Optional splitting formula in a progressively enlarged filtration

Shiqi Song

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 829-853
  • ISSN: 1292-8100

Abstract

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Let 𝔽 F be a filtration andτbe a random time. Let 𝔾 G be the progressive enlargement of 𝔽 F withτ. We study the following formula, called the optional splitting formula: For any 𝔾 G-optional processY, there exists an 𝔽 F-optional processY′ and a function Y′′ defined on [0,∞] × (ℝ+ × Ω) being ℬ [ 0 , ∞ ] ⊗ 𝒪 ( 𝔽 ) ℬ[0,∞]⊗x1d4aa;(F) measurable, such that Y = Y ' 1 [ 0 , τ ) + Y ' ' ( τ ) 1 [ τ , ∞ ) . Y=Y′1[0,τ)+Y′′(τ)1[τ,∞). (This formula can also be formulated for multiple random timesτ1,...,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.

How to cite

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Song, Shiqi. "Optional splitting formula in a progressively enlarged filtration." ESAIM: Probability and Statistics 18 (2014): 829-853. <http://eudml.org/doc/274352>.

@article{Song2014,
abstract = {Let $\mathbb \{F\}^\{\}_\{\}$F be a filtration andτbe a random time. Let $\mathbb \{G\}^\{\}_\{\}$G be the progressive enlargement of $\mathbb \{F\}^\{\}_\{\}$F withτ. We study the following formula, called the optional splitting formula: For any $\mathbb \{G\}^\{\}_\{\}$G-optional processY, there exists an $\mathbb \{F\}^\{\}_\{\}$F-optional processY′ and a function Y′′ defined on [0,∞] × (ℝ+ × Ω) being $\mathcal \{B\}[0,\infty ]\otimes \mathcal \{O\}(\mathbb \{F\}^\{\}_\{\})$ℬ[0,∞]⊗x1d4aa;(F) measurable, such that $Y=Y^\{\prime \}\{1\}^\{\}_\{[0,\tau )\}+Y^\{\prime \prime \}(\tau )\mathbb \{1\}^\{\}_\{[\tau ,\infty )\}.$Y=Y′1[0,τ)+Y′′(τ)1[τ,∞). (This formula can also be formulated for multiple random timesτ1,...,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.},
author = {Song, Shiqi},
journal = {ESAIM: Probability and Statistics},
keywords = {optional process; progressive enlargement of filtration; credit risk modeling; conditional density hypothesis},
language = {eng},
pages = {829-853},
publisher = {EDP-Sciences},
title = {Optional splitting formula in a progressively enlarged filtration},
url = {http://eudml.org/doc/274352},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Song, Shiqi
TI - Optional splitting formula in a progressively enlarged filtration
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 829
EP - 853
AB - Let $\mathbb {F}^{}_{}$F be a filtration andτbe a random time. Let $\mathbb {G}^{}_{}$G be the progressive enlargement of $\mathbb {F}^{}_{}$F withτ. We study the following formula, called the optional splitting formula: For any $\mathbb {G}^{}_{}$G-optional processY, there exists an $\mathbb {F}^{}_{}$F-optional processY′ and a function Y′′ defined on [0,∞] × (ℝ+ × Ω) being $\mathcal {B}[0,\infty ]\otimes \mathcal {O}(\mathbb {F}^{}_{})$ℬ[0,∞]⊗x1d4aa;(F) measurable, such that $Y=Y^{\prime }{1}^{}_{[0,\tau )}+Y^{\prime \prime }(\tau )\mathbb {1}^{}_{[\tau ,\infty )}.$Y=Y′1[0,τ)+Y′′(τ)1[τ,∞). (This formula can also be formulated for multiple random timesτ1,...,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.
LA - eng
KW - optional process; progressive enlargement of filtration; credit risk modeling; conditional density hypothesis
UR - http://eudml.org/doc/274352
ER -

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