioning object as an exogenous measure-valued process $p_t$, designed to coincide at equilibrium with a conditional law (Shao et al., 2023, Agram et al., 2021, Bao et al., 2 Sep 2025, Hernández-Hernández et al., 2023).

Framework	Conditioning object	Jump component
Regime-switching conditional MKV SDE	$\mathcal F_t^0$ generated by $A_t$	Compensated Poisson random measure
Brownian common-noise control model	$$\mu_t=\mathcal L(X_t\mid \mathcal F_t^0) \quad\text{or}\quad \mu_t=\mathcal L(X(t)\mid \mathcal F_t^{1}),$$0	Poisson random measure
Jump-common-noise conditional MKV SDE	$$\mu_t=\mathcal L(X_t\mid \mathcal F_t^0) \quad\text{or}\quad \mu_t=\mathcal L(X(t)\mid \mathcal F_t^{1}),$$1 generated by common Lévy noise	Idiosyncratic and common Lévy jumps
Random-environment FBSDE	Measure-valued process $$\mu_t=\mathcal L(X_t\mid \mathcal F_t^0) \quad\text{or}\quad \mu_t=\mathcal L(X(t)\mid \mathcal F_t^{1}),$$2	Marked Poisson noise with stochastic intensity

This class is broader than a single canonical equation. Some works emphasize Lipschitz dependence on the conditional law in Wasserstein distance, others use Fréchet calculus on spaces of random Radon measures, and others encode the conditional law through a random directing measure arising as the limit of interacting particles. A common structural point is that the conditional law is itself stochastic, so the mean-field term remains random even in the infinite-population limit (Shao et al., 2023, Agram et al., 2021, Erny et al., 2021).

2. Canonical equations and measure-valued formulations

A standard jump-diffusion specification is the regime-switching equation

$$\mu_t=\mathcal L(X_t\mid \mathcal F_t^0) \quad\text{or}\quad \mu_t=\mathcal L(X(t)\mid \mathcal F_t^{1}),$$3

with $$\mu_t=\mathcal L(X_t\mid \mathcal F_t^0) \quad\text{or}\quad \mu_t=\mathcal L(X(t)\mid \mathcal F_t^{1}),$$4 a common Markov chain, $$\mu_t=\mathcal L(X_t\mid \mathcal F_t^0) \quad\text{or}\quad \mu_t=\mathcal L(X(t)\mid \mathcal F_t^{1}),$$5 a Brownian motion, and $$\mu_t=\mathcal L(X_t\mid \mathcal F_t^0) \quad\text{or}\quad \mu_t=\mathcal L(X(t)\mid \mathcal F_t^{1}),$$6 a compensated Poisson random measure. Here the coefficients are random through the pair $$\mu_t=\mathcal L(X_t\mid \mathcal F_t^0) \quad\text{or}\quad \mu_t=\mathcal L(X(t)\mid \mathcal F_t^{1}),$$7, where $$\mu_t=\mathcal L(X_t\mid \mathcal F_t^0) \quad\text{or}\quad \mu_t=\mathcal L(X(t)\mid \mathcal F_t^{1}),$$8 (Shao et al., 2023).

A control-oriented formulation writes

$$\mu_t=\mathcal L(X_t\mid \mathcal F_t^0) \quad\text{or}\quad \mu_t=\mathcal L(X(t)\mid \mathcal F_t^{1}),$$9

with

$A_t$0

In this setting, the pair $A_t$1 becomes Markov after augmenting the state space to include the conditional law, and this Markovianization underpins HJB and quasi-variational formulations (Agram et al., 2021, Agram et al., 2023).

The law process itself satisfies a stochastic evolution equation. For the Brownian-common-noise case,

$A_t$2

where $A_t$3 is an integro-differential adjoint operator incorporating drift, diffusion, and jumps, while $A_t$4 captures the common-noise transport term. When $A_t$5 has a density $A_t$6 and the jump coefficient is independent of $A_t$7, this reduces to an SPDE for $A_t$8 (Agram et al., 2021).

