Optimal control problem for reflected McKean--Vlasov stochastic differential equations with Poisson jumps

Abstract: In this paper, we consider the optimal relaxed control problem for a class of one-dimensional reflected McKean--Vlasov stochastic differential equations with Poisson jumps. Due to the presence of the jump term, the state process generally belongs to the Skorokhod space $D([0,T],\Rp)$, which makes the proof of tightness and the passage to the limit more complicated. Under Lipschitz conditions and suitable growth conditions, we establish uniform moment estimates for the state process and the reflecting process. Then, by using Aldous' tightness criterion, the continuity of the Skorokhod map, and the stability results for stochastic integrals, we prove the existence of an optimal relaxed control. Furthermore, under the Roxin convexity condition, we prove the existence of a strict optimal control. In the general case, we show that relaxed controls can be approximated by a sequence of strict controls.

• The paper establishes the existence of optimal relaxed controls for reflected McKean–Vlasov SDEs with Poisson jumps.
• It demonstrates uniform moment bounds and tightness in Skorokhod space using Aldous’ criterion and a Skorokhod reflection mapping.
• Under convexity assumptions, the relaxed controls can be represented by strict controls, ensuring equivalence in control performance.

Optimal Control for Reflected McKean–Vlasov SDEs with Poisson Jumps: Existence Theory

Introduction and Problem Setting

The paper "Optimal control problem for reflected McKean–Vlasov stochastic differential equations with Poisson jumps" (2607.00799) addresses the existence and structural properties of optimal control in a non-classical stochastic framework combining mean-field interactions, boundary reflection, and jump perturbations. Specifically, the system under consideration is a one-dimensional reflected McKean–Vlasov SDE driven by both Brownian noise and a Poisson random measure, in which the coefficients depend not only on the current state but also on its distribution, consistent with mean-field (McKean–Vlasov) models.

A central focus of the paper is the optimal control problem where the control—initially allowed to be "strict" (i.e., deterministic, adapted process)—acts on the drift coefficient, and the state is constrained to remain in the non-negative half-line via a Skorokhod-type reflection term. The system is further complicated by the presence of Poisson jumps, necessitating analysis in the Skorokhod space of càdlàg paths, which demands more intricate tightness and limit-passage techniques relative to the continuous-path case.

The main technical aims are threefold:

1. Establishment of existence and uniqueness for the controlled reflected McKean–Vlasov SDE with Poisson jumps.
2. Existence of an optimal relaxed control, leveraging compactness due to probabilistic (measure-valued) controls.
3. Conditions under which a strict optimal control exists or when relaxed solutions can be approximated by strict controls, particularly under convexity assumptions on the control set (Roxin's condition).

Technical Framework and Main Results

Model Specification

The controlled system evolves as

$$\begin{aligned} dX_t &= \int_A b(t, X_t, \mathcal{L}_{X_t}, a)\, q_t(da)\,dt + \sigma(t, X_t, \mathcal{L}_{X_t})\, dB_t + \int_{\mathbb{R}_0} \gamma(t, X_{t-}, \mathcal{L}_{X_{t-}}, z)\, \widetilde{N}(dt, dz) + dK_t, \ X_0 &= x, \quad X_t \ge 0, \quad K_0 = 0, \quad \int_0^T \mathbb{I}_{\{X_s>0\}}\, dK_s = 0, \end{aligned}$$

where $q_t$ is a measure-valued relaxed control, and $K_t$ is the reflection process (minimal regulator to stay in the domain $[0, \infty)$).

The cost functional is

$$J(r) = \mathbb{E} \left[ \int_0^T \int_A f(s, X_s, \mathcal{L}_{X_s}, a)\, q_s(da)\,ds + \int_0^T c(s, X_s, \mathcal{L}_{X_s})\, dK_s + g(X_T, \mathcal{L}_{X_T}) \right].$$

The aim is to find $r^*\in \mathcal{R}$ achieving the infimum of $J$ over all admissible (relaxed) controls.

