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Well-posedness of state-dependent rank-based interacting systems

Hélène Guérin, Nathalie Krell

Abstract

We study the existence and uniqueness of rank-based interacting systems of stochastic differential equations. These systems can be seen as modifications with state-dependent coefficients of the Atlas model in mathematical finance. The coefficients of the underlying SDEs are possibly discontinuous. We first establish strong well-posedness for a planar system with rank-dependent drift coefficients, and non-rank-dependent and non-uniformly elliptic diffusion coefficients. We then state weak well-posedness for two classes of high-dimensional rank-based interacting SDEs with elliptic diffusion coefficients. Finally, we address the positivity of solutions in the case where the diffusion coefficients vanish at zero.

Well-posedness of state-dependent rank-based interacting systems

Abstract

We study the existence and uniqueness of rank-based interacting systems of stochastic differential equations. These systems can be seen as modifications with state-dependent coefficients of the Atlas model in mathematical finance. The coefficients of the underlying SDEs are possibly discontinuous. We first establish strong well-posedness for a planar system with rank-dependent drift coefficients, and non-rank-dependent and non-uniformly elliptic diffusion coefficients. We then state weak well-posedness for two classes of high-dimensional rank-based interacting SDEs with elliptic diffusion coefficients. Finally, we address the positivity of solutions in the case where the diffusion coefficients vanish at zero.
Paper Structure (9 sections, 6 theorems, 38 equations)

This paper contains 9 sections, 6 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Strong well-posedness of a planar rank-based interacting system
  3. Properties of the function $G$
  4. The stochastic differential equation satisfied by $G(\mathbf{X})$
  5. Proof of Theorem \ref{['thm:main result']}
  6. Rank-based interacting systems in higher dimensions
  7. Weak well-posedness of the SDE \ref{['eq:EDS-poisson']} with positive and constant diffusion coefficients
  8. Weak well-posedness of the SDE \ref{['eq:EDS-poisson-2']}
  9. Non-negativity of the solution

Key Result

Theorem 2.5

We suppose that Assumptions ${{\left(A_{b,\sigma}\right)}}$ and ${{\left(A_\alpha\right)}}$ are satisfied. Let $\mathbf{X}_0$ be a random variable independent of $W$. There is a unique strong solution $\mathbf{X}$ to the SDE eq:EDSX with initial condition $\mathbf{X}_0$, up to a potential explosion

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Example 2.7
  • Proposition 2.8
  • proof
  • Theorem 2.9
  • ...and 10 more