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January  2018, 3: 3 doi: 10.1186/s41546-018-0029-8

Information uncertainty related to marked random times and optimal investment

Ying Jiao 1, and  Idris Kharroubi 2,,

1. Université Claude Bernard-Lyon 1, Institut de Science Financière et d'Assurances, 50 Avenue Tony Garnier, 69007 Lyon, France;

2. Sorbonne Université, Sorbonne Paris Cité, CNRS, Laboratoire de Probabilités Statistique et Modelisation, LPSM, F-75005 Paris, France

Received  January 19, 2017 Revised  April 18, 2018 Published  May 2018

Fund Project: The authors are grateful to the anonymous referees for their careful reading and their many insightful comments and suggestions.

We study an optimal investment problem under default risk where related information such as loss or recovery at default is considered as an exogenous random mark added at default time. Two types of agents who have different levels of information are considered. We first make precise the insider's information flow by using the theory of enlargement of filtrations and then obtain explicit logarithmic utility maximization results to compare optimal wealth for the insider and the ordinary agent.
Keywords: Information uncertainty, Marked random times, Enlargement of filtrations, Utility maximization.
Mathematics Subject Classification: 60G20;91G40;93E20.
Citation: Ying Jiao, Idris Kharroubi. Information uncertainty related to marked random times and optimal investment. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 3-. doi: 10.1186/s41546-018-0029-8
References:
[1]

Aksamit, A, Jeanblanc, M:Enlargement of Filtration with Finance in View. SpringerBriefs in Quantitative Finance. Springer, Cham (2017)

[2]

Amendinger, J:Martingale representation theorems for initially enlarged filtrations. Stoch. Process. Appl. 89, 101-116 (2000)

[3]

Amendinger, J, Becherer, D, Schweizer, M:A monetary value for initial information in portfolio optimization. Finance Stochast. 7(1), 29-46 (2003)

[4]

Amendinger, J, Imkeller, P, Schweizer, M:Additional logarithmic utility of an insider. Stoch. Process. Appl. 75, 263-286 (1998)

[5]

Bakshi, G, Madan, D, Zhang, F:Understanding the role of recovery in default risk models:Empirical comparisons and implied recovery rates (2006). preprint, University of Maryland

[6]

Bielecki, TR, Rutkowski, M:Credit risk:modelling, valuation and hedging. Springer-Verlag, Berlin (2002)

[7]

Blanchet-Scalliet, C, El Karoui, N, Jeanblanc, M, Martellini, L:Optimal investment decisions when timehorizon is uncertain. J. Math. Econ. 44(11), 1100-1113 (2008)

[8]

Callegaro, G, Jeanblanc, M, Zargari, B:Carthaginian enlargement of filtrations. ESAIM. Probab. Stat. 17, 550-566 (2013)

[9]

Campi, L, Polbennikov, S, A, Sbuelz:Systematic equity-based credit risk:A CEV model with jump to default. J. Econ. Dynamics Control. 33(1), 93-101 (2009)

[10]

Carr, P, Linetsky, V:A jump to default extended CEV model:An application of Bessel processes. Finance Stochast. 10(3), 303-330 (2006)

[11]

Dellacherie, C, Meyer, P-A:Probabilités et potentiel, Chapitres I à IV. Hermann, Paris (1975)

[12]

Dellacherie, C, Meyer, P-A:Probabilités et potentiel. Chapitres V à VIII. Hermann, Paris (1980). Théorie des martingales

[13]

Duffie, D, Singleton, KJ:Credit risk:pricing, measurement and management. Princeton University Press, Princeton (2003). Princeton Series in Finance

[14]

Elliott, RJ, Jeanblanc, M, Yor, M:On models of default risk. Math. Finance. 10(2), 179-195 (2000)

[15]

Föllmer, H, Imkeller, P:Anticipation cancelled by a Girsanov transformation:a paradox on Wiener space. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques. 29(4), 569-586 (1993)

[16]

Grorud, A, Pontier, M:Insider trading in a continuous time market model. Int. J. Theor. Appl. Finance. 1(3), 331-347 (1998)

[17]

Guo, X, Jarrow, R, Zeng, Y:Modeling the recovery rate in a reduced form model. Math. Finance. 19(1), 73-97 (2009)

[18]

Jacod, J:Grossissement initial, hypothèse (H') et théorème de Girsanov. Grossissements de filtrations:exemples et applications, volume 1118 of Lecture Notes in Mathematics, pp. 15-35. Springer-Verlag, Berlin (1985)

[19]

Jacod, J, Shiryaev, A:Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], second edition. Springer-Verlag, Berlin (2003)

[20]

Jeanblanc, M, Mastrolia, T, Possamaï, D, Réveillac, A:Utility maximization with random horizon:a BSDE approach. Int. J. Theoret. Appl. Finance. 18(7), 1550045,43 (2015)

[21]

Jeulin:Semi-martingales et grossissement d'une filtration, volume 833 of Lecture Notes in Mathematics. Springer, Berlin (1980)

[22]

Jiao, Y, Kharroubi, I, Pham, H:Optimal investment under multiple defaults risk:a BSDE-decomposition approach. Ann. Appl. Probabil. 23(2), 455-491 (2013)

[23]

