The Annals of Probability

The Shannon information of filtrations and the additional logarithmic utility of insiders

Stefan Ankirchner, Steffen Dereich, and Peter Imkeller

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Abstract

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.

Article information

Source
Ann. Probab., Volume 34, Number 2 (2006), 743-778.

Dates
First available in Project Euclid: 9 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.aop/1147179988

Digital Object Identifier
doi:10.1214/009117905000000648

Mathematical Reviews number (MathSciNet)
MR2223957

Zentralblatt MATH identifier
1098.60065

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

Keywords
Enlargement of filtration logarithmic utility utility maximization heterogeneous information insider model Shannon information information difference entropy differential entropy

Citation

Ankirchner, Stefan; Dereich, Steffen; Imkeller, Peter. The Shannon information of filtrations and the additional logarithmic utility of insiders. Ann. Probab. 34 (2006), no. 2, 743--778. doi:10.1214/009117905000000648. https://projecteuclid.org/euclid.aop/1147179988


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