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Weak uniqueness for the PDE governing the joint law of a diffusion and its running supremum

Laure Coutin, Lorick Huang, Monique Pontier

TL;DR

The paper proves the weak uniqueness of the non-standard PDE governing the joint density $p_V$ of a diffusion and its running maximum, by transforming the PDE into a parabolic problem for the diagonal-projected quantity $q_H$, applying Ball's theorem, and using a diagonal localization to show the off-diagonal components vanish. This establishes that the series expansion for the joint law found in prior work is the unique weak solution in the function space ${\cal X}$, thereby confirming the regularity and structure of the joint density. The results have implications for precisely pricing path-dependent payoffs such as lookback options by providing a rigorous foundation for the joint density’s existence and smoothness. The work also outlines potential generalizations to diffusion coefficients, path-dependent dynamics, and low-regularity coefficients, connecting to broader stochastic-analytic frameworks.

Abstract

In a previous work [8], it was shown that the joint law of a diffusion process and the running supremum of its first component is absolutely continuous, and that its density satisfies a non standard weak partial differential equation (PDE). In this paper, we establish the uniqueness of the solution to this PDE, providing a more complete understanding of the system's behavior and further validating the approach introduced in [8].

Weak uniqueness for the PDE governing the joint law of a diffusion and its running supremum

TL;DR

The paper proves the weak uniqueness of the non-standard PDE governing the joint density of a diffusion and its running maximum, by transforming the PDE into a parabolic problem for the diagonal-projected quantity , applying Ball's theorem, and using a diagonal localization to show the off-diagonal components vanish. This establishes that the series expansion for the joint law found in prior work is the unique weak solution in the function space , thereby confirming the regularity and structure of the joint density. The results have implications for precisely pricing path-dependent payoffs such as lookback options by providing a rigorous foundation for the joint density’s existence and smoothness. The work also outlines potential generalizations to diffusion coefficients, path-dependent dynamics, and low-regularity coefficients, connecting to broader stochastic-analytic frameworks.

Abstract

In a previous work [8], it was shown that the joint law of a diffusion process and the running supremum of its first component is absolutely continuous, and that its density satisfies a non standard weak partial differential equation (PDE). In this paper, we establish the uniqueness of the solution to this PDE, providing a more complete understanding of the system's behavior and further validating the approach introduced in [8].
Paper Structure (11 sections, 27 theorems, 43 equations)

This paper contains 11 sections, 27 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Uniqueness for weak solution of PDE in the set ${\cal X}$
  3. The function $q_H$ is solution of a parabolic partial differential equation
  4. Localisation on the diagonal
  5. Proof of Theorem \ref{['unique0']}
  6. Checking assumptions of Theorem \ref{['ball']} for $q_H$
  7. The function $f$ belongs to $L^1([0,T], X)$
  8. The function $u=q_H$ belongs to $\mathcal{C}([0,T],X),$ where $X=L^2(\mathbb R^d)$.
  9. The density $p_V$ is an element of space ${\mathcal{X}}$
  10. Tools
  11. Conclusion and perspectives

Key Result

Theorem 1

Assume that $B \in C^1_b(\mathbb R^d, \mathbb R^d).$ Let $T>0,$ and $f_0 \in L^1(\mathbb R^d)\cap L^2(\mathbb R^d)$ with positive values such that $\int_{\mathbb R^d} f_0(x) dx=1.$ Then, $p_V$ belongs to ${\cal X}$ and is the unique solution in ${\cal X}$ of edp1m in a weak sense.

Theorems & Definitions (32)

  • Remark 1: Notations
  • Definition 1
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Proposition 2.1
  • Remark 3
  • Theorem 3
  • Corollary 1
  • Proposition 2.2
  • ...and 22 more