For jump common noise, the conditional distribution flow $A_t$9 is a measure-valued process satisfying a stochastic partial integro-differential equation driven by a Poisson random measure. In weak form, for $\mathcal L(Y_t\mid\mathcal F_t^0)$0, the equation contains the drift transport term, idiosyncratic jump drift, common jump drift, and a stochastic term driven by the common jump noise; the last term is exactly what makes the Fokker–Planck equation stochastic in the jump-common-noise setting (Bao et al., 2 Sep 2025).

A related random-environment FBSDE formalism fixes a measure-valued process $\mathcal L(Y_t\mid\mathcal F_t^0)$1 and lets both coefficients and jump intensities depend on it. At equilibrium, one imposes $\mathcal L(Y_t\mid\mathcal F_t^0)$2, yielding a conditional McKean–Vlasov jump diffusion with environment-dependent stochastic intensity (Hernández-Hernández et al., 2023).

3. Sources of jumps: explicit, simultaneous, and endogenous

One important distinction is between explicit jump drivers and endogenous jump phenomena. In the explicit case, the state equation contains a Poisson or Lévy term directly, as in the regime-switching model above or in Lévy-driven common-noise systems (Shao et al., 2023, Bao et al., 2 Sep 2025). In these models, the jump structure is exogenous at the level of the state equation.

A second mechanism arises from simultaneous microscopic jumps. One particle system is driven by Brownian motion and Poisson random measures, with every particle jumping at a rate depending on its position and the empirical measure, and with jump heights centered and scaled in $\mathcal L(Y_t\mid\mathcal F_t^0)$3. The limit empirical measure is then random rather than deterministic, and the limit system is driven by martingale measures and white noises whose intensity depends on the conditional law. In this sense, the common noise is emergent rather than primitive, and the appropriate limit concept is a conditional McKean–Vlasov limit (Erny et al., 2021).

A related simultaneous-jump mean-field model lets a jump of particle $\mathcal L(Y_t\mid\mathcal F_t^0)$4 cause collateral jumps of all other particles. In the McKean–Vlasov limit, the collateral effect averages into a law-dependent drift term, while the limit particle retains only its own jump mechanism. The finite system therefore has synchronous jumps, but the limit equation transforms part of that synchronization into deterministic mean-field transport (Andreis et al., 2017).

A third mechanism is endogenous jump formation without primitive jump noise. For absorbing diffusions on the half-line, the empirical loss process

$\mathcal L(Y_t\mid\mathcal F_t^0)$5

is a càdlàg monotone step process, and the limiting loss process $\mathcal L(Y_t\mid\mathcal F_t^0)$6 may retain macroscopic jumps even though each particle is a continuous diffusion up to absorption. The limit admits both an SPDE formulation and a stochastic McKean–Vlasov representation, with the conditional loss

$\mathcal L(Y_t\mid\mathcal F_t^0)$7

playing the role of the mean-field variable (Hambly et al., 2016).

The most striking endogenous-jump example is the positive-feedback blow-up model with common Brownian noise,

$\mathcal L(Y_t\mid\mathcal F_t^0)$8

where a blow-up is a jump discontinuity of $\mathcal L(Y_t\mid\mathcal F_t^0)$9. The jump size is selected by the minimality constraint

$B^1$0

so the jump is determined by the pre-blow-up conditional law. This is a canonical correction to the misconception that “jump diffusions” in the mean-field setting must be driven by exogenous Poisson noise: here the primitive noises are continuous Brownian motions, yet the conditional law and the state process acquire jump behavior through the feedback mechanism (Ledger et al., 2018).

4. Particle systems and conditional propagation of chaos

Conditional McKean–Vlasov jump diffusions are typically obtained as limits of interacting particle systems with common randomness. In the regime-switching jump model, the particle approximation has empirical measure

$B^1$1

and each particle is driven by its own Brownian motion and Poisson random measure, while all particles share the same common chain $B^1$2. Conditioned on $B^1$3, the limit particles are i.i.d., and the convergence rate is explicit: $B^1$4 with

$B^1$5

The same rate governs

$B^1$6

This is propagation of chaos conditional on the common noise, not unconditional chaos (Shao et al., 2023).

In the white-noise limit of simultaneous small jumps, the empirical measures converge in law to a random directing measure

$B^1$7

and the limit particles are conditionally i.i.d. given the common white noise $B^1$8. The randomness of the limit empirical measure is therefore structural rather than a finite-sample artifact (Erny et al., 2021).