Kchia, Y, Larsson, M, Protter, P:Linking progressive and initial filtration expansions. Malliavin calculus and stochastic analysis, volume 34 of Springer Proc. Math. Stat, pp. 469-487. Springer, New York(2013)

[24]

Kchia, Y, Protter, P:On progressive filtration expansions with a process; applications to insider trading. Int. J. Theor. Appl. Finance. 18, 1550027,48 (2015)

[25]

Kharroubi, I, Lim, T, Ngoupeyou, A:Mean-variance hedging on uncertain time horizon in a market with a jump. Appl. Math. Optimization. 68(3), 413-444 (2013)

[26]

Lim, T, Quenez, M-C:Exponential utility maximization in an incomplete market with defaults. Electron. J. Probabil. 16(53), 1434-1464 (2011)

show all references

References:
[1]

Aksamit, A, Jeanblanc, M:Enlargement of Filtration with Finance in View. SpringerBriefs in Quantitative Finance. Springer, Cham (2017)

[2]

Amendinger, J:Martingale representation theorems for initially enlarged filtrations. Stoch. Process. Appl. 89, 101-116 (2000)

[3]

Amendinger, J, Becherer, D, Schweizer, M:A monetary value for initial information in portfolio optimization. Finance Stochast. 7(1), 29-46 (2003)

[4]

Amendinger, J, Imkeller, P, Schweizer, M:Additional logarithmic utility of an insider. Stoch. Process. Appl. 75, 263-286 (1998)

[5]

Bakshi, G, Madan, D, Zhang, F:Understanding the role of recovery in default risk models:Empirical comparisons and implied recovery rates (2006). preprint, University of Maryland

[6]

Bielecki, TR, Rutkowski, M:Credit risk:modelling, valuation and hedging. Springer-Verlag, Berlin (2002)

[7]

Blanchet-Scalliet, C, El Karoui, N, Jeanblanc, M, Martellini, L:Optimal investment decisions when timehorizon is uncertain. J. Math. Econ. 44(11), 1100-1113 (2008)

[8]

Callegaro, G, Jeanblanc, M, Zargari, B:Carthaginian enlargement of filtrations. ESAIM. Probab. Stat. 17, 550-566 (2013)

[9]

Campi, L, Polbennikov, S, A, Sbuelz:Systematic equity-based credit risk:A CEV model with jump to default. J. Econ. Dynamics Control. 33(1), 93-101 (2009)

[10]

Carr, P, Linetsky, V:A jump to default extended CEV model:An application of Bessel processes. Finance Stochast. 10(3), 303-330 (2006)

[11]

Dellacherie, C, Meyer, P-A:Probabilités et potentiel, Chapitres I à IV. Hermann, Paris (1975)

[12]

Dellacherie, C, Meyer, P-A:Probabilités et potentiel. Chapitres V à VIII. Hermann, Paris (1980). Théorie des martingales

[13]

Duffie, D, Singleton, KJ:Credit risk:pricing, measurement and management. Princeton University Press, Princeton (2003). Princeton Series in Finance

[14]

Elliott, RJ, Jeanblanc, M, Yor, M:On models of default risk. Math. Finance. 10(2), 179-195 (2000)

[15]

Föllmer, H, Imkeller, P:Anticipation cancelled by a Girsanov transformation:a paradox on Wiener space. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques. 29(4), 569-586 (1993)

[16]

Grorud, A, Pontier, M:Insider trading in a continuous time market model. Int. J. Theor. Appl. Finance. 1(3), 331-347 (1998)

[17]

Guo, X, Jarrow, R, Zeng, Y:Modeling the recovery rate in a reduced form model. Math. Finance. 19(1), 73-97 (2009)

[18]

Jacod, J:Grossissement initial, hypothèse (H') et théorème de Girsanov. Grossissements de filtrations:exemples et applications, volume 1118 of Lecture Notes in Mathematics, pp. 15-35. Springer-Verlag, Berlin (1985)

[19]

Jacod, J, Shiryaev, A:Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], second edition. Springer-Verlag, Berlin (2003)

[20]

Jeanblanc, M, Mastrolia, T, Possamaï, D, Réveillac, A:Utility maximization with random horizon:a BSDE approach. Int. J. Theoret. Appl. Finance. 18(7), 1550045,43 (2015)

[21]

Jeulin:Semi-martingales et grossissement d'une filtration, volume 833 of Lecture Notes in Mathematics. Springer, Berlin (1980)

[22]

Jiao, Y, Kharroubi, I, Pham, H:Optimal investment under multiple defaults risk:a BSDE-decomposition approach. Ann. Appl. Probabil. 23(2), 455-491 (2013)

[23]

Kchia, Y, Larsson, M, Protter, P:Linking progressive and initial filtration expansions. Malliavin calculus and stochastic analysis, volume 34 of Springer Proc. Math. Stat, pp. 469-487. Springer, New York(2013)

[24]

Kchia, Y, Protter, P:On progressive filtration expansions with a process; applications to insider trading. Int. J. Theor. Appl. Finance. 18, 1550027,48 (2015)

[25]

Kharroubi, I, Lim, T, Ngoupeyou, A:Mean-variance hedging on uncertain time horizon in a market with a jump. Appl. Math. Optimization. 68(3), 413-444 (2013)

[26]

Lim, T, Quenez, M-C:Exponential utility maximization in an incomplete market with defaults. Electron. J. Probabil. 16(53), 1434-1464 (2011)

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