Term Structure Models Driven by Wiener Process and Poisson Measures: Existence and Positivity

TL;DR

This paper establishes existence, uniqueness, and positivity conditions for advanced term structure models driven by Wiener processes and Poisson measures, including applications to interest rate equations with jumps.

Contribution

It provides a comprehensive existence and positivity characterization for jump-diffusion term structure models, extending prior work to include jump processes and the Brody-Hughston equation.

Findings

Proved existence and uniqueness of the HJM type equation with jumps.

Characterized positivity preserving models via characteristic coefficients.

Analyzed the Brody-Hughston interest rate equation with jumps.

Abstract

In the spirit of Bj\"ork-DiMasi-Kabanov-Runggaldier, we investigate term structure models driven by Wiener process and Poisson measures with forward curve dependent volatilities. This includes a full existence and uniqueness proof for the corresponding Heath--Jarrow--Morton type term structure equation. Furthermore, we characterize positivity preserving models by means of the characteristic coefficients, which was open for jump-diffusions. Additionally we treat existence, uniqueness and positivity of the Brody-Hughston equation of interest rate theory with jumps, an equation which we believe to be very useful for applications. A key role in our investigation is played by the method of the moving frame, which allows to transform the Heath--Jarrow--Morton--Musiela equation to a time-dependent SDE.