For interacting systems with simultaneous jumps of all particles, one can couple the original system to an intermediate system without simultaneous jumps. In the globally Lipschitz setting, the pathwise error between the original particle $B^1$9 and the intermediate particle $\mathcal L(X(t)\mid\mathcal F_t^1)$0 is of order $\mathcal L(X(t)\mid\mathcal F_t^1)$1: $\mathcal L(X(t)\mid\mathcal F_t^1)$2 The remaining propagation-of-chaos error is then reduced to standard empirical $\mathcal L(X(t)\mid\mathcal F_t^1)$3 fluctuations on path space (Andreis et al., 2017).

For conditional systems with jump common noise, Bao–Liu–Wang construct an asymptotic coupling by reflection between non-interacting and interacting particles, then pass from particle-level estimates to contraction of the law of the measure-valued conditional distribution flow. This coupling is specific to the jump structure: small jumps are reflected, large jumps are synchronously coupled, and the cutoff is controlled by an auxiliary function built from the Lévy measures (Bao et al., 2 Sep 2025).

5. Well-posedness, uniqueness, ergodicity, and stationary behavior

A central analytical issue is the simultaneous treatment of state dependence, measure dependence, common noise, and jumps. In the regime-switching setting, strong pathwise uniqueness follows from $\mathcal L(X(t)\mid\mathcal F_t^1)$4-Lipschitz estimates and the inequality

$\mathcal L(X(t)\mid\mathcal F_t^1)$5

Strong existence is then obtained by a fixed-point argument on a path space of càdlàg processes. However, the existence proof requires the jump coefficient $\mathcal L(X(t)\mid\mathcal F_t^1)$6 to be independent of $\mathcal L(X(t)\mid\mathcal F_t^1)$7, so the jump term is treated as additive Lévy noise. The same work explicitly notes that extending full McKean–Vlasov dependence in the jump coefficient with $\mathcal L(X(t)\mid\mathcal F_t^1)$8 remains challenging (Shao et al., 2023).

A different route uses weak monotonicity and weak coercivity. For Lévy-driven McKean–Vlasov SDEs, strong well-posedness is proved under one-sided monotonicity in the state variable and Lipschitz continuity in the measure variable with respect to $\mathcal L(X(t)\mid\mathcal F_t^1)$9, $Z^0$0. The theory simultaneously treats weak and strong propagation of chaos, and as a potential extension includes strong well-posedness and conditional propagation of chaos for common-noise versions. This framework is explicitly designed to handle non-globally Lipschitz drift and jump coefficients under monotone conditions (Bao et al., 2024).

Long-time behavior has also been analyzed at the level of the conditional law itself. For one-dimensional conditional McKean–Vlasov jump diffusions with jump idiosyncratic noise and jump common noise, the conditional distribution flow is a measure-valued process satisfying a SPIDE driven by a Poisson random measure. Under a partially dissipative condition in space and a smallness condition on the law-Lipschitz constant, the law of that measure-valued process contracts exponentially in the metric

$Z^0$1

and the paper shows that the intensity of the jump common noise and the jump idiosyncratic noise can simultaneously enhance the convergence rate of the exponential ergodicity (Bao et al., 2 Sep 2025).

For pure jump Lévy-driven McKean–Vlasov SDEs without conditioning, existence, uniqueness, and multiplicity of stationary distributions have been established via Schauder fixed point arguments, weighted total variation contraction, Wasserstein continuity, and local dissipativity. This suggests a route to random invariant measures or quenched stationary laws in conditional settings, although that extension is not part of the theorem itself (Bao et al., 22 Apr 2025).

A recurrent misconception is that the conditional setting merely adds notation to classical McKean–Vlasov theory. The literature shows otherwise: the infinite-dimensional state variable is genuinely random, the law equation is stochastic rather than deterministic, particle limits produce conditional rather than ordinary chaos, and even the long-time ergodic object is a law of random probability measures rather than a single deterministic invariant law (Shao et al., 2023, Bao et al., 2 Sep 2025).

