Forward rate models with linear volatilities

TL;DR

This paper investigates the conditions under which solutions exist for the Heath-Jarrow-Morton bond market model with linear volatility driven by Lévy processes, highlighting the necessity of no Gaussian part or negative jumps.

Contribution

It provides necessary and sufficient conditions for the existence of solutions in the linear volatility model with Lévy noise, extending understanding of the model's mathematical properties.

Findings

Solutions exist only if the Lévy process has no Gaussian component.

Negative jumps in the Lévy process prevent solution existence.

Conditions depend on the Lévy measure near zero and the Laplace exponent at infinity.

Abstract

Existence of solutions to the Heath-Jarrow-Morton equation of the bond market with linear volatility and general L\'evy random factor is studied. Conditions for existence and non-existence of solutions in the class of bounded fields are presented. For the existence of solutions the L\'evy process should necessarily be without the Gaussian part and without negative jumps. If this is the case then necessary and sufficient conditions for the existence are formulated either in terms of the behavior of the L\'evy measure of the noise near the origin or the behavior of the Laplace exponent of the noise at infinity.