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...

The effectiveness of utility-maximization techniques for portfolio management relies on our ability to estimate correctly the parameters of the dynamics of the underlying financial assets. In the setting of complete or incomplete financial markets, we investigate whether small perturbations of the market coefficient processes lead to small changes in the agent's optimal behavior, as derived from the solution of the related utility-maximization problems. Specifically, we identify the topologies on the parameter process space and the solution space under which utility-maximization is a continuous operation, and we provide a counterexample showing that our results are best possible, in a certain sense. A novel result about the structure of the solution of the utility-maximization problem, where prices are modeled by continuous semimartingales, is established as an offshoot of the proof of our central theorem. Published by Elsevier B.V. MSC: 91B16; 91B28

This paper studies the problem of maximizing the expected utility of terminal wealth for a financial agent with an unbounded random endowment, and with a utility function which supports both positive and negative wealth. We prove the existence of an optimal trading strategy within a class of permissible strategies -those strategies whose wealth process is a supermartingale under all pricing measures with finite relative entropy. We give necessary and sufficient conditions for the absence of utility-based arbitrage, and for the existence of a solution to the primal problem.We consider two utility-based methods which can be used to price contingent claims. Firstly we investigate marginal utility-based price processes (MUBPP's). We show that such processes can be characterized as local martingales under the normalized optimal dual measure for the utility maximizing investor. Finally, we present some new results on utility indifference prices, including continuity properties and volume asymptotics for the case of a general utility function, unbounded endowment and unbounded contingent claims.

Abstract. We revisit the optimal investment and consumption model of Davis and Norman (1990) and Shreve and Soner (1994), following a shadow-price approach similar to that of Kallsen and Muhle-Karbe (2010). Making use of the completeness of the model without transaction costs, we reformulate and reduce the Hamilton-Jacobi-Bellman equation for this singular stochastic control problem to a non-standard free-boundary problem for a first-order ODE with an integral constraint. Having shown that the free boundary problem has a smooth solution, we use it to construct the solution of the original optimal investment/consumption problem in a self-contained manner and without any recourse to the dynamic programming principle. Furthermore, we provide an explicit characterization of model parameters for which the value function is finite.

We consider the problem of maximizing expected utility from consumption in a constrained incomplete semimartingale market with a random endowment process, and establish a general existence and uniqueness result using techniques from convex duality. The notion of "asymptotic elasticity" of Kramkov and Schachermayer is extended to the time-dependent case. By imposing no smoothness requirements on the utility function in the temporal argument, we can treat both pure consumption and combined consumptionterminal wealth problems in a common framework. To make the duality approach possible, we provide a detailed characterization of the enlarged dual domain which is reminiscent of the enlargement of L 1 to its topological bidual (L ∞ ) * , a space of finitely additive measures. As an application, we treat a constrained Itô process market model, as well as a "totally incomplete" model.