Paper: arXiv 2506.00762

Authors: Martin Larsson, Shukun Long

Abstract

Given an Itô semimartingale $X$, its Markovian projection is an Itô semimartingale $\widehat{X}$, with Markovian differential characteristics, that matches the one-dimensional marginal laws of $X$. One may even require certain functionals of the two processes to have the same fixed-time marginals, at the cost of enhancing the differential characteristics of $\widehat{X}$ but still in a Markovian sense. In the continuous case, the definitive result on existence of Markovian projections was obtained by Brunick and Shreve~\cite{MR3098443}. In this paper, we extend their result to the fully general setting of Itô semimartingales with jumps.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly rigorous mathematical extension of Markovian projection theory to Itô semimartingales with jumps using updating functions and canonical measure spaces. It is purely theoretical probability/stochastic analysis with no empirical backtesting or real-world financial data applications, placing it squarely in the High Math / Low Rigor quadrant.

Research Flowchart

  flowchart TD
    A["Research Goal: Extend Markovian projections to Itô semimartingales with jumps"] --> B["Input: Original Itô Semimartingale X with jump characteristics"]
    B --> C["Enhance Differential Characteristics to capture path-dependent functionals"]
    C --> D["Construct Markovian Process X̂ matching 1D marginal laws of X"]
    D --> E["Apply Stochastic Calculus & Mimicking Theorems for Jump Processes"]
    E --> F["Key Outcome: Generalized Brunick-Shreve Theorem for Jump Semimartingales"]

Cited References (1)

  1. Mimicking the marginal distributions of a semimartingale (2012) — link

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