Paper: arXiv 2502.09486

Authors: Nils Detering, Silvia Lavagnini

Abstract

We present a dynamic model for forward curves within the Heath-Jarrow-Morton framework under the Musiela parametrization. The forward curves take values in a function space H, and their dynamics follows a stochastic partial differential equation with state-dependent coefficients. In particular, the coefficients are defined through point-wise operating maps on H, resulting in a locally state-dependent structure. We first explore conditions under which these point-wise operators are well defined on H. Next, we determine conditions to ensure that the resulting coefficient functions satisfy local growth and Lipschitz properties, so to guarantee the existence and uniqueness of mild solutions. The proposed model captures the behavior of the entire forward curve through a single equation, yet retains remarkable simplicity. Notably, we demonstrate that certain one-dimensional projections of the model are Markovian and satisfy a one-dimensional stochastic differential equation. This connects our Hilbert-space approach to well established models for forward contracts with fixed delivery times, for which existing formulas and numerical techniques can be applied. This link allows us to examine also conditions for maintaining positivity of the solutions. As concrete examples, we analyze Hilbert-space valued variants of an exponential model and of a constant elasticity of variance model.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using SPDEs in Hilbert spaces (Filipović space) for forward curves, demonstrating strong mathematical complexity. However, it is purely theoretical in this excerpt with zero empirical validation, backtesting, or empirical data analysis.

Research Flowchart

  flowchart TD
    A[Goal: Develop locally state-dependent HJM forward curve model via SPDEs] --> B[Input: Musiela parametrization & Hilbert space H of forward curves]
    B --> C[Method: Define point-wise operating maps for state-dependent coefficients]
    C --> D[Analysis: Establish local growth & Lipschitz conditions for maps on H]
    D --> E[Outcome 1: Prove existence & uniqueness of mild SPDE solutions]
    D --> F[Outcome 2: Derive Markovian 1D projections for fixed delivery times]
    E & F --> G[Applications: Analyze positivity, Exponential, and CEV model variants]