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Existence of Lévy term structure models

Damir Filipović, Stefan Tappe

TL;DR

This paper establishes existence and uniqueness for Lévy-driven Heath–Jarrow–Morton–Musiela term-structure models by formulating the forward-curve dynamics as infinite-dimensional SDEs. It first derives the HJM drift condition ensuring no-arbitrage via cumulant generating functions of the Lévy drivers and then analyzes solution concepts on two forward-curve spaces: the Björk–Svensson space $H_{\beta,\gamma}$ for strong solutions and the larger weighted space $H_w$ for mild and weak solutions. While $H_{\beta,\gamma}$ is sometimes too small to accommodate the drift, the main existence/uniqueness results are obtained on $H_w$, applicable to a broad class of Lévy processes including mixed Brownian and jump components as well as purely discontinuous processes. The work integrates a rigorous SDE framework with càdlàg modification results for the Lévy integral, providing a solid foundation for arbitrage-free, Lévy-driven term-structure models with well-behaved forward curves in finance.

Abstract

Lévy driven term structure models have become an important subject in the mathematical finance literature. This paper provides a comprehensive analysis of the Lévy driven Heath-Jarrow-Morton type term structure equation. This includes a full proof of existence and uniqueness in particular, which seems to have been lacking in the finance literature so far.

Existence of Lévy term structure models

TL;DR

This paper establishes existence and uniqueness for Lévy-driven Heath–Jarrow–Morton–Musiela term-structure models by formulating the forward-curve dynamics as infinite-dimensional SDEs. It first derives the HJM drift condition ensuring no-arbitrage via cumulant generating functions of the Lévy drivers and then analyzes solution concepts on two forward-curve spaces: the Björk–Svensson space for strong solutions and the larger weighted space for mild and weak solutions. While is sometimes too small to accommodate the drift, the main existence/uniqueness results are obtained on , applicable to a broad class of Lévy processes including mixed Brownian and jump components as well as purely discontinuous processes. The work integrates a rigorous SDE framework with càdlàg modification results for the Lévy integral, providing a solid foundation for arbitrage-free, Lévy-driven term-structure models with well-behaved forward curves in finance.

Abstract

Lévy driven term structure models have become an important subject in the mathematical finance literature. This paper provides a comprehensive analysis of the Lévy driven Heath-Jarrow-Morton type term structure equation. This includes a full proof of existence and uniqueness in particular, which seems to have been lacking in the finance literature so far.

Paper Structure

This paper contains 8 sections, 23 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. The HJM drift condition
  3. Forward curve evolutions as strong solutions of infinite dimensional stochastic differential equations
  4. Forward curve evolutions as mild and weak solutions of infinite dimensional stochastic differential equations
  5. Conclusion
  6. Overview and notation
  7. Stochastic integration
  8. Stochastic differential equations

Key Result

Proposition 3.1

The space $(H_{\beta,\gamma},\langle \cdot,\cdot \rangle_{\beta,\gamma})$ is a separable Hilbert space and for each $x \in \mathbb{R}_+$, the point evaluation $h \mapsto h(x) : H_{\beta,\gamma} \rightarrow \mathbb{R}$ is a continuous linear functional.

Theorems & Definitions (52)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Example 3.4
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • ...and 42 more