Optional splitting formula in a progressively enlarged filtration

Let $𝔽$ be a filtration and τ be a random time. Let $𝔾$ be the progressive enlargement of $𝔽$ with τ. We study the following formula, called the optional splitting formula: For any $𝔾$-optional process Y, there exists an $𝔽$-optional process Y^′ and a function Y^′′ defined on [0,∞] × (ℝ₊ × Ω) being $ℬ[0,\infty ]\otimes 𝒪\left(𝔽\right)$ measurable, such that $Y=Y^{\text{'}}{\mathbb{1}}_{[0,\tau )}+Y^{\text{'}\text{'}}\left(\tau \right){\mathbb{1}}_{[\tau ,\infty )}$. (This formula can also be formulated for multiple random times τ₁,...,τ_k). 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.

[1] M. Barlow, Study of a filtration expanded to include an honest time. Probab. Theory Relat. Fields 44 (1978) 307-323. | MR 509204 | Zbl 0369.60047

[2] M. Barlow, M. Emery, F. Knight, S. Song and M. Yor, Autour d'un théorème de Tsirelson sur des filtrations browniennes et non browniennes. Séminaire de Probabilité, XXXII. Springer-Verlag Berlin (1998). | Numdam | MR 1655299 | Zbl 0914.60064

[3] M. Barlow and J. Pitman and M. Yor, On Walsh's Brownian motion. Séminaire de Probabilité, XXIII. Springer-Verlag Berlin (1989) | Numdam | Zbl 0747.60072

[4] A. Bélanger and S. Shreve and D. Wong, A unified model for credit derivatives. Working paper (2002).

[5] F. Biagini and A. Cretarola, Local risk-minimization for defaultable claims with recovery process. Appl. Math. Optim. 65 (2012) 293-314. | MR 2902694 | Zbl 1244.93152

[6] T. Bielecki and M. Jeanblanc and M. Rutkowski, Credit Risk Modelling. Osaka University Press (2009). | Zbl 1107.91351

[7] P. Billingsley, Convergence of probability measures. John Wiley & Sons (1968). | MR 233396 | Zbl 0944.60003

[8] P. Brémaud and M. Yor, Changes of filtrations and of probability measures. Prob. Theory Relat. Fields 4 (1978) 269-295. | MR 511775 | Zbl 0415.60048

[9] G. Callegaro and M. Jeanblanc and B. Zargari, Carthaginian enlargement of filtrations. ESAIM: PS 17 (2013) 550-566. | Numdam | MR 3085632 | Zbl 1296.60106

[10] L. Chaumont and M. Yor, Exercises in probability: a guide tour from measure theory to random processes, via conditioning. Cambridge University Press (2009). | MR 2016344 | Zbl 1180.60002

[11] C. Dellacherie and M. Emery, Filtrations indexed by ordinals; application to a conjecture of S. Laurent. Working paper (2012). | MR 3185912

[12] C. Dellacherie and P. Meyer, Probabilités et potentiel Chapitres I à IV. Hermann Paris (1975). | Zbl 0138.10402

[13] C. Dellacherie and P. Meyer, Probabilités et potentiel Chapitres XVII à XXIV. Hermann Paris (1992). | Zbl 0138.10402

[14] N. El. Karoui and M. Jeanblanc and Y. Jiao, What happens after a default: the conditional density approach. Stoch. Process. Appl. 120 (2010) 1011-1032. | MR 2639736 | Zbl 1194.91187

[15] M. Emery and W. Schachermayer, A remark on Tsirelson's stochastic differential equation. Séminaire de Probabilités XXXIII, Springer-Verlag Berlin (1999) 291-303. | Numdam | MR 1768002 | Zbl 0957.60064

[16] M. Emery and W. Schachermayer, On Vershik's standardness criterion and Tsirelson's notion of cosiness. Séminaire de Probabilités XXXV. Springer (2001) 265-305. | Numdam | MR 1837293 | Zbl 1001.60039

