CS05: Recent Advances in Interacting Brownian Particle Systems and Their Mean-Field Limits

Organizer: Armand Bernou (Université Claude Bernard Lyon 1)

Convex order and increasing convex order for McKean-Vlasov processes with common noise

Yating Liu (Paris-Dauphine University)

We establish results on the conditional and standard convex order, as well as the increasing convex order, for two processes \(X\) and \(Y\) defined by McKean-Vlasov equations with common Brownian noise \(B^0\). Under suitable conditions, for a (non-decreasing) convex functional \(F\) on the path space with polynomial growth, we show \(\mathbb{E}[ F(X) | B^0 ] \leq \mathbb{E}[ F(Y) | B^0 ]\) almost surely. Similar convex order results are also established for the corresponding particle system. Finally, we explore applications of these results to stochastic control problem - deducing in particular an associated comparison principle for Hamilton-Jacobi-Bellman equation with different coefficients - and to an interbank systemic risk model.

Bibliography

\([1]\) Armand Bernou, Théophile Le Gall, Yating Liu. Convex order and increasing convex order for McKean-Vlasov processes with common noise. (2025) In preparation.

\([2]\) Liu, Yating, and Gilles Pagès. "Monotone convex order for the McKean–Vlasov processes." Stochastic Processes and their Applications 152 (2022): 312-338.

\([3]\) Liu, Yating, and Gilles Pagès. "Functional convex order for the scaled McKean–Vlasov processes." The Annals of Applied Probability 33, no. 6A (2023): 4491-4527.