6. Control, stopping, FBSDEs, and computation

Control problems are naturally formulated on the augmented state $Z^0$2. In the Brownian-common-noise jump setting, the conditional law satisfies a stochastic Fokker–Planck PIDE, and the pair $Z^0$3 forms a Markov process on $Z^0$4. This permits a generator calculus with Fréchet derivatives in the measure argument and yields an HJB equation on the augmented space. When the conditional law is absolutely continuous, the measure-valued equation reduces to an SPDE for the density, and explicit solutions can be derived for linear-quadratic optimal control and optimal consumption from a cash-flow modelled as a conditional McKean–Vlasov differential equation with jumps (Agram et al., 2021).

Impulse control leads to a quasi-variational inequality on the same augmented space. The impulse acts not only on the physical state $Z^0$5 but also on the conditional law through the transformed measure

$Z^0$6

A verification theorem identifies sufficient conditions for a function to be the value function and for an impulse strategy to be optimal, and the framework is illustrated by an optimal stream of dividends under transaction costs, where an explicit barrier-type optimal strategy is obtained (Agram et al., 2023).

The random-environment FBSDE literature provides a complementary probabilistic representation. In that framework, the environment is a measure-valued process $Z^0$7, the jump term is driven by a marked Poisson process with stochastic intensity kernel $Z^0$8, and both coefficients and intensities may depend on $Z^0$9. This encompasses Cox-type intensities, Hawkes-type self-excitation, and regime-switching conditional McKean–Vlasov equations, and it is designed to close into a conditional McKean–Vlasov jump diffusion when one enforces the fixed-point identity $p_t$0 (Hernández-Hernández et al., 2023).

Numerically, Malliavin calculus on Poisson space has been used to represent conditional expectations of functionals of mean-field jump-diffusion solutions as weighted unconditional expectations. For $p_t$1 and $p_t$2, one obtains formulas of the form

$p_t$3

with explicit weights $p_t$4 built from jump Malliavin derivatives and the linearized flow. This is then applied to the numerical pricing of American put options in a jump-diffusion mean-field setting (Sojudi et al., 16 Feb 2026).

Optimal stopping for conditional McKean–Vlasov jump diffusions has also been studied directly (Agram et al., 2022). Taken together, these works show that the conditional law is not only a theoretical device for well-posedness and mean-field limits; it is also a computational state variable for HJB equations, quasi-variational inequalities, FBSDEs, and Monte Carlo schemes.

7. Conceptual synthesis

Across the literature, several structural themes recur. First, the conditional law flow is the central state variable: it is random, adapted to common noise, and typically governed by a stochastic measure-valued equation. Second, jumps enter in multiple ways: as explicit Lévy or Poisson drivers, as simultaneous microscopic jumps that generate random mean-field limits, or as endogenous discontinuities of loss and conditional law produced by absorption and feedback (Shao et al., 2023, Erny et al., 2021, Ledger et al., 2018).

Third, the appropriate chaos notion is conditional propagation of chaos. In the presence of common noise, particles do not become asymptotically independent unconditionally; they become asymptotically independent given the common environment, and the empirical measure converges to a random conditional law (Shao et al., 2023, Erny et al., 2021).

Fourth, the analytical toolkit is hybrid. Wasserstein fixed points, Sobolev or energy estimates, SPIDE techniques, asymptotic coupling by reflection, martingale-measure limits, and Fréchet calculus on spaces of random measures all appear, depending on whether the model emphasizes explicit jumps, common-noise conditioning, boundary absorption, or control (Hambly et al., 2016, Bao et al., 2 Sep 2025, Agram et al., 2021).

Finally, the topic does not reduce to a single canonical theorem. Conditional McKean–Vlasov jump diffusions form a family of formulations unified by three ingredients: a state equation with distributional feedback, a conditioning mechanism generated by common randomness, and a jump structure that may be exogenous, emergent, or endogenous. The literature now covers strong well-posedness, particle approximations, stochastic law equations, ergodicity, impulse control, and numerical conditional-expectation formulas, while leaving open difficult problems such as full measure-dependent jump coefficients under weak regularity and a general stationary theory in random environments (Shao et al., 2023, Bao et al., 2024, Bao et al., 22 Apr 2025).