×
Ankirchner, Stefan; Dereich, Steffen; Imkeller, Peter

The Shannon information of filtrations and the additional logarithmic utility of insiders. (English) Zbl 1098.60065

Ann. Probab. 34, No. 2, 743-778 (2006).
Summary: The background for the general mathematical link between utility and information theory investigated in this paper is a simple financial market model with two kinds of small traders: less informed traders and insiders, whose extra information is represented by an enlargement of the other agents’ filtration. The expected logarithmic utility increment, that is, the difference of the insider’s and the less informed trader’s expected logarithmic utility is described in terms of the information drift, that is, the drift one has to eliminate in order to perceive the price dynamics as a martingale from the insider’s perspective. On the one hand, we describe the information drift in a very general setting by natural quantities expressing the probabilistic better informed view of the world. This, on the other hand, allows us to identify the additional utility by entropy related quantities known from information theory. In particular, in a complete market in which the insider has some fixed additional information during the entire trading interval, its utility increment can be represented by the Shannon information of his extra knowledge. For general markets, and in some particular examples, we provide estimates of maximal utility by information inequalities.

Cited in 33 Documents

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
94A17 Measures of information, entropy
91B16 Utility theory
60G44 Martingales with continuous parameter

Keywords:

enlargement of filtration; utility maximization; heterogeneous information; insider model; information difference; entropy; differential entropy
PDF BibTeX XML Cite
\textit{S. Ankirchner} et al., Ann. Probab. 34, No. 2, 743--778 (2006; Zbl 1098.60065)
Full Text: DOI arXiv

References:

[1] Amendinger, J., Becherer, D. and Schweizer, M. (2003). A monetary value for initial information in portfolio optimization. Finance Stoch. 7 29–46. · Zbl 1035.60069
[2] Ankirchner, S. and Imkeller, P. (2005). Finite utility on financial markets with asymmetric information and structure properties of the price dynamics. Ann. Inst. H. Poincaré Probab. Statist. 41 479–503. · Zbl 1115.91024
[3] Amendinger, J., Imkeller, P. and Schweizer, M. (1998). Additional logarithmic utility of an insider. Stochastic Process. Appl. 75 263–286. · Zbl 0934.91020
[4] Ankirchner, S. (2005). Information and semimartingales. Ph.D. thesis, Humboldt Univ., Berlin. · Zbl 1189.60003
[5] Baudoin, F. (2001). Conditioning of Brownian functionals and applications to the modelling of anticipations on a financial market. Ph.D. thesis, Univ. Pierre et Marie Curie.
[6] Biagini, F. and Oksendal, B. (2005). A general stochastic calculus approach to insider trading. Appl. Math. Optim. 52 167–181. · Zbl 1093.60044
[7] Campi, L. (2003). Some results on quadratic hedging with insider trading. Stochastics 77 327–348. · Zbl 1103.91032
[8] Corcuera, J., Imkeller, P., Kohatsu-Higa, A. and Nualart, D. (2003). Additional utility of insiders with imperfect dynamical information. · Zbl 1064.60087
[9] Csiszar, I. (1975). I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3 146–158. · Zbl 0318.60013
[10] Duffie, D. and Huang, C. (1986). Multiperiod security markets with differential information: Martingales and resolution times. J. Math. Econom. 15 283–303. · Zbl 0608.90006
[11] Dellacherie, C. and Meyer, P.-A. (1978). Probabilities and Potential . North-Holland, Amsterdam. · Zbl 0494.60001
[12] Delbaen, F. and Schachermayer, W. (1995). The existence of absolutely continuous local martingale measures. Ann. Appl. Probab. 5 926–945. · Zbl 0847.90013
[13] Elstrodt, J. (1996). Maß- und Integrationstheorie . ( Measure and Integration Theory ). Springer, Berlin. · Zbl 0861.28001
[14] Grorud, A. and Pontier, M. (1998). Insider trading in a continuous time market model. Internat. J. Theoret. Appl. Finance 1 331–347. · Zbl 0909.90023
[15] Gasbarra, D. and Valkeila, E. (2003). Initial enlargement: A Bayesian approach. · Zbl 1063.62030
[16] Ihara, S. (1993). Information Theory for Continuous Systems . World Scientific, Singapore. · Zbl 0798.94001
[17] Imkeller, P. (1996). Enlargement of the Wiener filtration by an absolutely continuous random variable via Malliavin’s calculus. Probab. Theory Related Fields 106 105–135. · Zbl 0855.60045
[18] Imkeller, P. (2002). Random times at which insiders can have free lunches. Stochastics Stochastics Rep. 74 465–487. · Zbl 1041.91034
[19] Imkeller, P. (2001). Malliavin’s calculus in insider models: Additional utility and free lunches. Math. Finance 13 153–169. · Zbl 1071.91017
[20] Imkeller, P., Pontier, M. and Weisz, F. (2001). Free lunch and arbitrage possibilities in a financial market model with an insider. Stochastic Process. Appl. 92 103–130. · Zbl 1047.60041
[21] Jeulin, Th. and Yor, M., eds. (1985). Grossissements de Filtrations : Exemples et Applications . Springer, Berlin. · Zbl 0547.00034
[22] Parthasarathy, K. R. (1977). Introduction to Probability and Measure. MacMillan Co. of India Ltd. Delhi. · Zbl 0395.28001
[23] Pikovsky, I. and Karatzas, I. (1996). Anticipative portfolio optimization. Adv. in Appl. Probab. 28 1095–1122. JSTOR: · Zbl 0867.90013
[24] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Springer, Berlin. · Zbl 0917.60006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.