A dual characterization of self-generation and exponential forward performances

arXiv:0809.0739v4 [q-fin.CP] 10 Dec 2009 The Annals of Applied Probability 2009, Vol. 19, No. 6, 2176–2210 DOI: 10.1214/09-AAP607 c ⃝Institute of Mathematical Statistics, 2009 A DUAL CHARACTERIZATION OF SELF-GENERATION AND EXPONENTIAL FORWARD PERFORMANCES By Gordan ˇZitkovi´c1 University of Texas at Austin We propose a mathematical framework for the study of a fam- ily of random fields—called forward performances—which arise as numerical representation of certain rational preference relations in mathematical finance. Their spatial structure corresponds to that of utility functions, while the temporal one reflects a Nisio-type semi- group property, referred to as self-generation. In the setting of semi- martingale financial markets, we provide a dual formulation of self- generation in addition to the original one, and show equivalence be- tween the two, thus giving a dual characterization of forward perfor- mances. Then we focus on random fields with an exponential struc- ture and provide necessary and sufficient conditions for self-generation in that case. Finally, we illustrate our methods in financial markets driven by Itˆo-processes, where we obtain an explicit parametrization of all exponential forward performances. 1. Introduction. The present paper aims to contribute to the fruitful and successful literature on utility maximization and optimal investment in stochastic financial markets. Born in the seminal work of Merton [24, 25], the theory has been further developed by Pliska [33], Cox and Huang [6], Karatzas et al. [19], He and Pearson [16], Kramkov and Schachermayer [22], Cvitani´c, Schachermayer and Wang [7], Karatzas and ˇZitkovi´c [21] and many others. In the setting similar to the one employed in here—namely, incomplete semimartingale markets with utility functions defined on the whole real line—the pertinent contributions include those of Frittelli [14], Bellini and Frittelli [4], Schachermayer [36], Owen and ˇZitkovi´c [32] and others. Received November 2008; revised March 2009. 1Supported in part by the NSF under award number DMS-07-06947. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect those of the National Science Foundation. AMS 2000 subject classifications. Primary 91B16; secondary 91B28. Key words and phrases. Exponential utility, forward performances, incomplete mar- kets, utility maximization, convex duality, random fields, mathematical finance. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2009, Vol. 19, No. 6, 2176–2210. This reprint differs from the original in pagination and typographic detail. 1 2 G. ˇZITKOVI´C The notion of forward performance or forward utility has appeared in the literature recently, and in various forms, in the work of Choulli, Henderson, Hobson, Li, Musiela, Stricker and Zariphopoulou (see [5, 17, 26, 27, 28, 29, 30]). It refers to a family of interrelated state-dependent utility functions parametrized by the positive time axis [0,∞). The glue holding these utility functions together is the following economic principle of consistency: a ra- tional economic agent should be indifferent between two random pay-offs as long as one can be produced from the other using a costless dynamic trading strategy in a financial market. We lay no claim to any originality in its for- mulation. In fact, it has existed in various forms in the financial literature for a long time. Recently, it has been used in the context of risk measures and their generalizations (see [13] and [15], among many other instances). An axiomatic treatment of a class of forward performances by Zariphopoulou and ˇZitkovi´c in [38] is based on an implenetation of this idea in the context of the risk-measure theory, but without a fixed finite investment horizon. The main goal of the present manuscript is to establish a solid mathemat- ical footing for the notion of forward performances, provide a dual charac- terization and illustrate the obtained results. Mathematically, the economic consistency criterion described above translates into a Nisio-type semigroup property which we call self-generation. The obstacles in the analysis, con- struction and characterization of self-generating random fields come from several directions. First, the level of generality needed for financial applica- tions usually surpasses that of a finite-state-variable (i.e., finite-dimensional Markov setting) and deals with random fields of utilities whose depen- dence structure is quite general. Therefore, the classical PDE-based control- theoretic tools no longer apply. Second, the market models we consider are typically incomplete, as the complete case degenerates in a certain sense, and lacks interesting mathematical or economic content. Incompleteness or, in analytic language, lack of strict ellipticity renders the analysis much more delicate; in particular, a

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