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

Information uncertainty related to marked random times and optimal investment

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.
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) Google Scholar

[2]

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

[3]

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

[4]

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

[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 Google Scholar

[6]

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

[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) Google Scholar

[8]

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

[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) Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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) Google Scholar

[16]

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

[17]

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

[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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[21]

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

[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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[26]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[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 Google Scholar

[6]

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

[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) Google Scholar

[8]

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

[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) Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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) Google Scholar

[16]

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

[17]

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

[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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[21]

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

[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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[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) Google Scholar

[26]

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

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