We consider the problem of optimal investment with intermediate consumption in a general semimartingale model of an incomplete market, with preferences being represented by a utility stochastic field. We show that the key conclusions of the utility maximization theory hold under the assumptions of no unbounded profit with bounded risk (NUPBR) and of the finiteness of both primal and dual value functions.2010 Mathematics Subject Classification. 91G10, 93E20. JEL Classification: C60, G1. Key words and phrases. Utility maximization, arbitrage of the first kind, local martingale deflator, duality theory, semimartingale, incomplete market.

In this paper we estimate the expected tracking error of a fixed gain stochastic approximation scheme. The underlying process is not assumed Markovian, a mixing condition is required instead. Furthermore, the updating function may be discontinuous in the parameter.MSC 2010 subject classification: Primary: 62L20; secondary: 93E15, 93E35 with some fixed measurable function g : X − → m . Clearly, X is a (strongly) stationary m -valued process, see Lemma 10.1 of [24].Remark 3.1. We remark that, in the present setting, the CLC property holds if, for all θ 1 , θ 2 ∈ D,due to the fact that the law of (X k+1 , ǫ k , ǫ k−1 , . . .) is the same as that of (X 1 , ǫ 0 , ǫ −1 , . . .), for all k ∈ .Define G(θ ) := EH(θ , X 0 ). Note that, by stationarity of X , G(θ ) = EH(θ , X t ) for all t ∈ . We need some stability hypotheses formulated in terms of an ordinary differential equation related to G. Assumption 3.2. On D, the function G is twice continuously differentiable and bounded, together with its first and second derivatives.

We consider a continuous-time market with proportional transaction costs. Under appropriate assumptions we prove the existence of optimal strategies for investors who maximize their worst-case utility over a class of possible models. We consider utility functions defined either on the positive axis or on the whole real line.

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