The Shannon information of filtrations and the additional logarithmic utility of insiders
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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.
THE SHANNON INFORMATION OF FILTRATIONS AND THE ADDITIONAL LOGARITHMIC UTILITY OF INSIDERS By Stefan Ankirchner, Steffen Dereich and Peter Imkeller Humboldt-Universit¨ at zu Berlin, Technische Universit¨ at Berlin and Humboldt-Universit¨ at zu Berlin
The Annals of Probability 2006, Vol. 34, No. 2, 743–778 DOI: 10.1214/009117905000000648 c Institute of Mathematical Statistics, 2006
arXiv:math/0503013v2 [math.PR] 25 May 2006
Received July 2004; revised February 2005. AMS 2000 subject classifications. Primary 60H30, 94A17; secondary 91B16, 60G44. Key words and phrases. Enlargement of filtration, logarithmic utility, utility maximization, heterogeneous information, insider model, Shannon information, information difference, entropy, differential entropy.
2
S. ANKIRCHNER, S. DEREICH AND P. IMKELLER
make their portfolio decisions: the less informed agent, at time t, just has the σ -field Ft , corresponding to the natural evolution of the market up to this time, at his disposal for deciding about future investments, while the insider is able to make better decisions, taking his knowledge from a bigger σ -field Gt ⊃ Ft . We give a short selection from among the many papers dealing with this model, just indicating the most important mathematical techniques used for its investigation. Methods are focused on martingale and stochastic control theory, and techniques of enlargement of filtrations (see [21]), starting with the conceptual paper by Duffie and Huang [10], mostly in the initial enlargement setting, that is, the insider gets some fixed extra information at the beginning of the trading interval. The model is successively studied on stochastic bases with increasing complexity: that is, Karatzas and Pikovsky [23] on Wiener space, Grorud and Pontier [14] allow Poissonian noise and Biagini and Oksendal [6] employ anticipative calculus techniques. In the same setting, Amendinger, Becherer and Schweizer [1] calculate the value of insider information from the perspective of specific utilities. Baudoin [5] introduces the concept of weak additional information consisting in the knowledge of the law of some random element. Campi [7] considers hedging techniques for insiders in the incomplete market setting. It is clear that the expected utility the insider is able to gain from final wealth in this simple model will be bigger than the uninformed traders’ utility, for every utility function. And, in fact, many of the quoted papers deal with the calculation of a better informed agent’s additional utility. In [3], in the setting of initial enlargements and logarithmic utility, a crucial and natural link between the additional expected logarithmic utility and information theoretic concepts was made. The insider’s logarithmic utility advantage is identified with the Shannon entropy of the additional information. In the same setting, Gasbarra and Valkeila [15] extended this link by interpreting the logarithmic utility increment by the Kullback–Leibler information of the insider’s additional knowledge from the perspective of Bayesian modeling. In the environment of this utility-information paradigm, the papers [2, 8, 17, 18, 19, 20] describe additional utility and treat arbitrage questions and their interpretation in information theoretic terms in increasingly complex models of the same base structure, including some simple examples of progressive enlargements. It is clear that utility concepts different from the logarithmic one correspond on the information theoretic side to the generalized entropy concepts of f -divergences. In this paper we shall continue the inຫໍສະໝຸດ Baiduestigation of mathematical questions related to the link between utility and information theory in the most general setting of enlargements of filtrations: besides assuming eventually that the base space be standard, to ensure the existence of regular conditional probabilities, we shall let the filtration of the better informed agent just contain the one of the natural evolution of knowledge. To concentrate
0. Introduction. A simple mathematical model of two agents on a financial market taking their portfolio decisions on the basis of different information horizons has attracted much attention in recent years. Both agents are small, and unable to influence the price dynamics of the risky assets constituting the market. One agent just acts on the basis of the evolution of the market, the other one, the insider, possesses some additional knowledge at every instant of the continuous trading interval. This basic fact is modeled by associating two different filtrations with each agent, from which they
This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2006, Vol. 34, No. 2, 743–778. This reprint differs from the original in pagination and typographic detail. 1
THE SHANNON INFORMATION OF FILTRATIONS AND THE ADDITIONAL LOGARITHMIC UTILITY OF INSIDERS By Stefan Ankirchner, Steffen Dereich and Peter Imkeller Humboldt-Universit¨ at zu Berlin, Technische Universit¨ at Berlin and Humboldt-Universit¨ at zu Berlin
The Annals of Probability 2006, Vol. 34, No. 2, 743–778 DOI: 10.1214/009117905000000648 c Institute of Mathematical Statistics, 2006
arXiv:math/0503013v2 [math.PR] 25 May 2006
Received July 2004; revised February 2005. AMS 2000 subject classifications. Primary 60H30, 94A17; secondary 91B16, 60G44. Key words and phrases. Enlargement of filtration, logarithmic utility, utility maximization, heterogeneous information, insider model, Shannon information, information difference, entropy, differential entropy.