[17] C. Fontana and M. Jeanblanc and S. Song, On arbitrages arising with honest times. Finance Stoch. 18 (2014) 515-543. | MR 3232015

[18] R. Handel On the exchange of intersection and supremum of σ-fields in filtering theory. Israel J. Math. 192 (2012) 763. | MR 3009741 | Zbl 1271.60054

[19] S.W. He and J.G. Wang and J.A. Yan, Semimartingale Theory And Stochastic Calculus. Science Press CRC Press Inc (1992). | MR 1219534 | Zbl 0781.60002

[20] M. Jeanblanc and M. Rutkowski, Modeling default risk: an overview Math. Finance: Theory and Practice. Fudan University High Education Press (1999).

[21] M. Jeanblanc and S. Song, An explicit model of default time with given survival probability. Stoch. Process. Appl. 121 (2010) 1678-1704. | MR 2811019 | Zbl 1298.91176

[22] M. Jeanblanc and S. Song, Random times with given survival probability and their F-martingale decomposition formula. Stoch. Process. Appl. 121 (2010) 1389-1410. | MR 2794982 | Zbl 1230.60047

[23] M. Jeanblanc and S. Song, Martingale representation theorem in progressively enlarged filtrations (2012). Preprint arXiv:1203.1447.

[24] M. Jeanblanc and Y. Lecam, Reduced form modelling for credit risk (2008). Available on: defaultrisk.com

[25] T. Jeulin Semi-martingales et grossissement d'une filtration, vol. 833 of Lect. Notes Math. Springer (1980). | MR 604176 | Zbl 0444.60002

[26] T. Jeulin and M. Yor, Grossissement d'une filtration and semi-martingales: formules explicites. Séminaire de Probabilités XII (1978) 78-97. | Numdam | MR 519998 | Zbl 0411.60045

[27] Y. Jiao, Multiple defaults and contagion risks with global and default-free information. Working paper (2010).

[28] I. Kharroubi and T. Lim, Progressive enlargement of filtrations and backward SDEs with jumps. Working paper (2011).

[29] S. Kusuoka, A remark on default risk models. Adv. Math. Econ. 1 (1999) 69-82. | MR 1722700 | Zbl 0939.60023

[30] A. Nikeghbali and E. Platen, On honest times in financial modeling (2008). Preprint arXiv:0808.2892.

[31] H. Pham, Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management. Stoch. Process. Appl. 120 (2010) 1795-1820. | MR 2673975 | Zbl 1196.60141

[32] Ph. Protter, Stochastic integration and differential equations, 2nd edition. Springer (2004). | MR 2020294 | Zbl 0694.60047

[33] L. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales, vol. 1, Foundations. John Wiley and Sons (1994). | MR 1331599 | Zbl 0402.60003

[34] S. Song, Grossissement d'une filtration et problèmes connexes. Thesis Université Paris IV (1987).

[35] S. Song, Drift operator in a market affected by the expansion of information flow: a case study (2012). Preprint arXiv:1207.1662v1.

[36] S. Song, Local solution method for the problem of enlargement of filtration (2013). Preprint arXiv:1302.2862.

[37] C. Stricker and M. Yor, Calcul stochastique dépendant d'un paramètre. Probab. Theory Relat. Fields 45 (1978) 109-133. | MR 510530 | Zbl 0388.60056

[38] H. Von Weizsäcker, Exchanging the order of taking suprema and countable intersections of σ-algebras. Ann. Inst. Henri Poincaré Section B, Tome 19 1 (1983) 91-100. | Numdam | MR 699981 | Zbl 0509.60002

[39] D. Wu, Dynamized copulas and applications to counterparty credit risk. Ph.D. Thesis, University of Evry (2012).

[40] K. Yano and M. Yor, Around Tsirelson's equation, or: The evolution process may not explain everything (2010). Preprint arXiv:0906.3442.