[Paper Review] Martingale representations in progressive enlargement setting: the role of the accessible jump times

Antonella Calzolari, Barbara Torti|arXiv (Cornell University)|Aug 19, 2017

Stochastic processes and financial applications9 references3 citations

TL;DR

This paper investigates martingale representations in progressive enlargement settings, showing that the multiplicity of FVH-martingales depends critically on the behavior of common accessible jump times of two martingales M and N. Under an equivalent decoupling measure, it extends Kusuoka's representation theorem by incorporating the role of these jump times, revealing how their predictability and alignment affect representation structure.

ABSTRACT

Let M and N be an F-martingale and an H-martingale respectively on the same probability space, both enjoying the predictable representation property. We discuss how, under the assumption of the existence of an equivalent decoupling measure for F and H, the nature of the jump times of M and N affects the representation of the FVH-martingales.~More precisely we show that the multiplicity in the sense of Davis and Varaiya of FVH depends on the behavior of the common accessible jump times of the two martingales. Then we propose an extension of Kusuoka's representation theorem.

Motivation & Objective

• To understand how the nature of accessible jump times influences the representation of FVH-martingales in progressive enlargement frameworks.
• To analyze the impact of common jump times between two F- and H-martingales on the multiplicity of representations.
• To extend Kusuoka's representation theorem by incorporating the role of accessible jump times under an equivalent decoupling measure.
• To clarify the conditions under which predictable representation properties are preserved in enlarged filtration settings.

Proposed method

• Assumes the existence of an equivalent decoupling measure for the filtrations F and H, enabling separation of dependence structures.
• Analyzes the predictable representation property (PRP) for F and H-martingales M and N, respectively.
• Examines the behavior of common accessible jump times of M and N to determine their influence on the multiplicity of FVH-martingales.
• Uses Davis and Varaiya's concept of multiplicity to characterize the structure of FVH-martingales in terms of jump time alignment.
• Derives an extended version of Kusuoka's representation theorem that accounts for the predictability and co-occurrence of accessible jumps.
• Applies stochastic calculus in progressively enlarged filtrations to derive representation formulas involving jump times and local martingale components.

Experimental results

Research questions

• RQ1 How do common accessible jump times between two martingales affect the multiplicity of FVH-martingales in progressive enlargement?
• RQ2 What role do accessible jump times play in determining the structure of representations for FVH-martingales under a decoupling measure?
• RQ3 In what way does the predictability of jump times modify the representation of FVH-martingales in the enlarged filtration?
• RQ4 How can Kusuoka's representation theorem be generalized to include the influence of accessible jump times in progressive enlargement?
• RQ5 Under what conditions is the predictable representation property preserved for FVH-martingales when jump times are shared or synchronized?

Key findings

• The multiplicity of FVH-martingales, as defined by Davis and Varaiya, is directly influenced by the behavior of common accessible jump times of the underlying F and H-martingales.
• When common accessible jump times are predictable and aligned, the structure of FVH-martingale representations becomes more constrained, reducing multiplicity.
• The existence of an equivalent decoupling measure allows for a clean separation of dependence, enabling the derivation of extended representation formulas.
• The proposed extension of Kusuoka's theorem incorporates the role of accessible jump times, providing a more comprehensive framework for representation in progressive enlargement.
• The representation of FVH-martingales depends not only on the individual PRP of M and N but also on the joint predictability of their accessible jump times.
• The paper establishes that the alignment and predictability of common accessible jumps are critical in determining whether FVH-martingales admit unique or multiple representations.