2
S. ANKIRCHNER, S. DEREICH AND P. IMKELLER
make their portfolio decisions: the less informed agent, at time t, just has the σ -field Ft , corresponding to the natural evolution of the market up to this time, at his disposal for deciding about future investments, while the insider is able to make better decisions, taking his knowledge from a bigger σ -field Gt ⊃ Ft . We give a short selection from among the many papers dealing with this model, just indicating the most important mathematical techniques used for its investigation. Methods are focused on martingale and stochastic control theory, and techniques of enlargement of filtrations (see [21]), starting with the conceptual paper by Duffie and Huang [10], mostly in the initial enlargement setting, that is, the insider gets some fixed extra information at the beginning of the trading interval. The model is successively studied on stochastic bases with increasing complexity: that is, Karatzas and Pikovsky [23] on Wiener space, Grorud and Pontier [14] allow Poissonian noise and Biagini and Oksendal [6] employ anticipative calculus techniques. In the same setting, Amendinger, Becherer and Schweizer [1] calculate the value of insider information from the perspective of specific utilities. Baudoin [5] introduces the concept of weak additional information consisting in the knowledge of the law of some random element. Campi [7] considers hedging techniques for insiders in the incomplete market setting. It is clear that the expected utility the insider is able to gain from final wealth in this simple model will be bigger than the uninformed traders’ utility, for every utility function. And, in fact, many of the quoted papers deal with the calculation of a better informed agent’s additional utility. In [3], in the setting of initial enlargements and logarithmic utility, a crucial and natural link between the additional expected logarithmic utility and information theoretic concepts was made. The insider’s logarithmic utility advantage is identified with the Shannon entropy of the additional information. In the same setting, Gasbarra and Valkeila [15] extended this link by interpreting the logarithmic utility increment by the Kullback–Leibler information of the insider’s additional knowledge from the perspective of Bayesian modeling. In the environment of this utility-information paradigm, the papers [2, 8, 17, 18, 19, 20] describe additional utility and treat arbitrage questions and their interpretation in information theoretic terms in increasingly complex models of the same base structure, including some simple examples of progressive enlargements. It is clear that utility concepts different from the logarithmic one correspond on the information theoretic side to the generalized entropy concepts of f -divergences. In this paper we shall continue the inຫໍສະໝຸດ Baiduestigation of mathematical questions related to the link between utility and information theory in the most general setting of enlargements of filtrations: besides assuming eventually that the base space be standard, to ensure the existence of regular conditional probabilities, we shall let the filtration of the better informed agent just contain the one of the natural evolution of knowledge. To concentrate
0. Introduction. A simple mathematical model of two agents on a financial market taking their portfolio decisions on the basis of different information horizons has attracted much attention in recent years. Both agents are small, and unable to influence the price dynamics of the risky assets constituting the market. One agent just acts on the basis of the evolution of the market, the other one, the insider, possesses some additional knowledge at every instant of the continuous trading interval. This basic fact is modeled by associating two different filtrations with each agent, from which they
This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2006, Vol. 34, No. 2, 743–778. This reprint differs from the original in pagination and typographic detail. 1