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Polynomial McKean-Vlasov SDEs

Christa Cuchiero, Janka Möller

TL;DR

The paper investigates a novel class of MV-SDEs where drift and diffusion depend on the current state and its marginal moments, potentially with a common-noise component. It builds a bridge to time-inhomogeneous polynomial processes to obtain well-posedness results via martingale problems and to compute conditional moments through nonlinear ODEs, with a Magnus expansion enabling explicit calculations when needed. The work presents primal and dual viewpoints, establishes existence and uniqueness under milder growth and Lipschitz conditions, and extends the framework to conditional moments under common noise, highlighting polynomial structure and forward–backward dynamics. These results enable analytic moment calculations and offer pathways for state-space confinement, with potential applications in areas such as population genetics and financial modeling.

Abstract

We study a new class of McKean-Vlasov stochastic differential equations (SDEs), possibly with common noise, applying the theory of time-inhomogeneous polynomial processes. The drift and volatility coefficients of these SDEs depend on the state variables themselves as well as their conditional moments in a way that mimics the standard polynomial structure. Our approach leads to new results on the existence and uniqueness of solutions to such conditional McKean-Vlasov SDEs which are, to the best of our knowledge, not obtainable using standard methods. Moreover, we show in the case without common noise that the moments of these McKean-Vlasov SDEs can be computed by non-linear ODEs. As a by-product, this also yields new results on the existence and uniqueness of global solutions to certain ODEs.

Polynomial McKean-Vlasov SDEs

TL;DR

The paper investigates a novel class of MV-SDEs where drift and diffusion depend on the current state and its marginal moments, potentially with a common-noise component. It builds a bridge to time-inhomogeneous polynomial processes to obtain well-posedness results via martingale problems and to compute conditional moments through nonlinear ODEs, with a Magnus expansion enabling explicit calculations when needed. The work presents primal and dual viewpoints, establishes existence and uniqueness under milder growth and Lipschitz conditions, and extends the framework to conditional moments under common noise, highlighting polynomial structure and forward–backward dynamics. These results enable analytic moment calculations and offer pathways for state-space confinement, with potential applications in areas such as population genetics and financial modeling.

Abstract

We study a new class of McKean-Vlasov stochastic differential equations (SDEs), possibly with common noise, applying the theory of time-inhomogeneous polynomial processes. The drift and volatility coefficients of these SDEs depend on the state variables themselves as well as their conditional moments in a way that mimics the standard polynomial structure. Our approach leads to new results on the existence and uniqueness of solutions to such conditional McKean-Vlasov SDEs which are, to the best of our knowledge, not obtainable using standard methods. Moreover, we show in the case without common noise that the moments of these McKean-Vlasov SDEs can be computed by non-linear ODEs. As a by-product, this also yields new results on the existence and uniqueness of global solutions to certain ODEs.

Paper Structure

This paper contains 17 sections, 26 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Time-inhomogeneous polynomial processes
  4. Existence and uniqueness of solutions to time-inhomogeneous diffusions
  5. Polynomial McKean-Vlasov equations
  6. Primal point of view
  7. Dual point of view
  8. Adding common noise
  9. ODE theory
  10. Proofs
  11. Proof of Theorem \ref{['thm:sol_MP']}
  12. Proof of Lemma \ref{['lem:new_ass_maps']}
  13. Proof of Proposition \ref{['prop:global_sol_ode']}
  14. Proof of Theorem \ref{['thm:dual_moment_formula']}
  15. Proof of Lemma \ref{['lem:fubini_type']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Consider a one-dimensional McKean-Vlasov SDE of the form where $(\mathbb{Z}_t)_{t\geq 0}:= (Z_t, Z^2_t, \dots, Z^N_t)_{t\geq 0}$ for $N\geq1$ and with fixed ${\mathbb E}[{\mathbb Z}_0]\in {\mathbb R}^N$, some maps $b,\, \beta, \,c, \,\gamma, \, \Gamma: {\mathbb R}^N \rightarrow {\mathbb R}$ and a Brownian motion $(W_t)_{t\geq 0}$. If $b, \beta, c, \gamma,

Theorems & Definitions (72)

  • Theorem 1.1
  • Definition 2.1: Markov Process
  • Definition 2.2: Transition Function
  • Definition 2.3: Transition Operator
  • Definition 2.4: Polynomial Process
  • Remark 2.5
  • Definition 2.6: Infinitesimal Generator
  • Proposition 2.7
  • proof
  • Definition 2.8: Polynomial Diffusion
  • ...and 62 more