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

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.