Let be a filtration and τ be a random time. Let be the progressive enlargement of with τ. We study the following formula, called the optional splitting formula: For any -optional process Y, there exists an -optional process Y′ and a function Y′′ defined on [0,∞] × (ℝ+ × Ω) being measurable, such that . (This formula can also be formulated for multiple random times τ1,...,τk). We are interested in this formula because of its fundamental role in many recent papers on credit risk modeling, and also because of the fact that its validity is limited in scope and this limitation is not sufficiently underlined. In this paper we will determine the circumstances in which the optional splitting formula is valid. We will then develop practical sufficient conditions for that validity. Incidentally, our results reveal a close relationship between the optional splitting formula and several measurability questions encountered in credit risk modeling. That relationship allows us to provide simple answers to these questions.
Mots clés : optional process, progressive enlargement of filtration, credit risk modeling, conditional density hypothesis
@article{PS_2014__18__829_0, author = {Song, Shiqi}, title = {Optional splitting formula in a progressively enlarged filtration}, journal = {ESAIM: Probability and Statistics}, pages = {829--853}, publisher = {EDP-Sciences}, volume = {18}, year = {2014}, doi = {10.1051/ps/2014003}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps/2014003/} }
TY - JOUR AU - Song, Shiqi TI - Optional splitting formula in a progressively enlarged filtration JO - ESAIM: Probability and Statistics PY - 2014 SP - 829 EP - 853 VL - 18 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2014003/ DO - 10.1051/ps/2014003 LA - en ID - PS_2014__18__829_0 ER -
Song, Shiqi. Optional splitting formula in a progressively enlarged filtration. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 829-853. doi : 10.1051/ps/2014003. http://www.numdam.org/articles/10.1051/ps/2014003/
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