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Classical solution of path-dependent mean-field semilinear PDEs

Shanjian Tang, Huilin Zhang

Abstract

The paper concerns classical solution of path-dependent partial differential equations (PPDEs) with coefficients depending on both variables of path and path-valued measure, which are crucial to understanding large-scale mean-field interacting systems in a non-Markovian setting. We construct classical solutions of the PPDEs via solution of the forward and backward stochastic differential equations. To accommodate the intricacies introduced by the appearance of the path in the coefficients, we develop a novel technique known as the ``parameter frozen'' approach to the PPDEs.

Classical solution of path-dependent mean-field semilinear PDEs

Abstract

The paper concerns classical solution of path-dependent partial differential equations (PPDEs) with coefficients depending on both variables of path and path-valued measure, which are crucial to understanding large-scale mean-field interacting systems in a non-Markovian setting. We construct classical solutions of the PPDEs via solution of the forward and backward stochastic differential equations. To accommodate the intricacies introduced by the appearance of the path in the coefficients, we develop a novel technique known as the ``parameter frozen'' approach to the PPDEs.

Paper Structure

This paper contains 16 sections, 30 theorems, 283 equations.

Table of Contents

  1. Introduction
  2. Basic setup and Itô calculus for functionals of both path and path-measure
  3. The canonical setup
  4. Strong vertical derivatives with respect to path and path-measure
  5. Itô-Dupire formula
  6. Solution of semilinear path-dependent master equations
  7. The decoupling field and its regularity
  8. Solution of BSDEs as solution of path-dependent mean-field PDEs
  9. Classical solution of semi-linear path-dependent PDEs
  10. Some typical cases
  11. Differentiability of solutions of path-dependent mean-field BSDEs
  12. First-order differentiability
  13. Second-order differentiability
  14. Appendix
  15. Proof of Lemma \ref{['unibd']}
  16. ...and 1 more sections

Key Result

Lemma 2.5

Suppose that $f:[0,T] \times \mathbb{D}_{T,d} \mapsto \mathbb{R}$ is strongly vertically differentiable, and uniformly Lipschitz continuous in $\omega:$ Then we have $|\partial_{\omega_\tau} f(t,\omega)| \le C$ for any $(t,\omega)\in [0,T] \times \mathbb{D}_{T,d}$ and $\tau\le t.$

Theorems & Definitions (73)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 63 more