[Paper Review] Martingale representation property in progressively enlarged filtrations

Monique Jeanblanc, S. Song|arXiv (Cornell University)|Mar 7, 2012

Stochastic processes and financial applications10 references4 citations

TL;DR

This paper develops a general methodology to determine when the martingale representation property (MRP) holds in progressively enlarged filtrations, extending the MRP from a base filtration 𝔽 to 𝔾 = 𝔽 ∨ 𝜏. By analyzing the projection of 𝔾-local martingales onto the stable space generated by the compensated jump martingales and the 𝔽-martingale parts, the authors unify and extend known results under various conditions—such as honest times, Jacod's criterion, and immersion—providing a systematic framework applicable even to the evolution model of [21].

ABSTRACT

Consider $\mathbb{G}$ the progressive enlargement of a filtration $\mathbb{F}$ with a random time $τ$. Assuming that, in $\mathbb{F}$, the martingale representation property holds, we examine conditions under which the martingale representation property holds also in $\mathbb{G}$. A general methodology is developed in this paper, with results covering every known (classical or recent) examples.

Motivation & Objective

• To determine conditions under which the martingale representation property (MRP) holds in a progressively enlarged filtration 𝔾, given that it holds in the base filtration 𝔽.
• To unify and generalize existing results on MRP in 𝔾 under various assumptions, including honest times, Jacod's criterion, and the immersion condition.
• To develop a systematic method for computing projections of 𝔾-local martingales onto the stable space generated by the 𝔽-martingale parts and jump martingales in 𝔾.
• To apply the methodology to the evolution model introduced in [21], which had resisted prior techniques, and establish MRP for this class of models.
• To provide a rigorous framework for credit risk modeling where the default time 𝜏 is not necessarily predictable or independent, and where the intensity is calibrated from market data.

Proposed method

• The method decomposes the analysis of 𝔾-martingales into two time intervals: [0,𝜏] and (𝜏,∞), exploiting the fact that 𝔾-predictable sets coincide with 𝔽-predictable sets on [0,𝜏] for any random time 𝜏.
• It uses the key identity (1) from [5] to express the restriction of a bounded 𝔾-martingale N to [0,𝜏] as a function of the 𝔽-conditional expectation and the survival probability process Z_t = ℚ(t < 𝜏 | 𝔽_t).
• The projection of a 𝔾-local martingale onto the stable space generated by the 𝔾-martingale parts of 𝔽-martingales and the jump martingales of the form v1_{[𝜏,∞)} − (v1_{[𝜏,∞)})^{𝔾·p} is computed via integration by parts on the interval [0,𝜏].
• The approach relies on the use of the (H′)-hypothesis and the immersion condition to ensure compatibility between 𝔽 and 𝔾-martingales, particularly in the context of change of probability measures.
• The methodology is validated through a series of lemmas (A.7–A.12) establishing the equivalence of predictability, integrability, and dual predictable projection between the filtrations 𝔾 and 𝔾^{(S,T]}, enabling the transfer of stochastic calculus tools.
• The core technical innovation lies in the use of the monotone class theorem to extend results from left-continuous processes to general predictable processes, ensuring broad applicability.

Experimental results

Research questions

• RQ1 Under what conditions does the martingale representation property in the base filtration 𝔽 extend to the progressively enlarged filtration 𝔾 = 𝔽 ∨ 𝜏?
• RQ2 How can the projection of a general 𝔾-local martingale onto the stable space generated by the 𝔽-martingale parts and jump martingales be systematically computed in 𝔾?
• RQ3 Can the proposed methodology unify and generalize known results under classical assumptions such as honest time, Jacod’s criterion, and the immersion condition?
• RQ4 Does the methodology apply to the evolution model of [21], which is known to be outside the scope of prior techniques?
• RQ5 What is the role of the (H′)-hypothesis and the immersion condition in ensuring the stability and predictability of the representation space in 𝔾?

Key findings

• The paper establishes a general projection formula for 𝔾-local martingales restricted to [0,𝜏], which allows the representation of any such martingale as a stochastic integral with respect to the 𝔽-martingale parts and the jump martingales in 𝔾, under no additional assumptions beyond the MRP in 𝔽.
• The methodology unifies and extends previous results: it recovers the MRP under the honest time assumption (as in [2]), the Jacod criterion (as in [18]), and the immersion condition (as in [27]), providing a single framework.
• The MRP holds in 𝔾 if and only if the projection of any 𝔾-local martingale onto the stable space generated by the 𝔾-martingale parts of 𝔽-martingales and the jump martingales is well-defined and recovers the original process, which is guaranteed under the developed framework.
• The evolution model of [21] is shown to satisfy the MRP in 𝔾, resolving a long-standing open problem where prior techniques failed due to lack of independence or predictability.
• The paper proves that the (H′)-hypothesis and the immersion condition are sufficient for the MRP to hold in 𝔾, and that the structure of the 𝔾-predictable sets on [0,𝜏] is key to the stability of the representation.
• The results are formally stated and proven in Theorem 3.1, Theorem 4.2, Theorem 5.1, and Theorem 7.2, which provide necessary and sufficient conditions for the MRP in 𝔾 under various assumptions.