Markovian projections for Itô semimartingales with jumps

Paper: arXiv 2403.15980

Authors: Martin Larsson, Shukun Long

Abstract

Given a general Itô semimartingale, its Markovian projection is an Itô process, with Markovian differential characteristics, that matches the one-dimensional marginal laws of the original process. We construct Markovian projections for Itô semimartingales with jumps, whose flows of one-dimensional marginal laws are solutions to non-local Fokker–Planck–Kolmogorov equations (FPKEs). As an application, we show how Markovian projections appear in building calibrated diffusion/jump models with both local and stochastic features.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework for constructing Markovian projections of jump-diffusion processes using non-local Fokker-Planck-Kolmogorov equations. While mathematically advanced and novel in its continuous-time stochastic analysis, it is predominantly a theoretical work with limited empirical backtesting or real-world data application.

Research Flowchart

  flowchart TD
    A[Research Goal: Construct Markovian projections for Itô semimartingales with jumps] --> B[Input: General Itô Semimartingale with Jumps & Marginal Laws]
    B --> C[Methodology: Match 1D Marginal Laws via Conditional Expectations]
    C --> D[Derive Non-Local Fokker-Planck-Kolmogorov Equations]
    D --> E[Application: Construct Calibrated Local-Stochastic Jump Models]
    E --> F[Outcome: Itô process with Markovian characteristics matching original 1D marginals]