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

Armand Bernou, Théophile Le Gall, Yating Liu

TL;DR

The paper addresses convex-order relations between two McKean–Vlasov processes with common noise, establishing conditional and standard convex order as well as increasing convex order in one dimension. The authors develop a rigorous framework based on Euler-type discretizations and backward operators to propagate convexity through the dynamics, and they extend these results to the associated particle systems via propagation of chaos. Key contributions include the conditional functional convex order (and an extended version for joint path-law functionals), the standard convex order for particle systems under a suitable mean-field structure, and the one-dimensional increasing convex order achieved through a truncated Euler scheme with convergence analysis. The results provide practical tools for bounding costs and systemic-risk metrics in mean-field models with common noise, with implications for stochastic control, mean-field games, and interbank risk models, and they connect to and extend prior work on MKV equations without common noise.

Abstract

We establish results on the conditional and standard convex order, as well as the increasing convex order, for two processes $ X = (X_t)_{t \in [0, T]} $ and $ Y = (Y_t)_{t \in [0, T]} $, defined by the following McKean-Vlasov equations with common Brownian noise $ B^0 = (B_t^0)_{t \in [0, T]} $: $$ dX_t=b(t, X_t, \mathcal{L}^1(X_t))d t+σ(t, X_t, \mathcal{L}^1(X_t))d B_t+σ^0 (t, \mathcal{L}^1(X_t))d B^0_t$$ $$dY_t=\,β(t, Y_t, \mathcal{L}^1(Y_t\,))d t+\,θ(t, Y_t\,, \mathcal{L}^1(Y_t\,))d B_t\,+\,θ^0 (t, \mathcal{L}^1(Y_t\,))d B^0_t,$$ where $ \mathcal{L}^1(X_t) $ (respectively $ \mathcal{L}^1(Y_t) $) denotes a version of the conditional distribution of $ X_t $ (resp. $ Y_t $) given $ B^0 $. These results extend those established for standard McKean-Vlasov equations in [Liu-Pagès, 2023] and [Liu-Pagès, 2021]. 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. Moreover, for a (non-decreasing) convex functional $G$ defined on the product space of paths and their marginal distributions, we establish $$ \mathbb{E} \Big[\,G\big(X, (\mathcal{L}^1(X_t))_{t\in[0, T]}\big)\,\Big| \, B^0\,\Big]\leq \mathbb{E} \Big[\,G\big(Y, (\mathcal{L}^1(Y_t))_{t\in[0, T]}\big)\,\Big| \, B^0\,\Big] \quad \text{almost surely}. $$ Similar convex order results are also established for the corresponding particle system. Finally, we explore applications of these results to stochastic control problems and to the interbank systemic risk model introduced in [Carmona-Fouque-Sun, 2015].

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

TL;DR

The paper addresses convex-order relations between two McKean–Vlasov processes with common noise, establishing conditional and standard convex order as well as increasing convex order in one dimension. The authors develop a rigorous framework based on Euler-type discretizations and backward operators to propagate convexity through the dynamics, and they extend these results to the associated particle systems via propagation of chaos. Key contributions include the conditional functional convex order (and an extended version for joint path-law functionals), the standard convex order for particle systems under a suitable mean-field structure, and the one-dimensional increasing convex order achieved through a truncated Euler scheme with convergence analysis. The results provide practical tools for bounding costs and systemic-risk metrics in mean-field models with common noise, with implications for stochastic control, mean-field games, and interbank risk models, and they connect to and extend prior work on MKV equations without common noise.

Abstract

We establish results on the conditional and standard convex order, as well as the increasing convex order, for two processes and , defined by the following McKean-Vlasov equations with common Brownian noise : where (respectively ) denotes a version of the conditional distribution of (resp. ) given . These results extend those established for standard McKean-Vlasov equations in [Liu-Pagès, 2023] and [Liu-Pagès, 2021]. Under suitable conditions, for a (non-decreasing) convex functional on the path space with polynomial growth, we show almost surely. Moreover, for a (non-decreasing) convex functional defined on the product space of paths and their marginal distributions, we establish Similar convex order results are also established for the corresponding particle system. Finally, we explore applications of these results to stochastic control problems and to the interbank systemic risk model introduced in [Carmona-Fouque-Sun, 2015].

Paper Structure

This paper contains 23 sections, 25 theorems, 152 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definition of the (conditional) convex order and increasing convex order
  3. Definition of the McKean-Vlasov equation with common noise
  4. Aim, key techniques and structure of the paper
  5. Assumptions and main results
  6. Notations
  7. Literature review and comments on the assumptions
  8. Literature review
  9. Comments on the assumptions
  10. Functional convex order for d-dimensional McKean-Vlasov equation with common noise
  11. Conditional convex order for the McKean-Vlasov equation with common noise
  12. Preliminary results and marginal convex order result of the Euler scheme
  13. Global convex order of the Euler scheme
  14. Proof of Theorem \ref{['thm:convex_order_cond']}
  15. Standard convex order for the particle system
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume that Assumptions Ass:AssumptionI and Ass:AssumptionII hold for some $p\!\in [2, +\infty)$. Let $X\coloneqq (X_{t})_{t\in[0, T]}$, $Y\coloneqq (Y_{t})_{t\in[0, T]}$ denote the unique solutions of the conditional McKean-Vlasov equations with common noise (eq:defMKV-CN-X) and (eq:defMKV-CN-Y) re with $p$-polynomial growth, one has $(b)$Extended conditional functional convex order. For any fun

Figures (1)

  • Figure 1: Estimations of the ESD by Monte-Carlo method with $N_{MC} = 10^4$ iterations as $N$ and $a$ vary. The simulation uses an Euler scheme with $100$ time steps. The full line represents the expected size of default (ESD) for the system \ref{['eq:fig1CFSX']}, while the dotted lines represents the expected size of default for the system with variable volatility \ref{['eq:fig1CSFY']}. Note that both quantities only depend on the empirical mean of the system, which, as already noted by Carmona-Fouque-Sun Carmona_Fouque_Sun_2015, is independent of $a$ (the small variations observed here being due to Monte-Carlo errors).

Theorems & Definitions (46)

  • Theorem 1.1: Conditional functional convex order
  • Theorem 1.2
  • Remark 1.3
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3: Lemma A.1 in MR4116705
  • Lemma 3.4
  • Lemma 3.5
  • Proposition 3.6
  • proof : Proof of Proposition \ref{['prop:marginalEuler']}
  • ...and 36 more