Paper: arXiv 2311.07936

Authors: Valentin Tissot-Daguette

Abstract

A stochastic process $X$ becomes occupied when it is enlarged with its occupation flow $\mathcal{O}$ that tracks the time spent by the path at each level. When $X$ is Markov, the occupied process $(\mathcal{O},X)$ enjoys a Markov structure as well. We develop an Itô calculus for occupied processes that lies midway between Dupire’s functional Itô calculus and the classical version. We derive Itô formulae and, through Feynman-Kac, unveil a broad class of path-dependent PDEs where $\mathcal{O}$ plays the role of time. The space variable, given by the current value of $X$, remains finite-dimensional, thereby paving the way for standard elliptic PDE techniques and numerical methods. The framework’s benefits are illustrated via an optimal stopping problem involving local times, followed by financial applications. For the latter, we show how occupation flows provide unified Markovian lifts for exotic options and variance instruments, allowing financial institutions to price derivatives books with a single numerical solver. We finally explore an extension of forward variance models so as to leverage the entire forward occupation surface.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces a highly novel mathematical framework combining occupation flows, functional Itô calculus, and path-dependent PDEs to solve complex pricing and optimal stopping problems. While mathematically rigorous and theoretical, it lacks extensive empirical backtesting and datasets, placing it firmly in the theoretical domain.

Research Flowchart

  flowchart TD
    A[Goal: Develop Markovian Framework & Itô Calculus for Occupied Processes] --> B[Data/Inputs: Process X & Occupation Flow O]
    B --> C[Methodology: Extend X to Occupied Process O,X & Build Mid-level Itô Calculus]
    C --> D[Derive Feynman-Kac Formula for Path-Dependent PDEs]
    D --> E[Applications: Solve Optimal Stopping & Forward Variance Models]
    E --> F[Outcome: Unified Markovian Lift & Single Numerical Solver for Exotic Derivatives]

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1 of 104 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.