Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the well-posedness of a class of McKean Feynman-Kac equations

Jonas Lieber, Nadia Oudjane, Francesco Russo

Abstract

We analyze the well-posedness of a so called McKean Feynman-Kac Equation (MFKE), which is a McKean type equation with a Feynman-Kac perturbation. We provide in particular weak and strong existence conditions as well as pathwise uniqueness conditions without strong regularity assumptions on the coefficients. One major tool to establish this result is a representation theorem relating the solutions of MFKE to the solutions of a nonconservative semilinear parabolic Partial Differential Equation (PDE).

On the well-posedness of a class of McKean Feynman-Kac equations

Abstract

We analyze the well-posedness of a so called McKean Feynman-Kac Equation (MFKE), which is a McKean type equation with a Feynman-Kac perturbation. We provide in particular weak and strong existence conditions as well as pathwise uniqueness conditions without strong regularity assumptions on the coefficients. One major tool to establish this result is a representation theorem relating the solutions of MFKE to the solutions of a nonconservative semilinear parabolic Partial Differential Equation (PDE).

Paper Structure

This paper contains 23 sections, 13 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Basic assumptions
  3. Notations
  4. Assumptions
  5. Weak and mild solutions
  6. Main results and strategy of the proofs
  7. Well-posedness for McKean Feynman-Kac equation
  8. A relaxation of Assumption \ref{['A5']}
  9. Main results
  10. Strategy of the proof
  11. Proof of Item 1. of Theorem \ref{['thm:representation']}
  12. Sketch of the proof of Item 2. of Theorem \ref{['thm:representation']}
  13. Sketch of the proof of Theorem \ref{['thm:final']}
  14. Equivalence of weak and mild solutions of the nonlinear PDE
  15. Existence and uniqueness of a mild solution to the nonlinear PDE
  16. ...and 8 more sections

Key Result

Theorem 12

Assume that Assumptions A1, A2, A3, A4 hold.

Theorems & Definitions (41)

  • Remark 1
  • Definition 2
  • Remark 3
  • Remark 4
  • Definition 5
  • Definition 6
  • Remark 7
  • Definition 8
  • Definition 9
  • Remark 10
  • ...and 31 more