2012/08/21 by Shiqi Song, Song, Shiqi · 1 citation
Decision Sciences · Economics, Econometrics and Finance · #60G07 #60G44 #91G40 #97M30 #Credit Risk and Financial Regulations #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1208.4149
openalex publication_date 2012/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathbbF be a filtration and τ be a random time. Let \mathbbG be the progressive enlargement of \mathbbF with τ. We study the validity of the following formula, called optional splitting formula : For any \mathbbG-optional process Y, there exist a \mathbbF-optional process Y' and a function Y" defined on [0,∞]×(ℝ+×Ω) being B[0,∞]\otimesO(\mathbbF) measurable, such that Y=Y'\ind[0,τ)+Y"(τ)\ind[τ,∞) 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.