arXiv subjects

Johan Benedikt Spille

Publications and source records attributed to Johan Benedikt Spille.

3 recordsLinked to original sources

arXiv2026

Peng's Maximum Principle for McKean-Vlasov Stochastic Differential Equations with Common Noise

We study a stochastic optimal control problem for McKean-Vlasov stochastic differential equations (SDEs) with common noise, where the dynamics depend on the conditional law of the state. We derive a stochastic maximum principle of Peng type without imposing convexity assumptions on the control domain. In comparison to the standard McKean-Vlasov case, the maximum principle for the common noise case contains a third adjoint state, which is needed to dualize all second-order Lions derivatives in the Taylor expansion of the cost functional. The additional adjoint state is given by a conditional McKean-Vlasov backward SDE. All three adjoint states together allow for a complete linearization of all contributions in the second-order expansion, including interactions between conditionally independent copies of the first variational process. As part of our analysis, we also prove a general well-posedness result for conditional McKean-Vlasov backward SDEs.

math.PR↗

arXiv2026

A Novel Approach to Peng's Maximum Principle for McKean-Vlasov Stochastic Differential Equations

We present a novel approach to the proof of Peng's maximum principle for McKean-Vlasov stochastic differential equations (SDE). The main step is the introduction of a third adjoint equation, a conditional McKean-Vlasov backward SDE, to accommodate the dualization of quadratic terms containing two independent copies of the first-order variational process. This is an intrinsic extension of the maximum principle from Peng for standard SDE and gives a conceptually consistent proof. Our approach will be useful in further extensions to the common noise setting and the infinite dimensional setting.

math.PR↗

arXiv2025

Pontryagin Maximum Principle for McKean-Vlasov Stochastic Reaction-Diffusion Equations

We consider the stochastic control of a semi-linear stochastic partial differential equations (SPDE) of McKean-Vlasov type. Based on a recent novel approach to the Lions derivative for Banach space valued functions, we prove the Gateaux differentiability of the control to state map and, using adjoint calculus, we derive explicit representations of the gradient of the cost functional and a Pontryagin maximum principle. On the way, we also prove a novel existence and uniqueness result for linear McKean-Vlasov backward SPDE. Furthermore, for deterministic controls, we prove the existence of optimal controls using a martingale approach and a novel compactness method. This result is complemented in the appendix with a rigorous proof of folklore results on the compactness method in the variational approach to SPDE. Our setting uses the variational approach to SPDE with monotone coefficients, allowing for a polynomial perturbation and allowing the drift and diffusion coefficients to depend on the state, the distribution of the state and the control.

math.PR↗