Robust utility functionals arise as numerical representations of investor preferences, when the investor is uncertain about the underlying probabilistic model and averse against both risk and model uncertainty. In this paper, we study the duality theory for the problem of maximizing the robust utility of the terminal wealth in a general incomplete market model. We also allow for very general sets of prior models. In particular, we do not assume that all prior models are equivalent to each other, which allows us to handle many economically meaningful robust utility functionals such as those defined by AVaR λ , concave distortions, or convex capacities. We also show that dropping the equivalence of prior models may lead to new effects such as the existence of arbitrage strategies under the least favorable model.

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We consider a class of Volterra linear transforms of Brownian motion associated to a sequence of Müntz Gaussian spaces and determine explicitly their kernels; the kernels take a simple form when expressed in terms of Müntz-Legendre polynomials. These are new explicit examples of progressive Gaussian enlargement of a Brownian filtration. We give a necessary and sufficient condition for the existence of kernels of infinite order associated to an infinite dimensional Müntz Gaussian space; we also examine when the transformed Brownian motion remains a semimartingale in the filtration of the original process. This completes some partial answers obtained in ([17], [21], [22]) to the aforementioned problems in the infinite dimensional case.

A class of Volterra transforms, preserving the Wiener measure, with kernels of Goursat type is considered. Such kernels satisfy a self-reproduction property. We provide some results on the inverses of the associated Gramian matrices which lead to a new self-reproduction property. A connection to the classical reproduction property is given. Results are then applied to the study of a class of singular linear stochastic differential equations together with the corresponding decompositions of filtrations. The studied equations are viewed as non-canonical decompositions of some generalized bridges.

Brownian motions defined as linear transformations of two independent Brownian motions are studied, together with certain orthogonal decompositions of Brownian filtrations.

This paper studies the role of production mode in determining the effects of an increase in uncertainty on the choice of investment outlay. In a continuous-time model of optimal capital investment with innovative "R&D" under demand uncertainty, we show that investments in both capital and innovative research decrease with an increase in uncertainty, and that such investments rise with the level of primary demand. Our result sheds light on the mode of production as a source of the negative investment-uncertainty relationship. Copyright (c) 2009 The Authors. Journal compilation (c) 2009 Economic Society of South Africa.

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