Well-Posedness and Moment Bounds

Under comprehensive Lipschitz and polynomial growth assumptions on coefficients, the paper proves that the reflected McKean–Vlasov SDE with Poisson jumps is well posed for both strict and relaxed controls. These results rely on mapping the relaxed control into an averaged drift and applying known results about McKean–Vlasov SDEs with reflection and jumps.

Further, uniform moment bounds of order four are established for both the state and reflection processes for any sequence of admissible controls, crucial for demonstrating tightness in the Skorokhod path space $D([0,T], \mathbb{R})$.

Tightness, Skorokhod Mapping, and Control Compactification

A principal technical challenge, stemming from the jump term, is to show that minimizing sequences of controls and associated state processes are tight jointly in control, state, Brownian, and jump spaces. The authors employ Aldous’ criterion for Skorokhod-space tightness, together with properties of the one-dimensional Skorokhod reflection map (notably, its continuity in the Skorokhod topology), to advance this argument.

The relaxed control space $\mathcal{V}$, with the stable topology, is compact and metrizable, enabling the passage to the limit in minimizing sequences via Prokhorov's theorem and the Skorokhod representation.

Existence of Optimal Controls and Structure Results

Main Existence Result:

Under the stated structural assumptions, an optimal relaxed control exists. That is, the infimum of the cost functional is attained in the broader space of measure-valued controls. The proof leverages tightness, the Skorokhod representation, stability of stochastic integrals under weak convergence (including for compensated Poisson integrals), and semi-continuity of the cost functional, especially the reflecting cost.

Strict Control Representation — Roxin Convexity:

Under a Roxin convexity condition (joint convexity of the set of drift-running cost pairs as a function of the control), the optimal relaxed control can be represented by a strict control. A measurable selection argument constructs a strict, adapted process that induces the same state trajectory and cost as the relaxed optimal, showing equivalence of the respective infima.

Approximation Result:

Even when the convexity condition fails and strict optimal controls may not exist, any relaxed control can be approximated by a sequence of strict controls (via a chattering lemma), with cost converging to that of the relaxed control. In particular, the optimal values of the strict and relaxed control problems coincide.

Numerical and Theoretical Strengths

While the paper does not report specific numerical experiments, it achieves the following strong theoretical results:

• Uniform moment bounds: Existence of a constant $C$ such that, for any minimizing sequence, $q_t$0.
• Existence theorems: Demonstrated for both optimal relaxed controls and strict optimal controls under convexity.
• Approximation of values: $q_t$1 without requiring convexity.

The results extend the existence theory of stochastic control to mean-field SDEs with both reflection and jumps, subsuming as particular cases the more restrictive settings of control without jumps or without reflection.

Implications and Future Directions

Theoretical

The paper provides a robust existence theory for control of mean-field reflected jump diffusions, an area with significant open questions owing to the inherent non-Markovianity, constrained domain, and discontinuities. The establishment of stability and approximation results for the control problem in the Skorokhod space advances the general well-posedness theory for McKean–Vlasov SDE control.

Further investigation could address:

• Characterization and regularity of the value function (e.g., via backward SPDEs on the Wasserstein space).
• Stochastic maximum principles and verification theorems in the mean-field jump-reflection context.
• Numerical schemes for such control problems, building on the tightness and approximation arguments.

Practical

From an applications perspective, the SDE and control structure analyzed here models systems in finance (e.g., risk processes with capital constraints and jump shocks), queueing, and various engineering scenarios requiring boundary constraints, mean-field coupling, and sudden external shocks. The existence and approximation results guarantee that optimal, or near-optimal, feedback policies can be theoretically constructed or approximated for such models, facilitating their use in stochastic optimal design and regulation.

Conclusion

This work rigorously establishes existence, approximation, and structural properties for optimal controls in reflected McKean–Vlasov SDEs with Poisson jumps (2607.00799). By leveraging techniques from Skorokhod space analysis, control compactification, and stochastic calculus with jumps, the results provide a foundation for further developments in the theory and application of constrained mean-field jump-diffusion control. The demonstrated equivalence in value between strict and relaxed control problems, under general assumptions, broadens the admissible settings where optimal regulation can be achieved in mean-field stochastic systems with reflection and jumps.