A dual characterization of self-generation and exponential forward performances

Abstract

We propose a mathematical framework for the study of a family 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 semigroup property, referred to as self-generation. In the setting of semimartingale financial markets, we provide a dual formulation of selfgeneration in addition to the original one, and show equivalence between the two, thus giving a dual characterization of forward performances. Then we focus on random fields with an exponential structure and provide necessary and sufficient conditions for self-generation in that case. Finally, we illustrate our methods in financial markets driven by Itô-processes, where we obtain an explicit parametrization of all exponential forward performances. . This reprint differs from the original in pagination and typographic detail. 1 2 G.ŽITKOVIĆThe 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 rational 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 formulation. 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 andŽitković 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 mathematical footing for the notion of forward performances, provide a dual characterization 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, construction and characterization of self-generating random fields come from several directions. First, the level of generality needed for financial applications usually surpasses that of a finite-state-variable (i.e., finite-dimensional Markov setting) and deals with random fields of utilities whose dependence structure is quite general. Therefore, the classical PDE-based controltheoretic tools no longer apply. Second, the market models we consider are typically incomplete, as the complete case degenerates in a certain sense, and lacks interes...

Portfolio Choice under Space-Time Monotone Performance Criteria

“…We do this because different cases for the domain and range of the functions u and h require appropriately modified and computationally tedious arguments which, if presented at this point, would obscure the clarity of the presentation. It is shown in section 4 (see Propositions 10,14,15,and 19) that there exists a one-to-one correspondence (modulo normalization constants) between increasing and strictly concave solutions to (14) with strictly increasing solutions to (19). It is also shown that the latter can be represented in terms of the bilateral Laplace transform of a positive finite Borel measure, denoted throughout by ν.…”

“…2 From a different perspective, one could seek an axiomatic construction of a forward performance process. Results in this direction, as well as on the dual formulation of the problem, can be found in [19] for the exponential case (see, also, [1] for a constrained case). We refer the reader to [10] for further discussion on the forward performance and its similarities to and differences from the classical value function.…”

“…This measure then emerges as the defining element in the entire analysis of the problem at hand. Its presence originates from the classical results of Widder (see Chapter XIV in [17]) on the construction of positive solutions to the heat equation (19). In the investment model we consider, the solution h of (19) is used to construct the optimal wealth which, however, can take arbitrary values.…”

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“…The model therein is incomplete binomial and the initial data is taken to be exponential. The exponential case was subsequently and extensively analyzed in [71] and [95].…”

“…This class of forward performance processes is analyzed in detail in [71] (see, also, [95]). Case 2: Let = ; > 1: Then (38) yields h (x; t) = 1 e The optimal wealth and portfolio processes are given, respectively, by…”