Optional splitting formula in a progressively enlarged filtration

Abstract

Let $\mathbb{F}$ be a filtration and $\tau$ be a random time. Let $\mathbb{G}$ be the progressive enlargement of $\mathbb{F}$ with $\tau$. We study the validity of the following formula, called optional splitting formula : For any $\mathbb{G}$-optional process $Y$, there exist a $\mathbb{F}$-optional process $Y^{\prime }$ and a function $Y"$ defined on $[0,\infty ]\times ({\mathbb{R}}_{+}\times \Omega )$ being $\mathcal{B}[0,\infty ]\otimes \mathcal{O}(\mathbb{F})$ measurable, such that $$ Y=Y'\ind_{[0,τ)}+Y"(τ)\ind_{[τ,\infty)} $$ We are interested in this formula, because it has been taken for granted in number of recent works in credit risk modeling, whilst such a formula can not be true in general. Sufficient conditions will be given for the validity of the above formula as well as of its extension in the case of multiple random times.

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• Local solution method for the problem of enlargement of filtration
• Drift operator in a market affected by the expansion of information flow : a case study
• On honest times in financial modeling
• Martingale representation property in progressively enlarged filtrations

Cited by in corpus (4)

• Martingale representation property in progressively enlarged filtrations
• An introduction of the enlargement of filtration
• An enlargement of filtration formula with application to progressive enlargement with multiple random times
• Random time with differentiable conditional distribution function