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Markovian projections for Itô semimartingales with jumps

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.

Markovian projections for Itô semimartingales with jumps

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.
Paper Structure (10 sections, 4 theorems, 56 equations)

This paper contains 10 sections, 4 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Prerequisites and Preliminary Results
  3. Transition Kernel
  4. Key Lemmas
  5. Differential Characteristics
  6. Main Results
  7. Examples
  8. Local Stochastic Volatility (LSV) Model.
  9. Local Stochastic Intensity (LSI) Model.
  10. Fake Hawkes Processes.

Key Result

Lemma 2.3

Let $X$ be an $\mathbb{R}^d$-valued measurable process, and $\alpha$ be a $C$-valued measurable process, where $C \subseteq \mathbb{R}^n$ is a closed convex set, satisfying Then, there exists a measurable function $a: \mathbb{R}_+ \times \mathbb{R}^d \to C$ such that for Lebesgue-a.e. $t \geq 0$,

Theorems & Definitions (19)

  • Definition 2.1: Transition kernel
  • Definition 2.2
  • Lemma 2.3: cf. MR3098443, Proposition 5.1
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 9 more