Optional splitting formula in a progressively enlarged filtration. (29th October 2014)

• Record Type: Journal Article
• Authors: Song, Shiqi
• Is Part Of: ESAIM. Volume 18(2014)
• Page Start: 829
• Page End: 853
• Publication Date: 2014-10-29
• Subjects: Probabilities -- Periodicals; Mathematical statistics -- Periodicals; 519.2
• DOI: 10.1051/ps/2014003
• Languages: English
• ISSNs: 1292-8100

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>Let <inline-formula><alternatives><tex-math id="tex_eq1"><![CDATA[\hbox{$\mathbb{F}^{}_{}$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxsg6" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq1" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>F</mml:mi></mml:msubsup></mml:math></alternatives></inline-formula> be a filtration and <italic>Ï„</italic> be a random time. Let <inline-formula><alternatives><tex-math id="tex_eq2"><![CDATA[\hbox{$\mathbb{G}^{}_{}$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxsfp" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq2" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>G</mml:mi></mml:msubsup></mml:math></alternatives></inline-formula> be the progressive enlargement of <inline-formula><alternatives><tex-math id="tex_eq3"><![CDATA[\hbox{$\mathbb{F}^{}_{}$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxsg6" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq3" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>F</mml:mi></mml:msubsup></mml:math></alternatives></inline-formula> with <italic>Ï„</italic>. We study the following formula, called the optional splitting formula: For any <inline-formula><alternatives><tex-math id="tex_eq4"><![CDATA[\hbox{$\mathbb{G}^{}_{}$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxsfp" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq4" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>G</mml:mi></mml:msubsup></mml:math></alternatives></inline-formula>-optional process <italic>Y</italic>, there exists an <inline-formula><alternatives><tex-math id="tex_eq5"><![CDATA[\hbox{$\mathbb{F}^{}_{}$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxsg6" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq5" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>F</mml:mi></mml:msubsup></mml:math></alternatives></inline-formula>-optional process <italic>Y</italic><sup>′</sup> and a function <italic>Y</italic><sup>′′</sup> defined on [0<italic>, </italic>∞] × (ℝ<sub>+</sub> × <italic>Ω</italic>) being <inline-formula><alternatives><tex-math id="tex_eq6"><![CDATA[\hbox{$\mathcal{B}[0, \infty]\otimes\mathcal{O}(\mathbb{F}^{}_{})$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxs7k" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq6" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">ℬ</mml:mi><mml:mo mathvariant="normal">[</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mi mathvariant="italic">, </mml:mi><mml:mi mathvariant="normal">∞</mml:mi><mml:mo mathvariant="normal">]</mml:mo><mml:mo mathvariant="normal">⊗</mml:mo><mml:mi mathvariant="normal">ð'ª</mml:mi><mml:mo mathvariant="normal">(</mml:mo><mml:msubsup><mml:mi>F</mml:mi></mml:msubsup><mml:mo mathvariant="normal">)</mml:mo></mml:math></alternatives></inline-formula> measurable, such that <inline-formula><alternatives><tex-math id="tex_eq7"><![CDATA[\hbox{$Y=Y'\mathds{1}^{}_{[0, \tau)}+Y''(\tau)\mathbb{1}^{}_{[\tau, \infty)}.$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxskr" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq7" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="italic">Y</mml:mi><mml:mo mathvariant="normal">=</mml:mo><mml:msup><mml:mi mathvariant="italic">Y</mml:mi><mml:mi mathvariant="normal">′</mml:mi></mml:msup><mml:msubsup><mml:mn>1</mml:mn><mml:mrow><mml:mo mathvariant="normal">[</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mi mathvariant="italic">, Ï„</mml:mi><mml:mo mathvariant="normal">)</mml:mo></mml:mrow></mml:msubsup><mml:mo mathvariant="normal">+</mml:mo><mml:msup><mml:mi mathvariant="italic">Y</mml:mi><mml:mi mathvariant="normal">′′</mml:mi></mml:msup><mml:mo mathvariant="normal">(</mml:mo><mml:mi mathvariant="italic">Ï„</mml:mi><mml:mo mathvariant="normal">)</mml:mo><mml:msubsup><mml:mn>1</mml:mn><mml:mrow><mml:mo mathvariant="normal">[</mml:mo><mml:mi mathvariant="italic">Ï„, </mml:mi><mml:mi mathvariant="normal">∞</mml:mi><mml:mo mathvariant="normal">)</mml:mo></mml:mrow></mml:msubsup><mml:mn mathvariant="italic">.</mml:mn></mml:math></alternatives></inline-formula> (This formula can also be formulated for multiple random times <italic>Ï„</italic><sub>1</sub><italic>, </italic><italic>...</italic><italic>, </italic><italic>Ï„</italic><sub><italic>k</italic></sub>). 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. </p> </abstract>