[Paper Review] Convergence of local supermartingales and Novikov-Kazamaki type conditions for processes with jumps

Martin Larsson, Johannes Ruf|arXiv (Cornell University)|Nov 23, 2014

Stochastic processes and financial applications43 references19 citations

TL;DR

This paper establishes necessary and sufficient conditions for the almost sure convergence of local supermartingales with jumps, using extended local integrability and predictable characteristics. It provides a novel proof of Novikov-Kazamaki-type conditions for the martingale property of nonnegative local martingales with jumps, linking convergence to stochastic exponential and Föllmer measure properties.

ABSTRACT

We characterize the event of convergence of a local supermartingale. Conditions are given in terms of its predictable characteristics and quadratic variation. The notion of extended local integrability plays a key role. We then apply these characterizations to provide a novel proof for the sufficiency and necessity of Novikov-Kazamaki type conditions for the martingale property of nonnegative local martingales with jumps.

Motivation & Objective

• To characterize the almost sure convergence of local supermartingales with jumps using predictable characteristics and quadratic variation.
• To introduce and employ the concept of extended local integrability to tame pathwise irregularity in jump processes.
• To provide a new proof of the necessity and sufficiency of Novikov-Kazamaki-type conditions for the uniform integrability of nonnegative local martingales with jumps.
• To connect convergence behavior to the stochastic exponential and Föllmer measure framework in the context of change of measure.
• To generalize continuous-time martingale convergence results to processes with jumps by identifying key structural conditions.

Proposed method

• The authors use extended local integrability as a key condition to control jump behavior in local supermartingales.
• They analyze convergence via the predictable characteristics and quadratic variation of the process, particularly focusing on the jump measure and compensator.
• The proof technique involves constructing a new probability measure (Föllmer measure) under which the stochastic exponential may explode, enabling analysis of convergence paths.
• A measurable mapping Φ is constructed from the original probability space to a canonical space, preserving stopping times and optional processes.
• The method relies on a monotone class argument and the use of a measure-determining family of functions to identify finite-dimensional distributions.
• The construction of a new filtration H and associated processes ensures that the transformed processes are predictable and adapted, enabling the application of martingale theory.

Experimental results

Research questions

• RQ1 Under what conditions does a local supermartingale with jumps converge almost surely?
• RQ2 How does extended local integrability refine the classical notion of local integrability in the context of jump processes?
• RQ3 What is the precise relationship between the convergence of a local supermartingale and the integrability of its stochastic exponential?
• RQ4 How can the Novikov-Kazamaki conditions be proven necessary and sufficient for the martingale property of nonnegative local martingales with jumps?
• RQ5 In what way does the Föllmer measure framework help characterize the convergence and explosion behavior of stochastic exponentials?

Key findings

• The event of almost sure convergence of a local supermartingale is characterized by conditions on its predictable characteristics and quadratic variation, particularly involving the compensator of the jump measure.
• Extended local integrability is identified as a necessary and sufficient condition to ensure that the limit of a local supermartingale exists almost surely, even in the presence of jumps.
• The paper provides a new proof of the necessity and sufficiency of Novikov-Kazamaki-type conditions for the uniform integrability of nonnegative local martingales with jumps, using the Föllmer measure framework.
• The stochastic exponential of a local martingale with jumps is a uniformly integrable martingale if and only if the exponential local martingale satisfies a moment condition involving the supremum over bounded stopping times.
• The construction of a canonical space via a measurable mapping Φ preserves the essential probabilistic structure, including stopping times and optional processes, enabling the transfer of convergence properties.
• The paper establishes that the convergence of a local supermartingale is equivalent to the finiteness of its quadratic variation and the extended local integrability of its compensator, under appropriate regularity conditions.