We consider the problem of utility maximization for small traders on incomplete financial markets. As opposed to most of the papers dealing with this subject, the investors' trading strategies we allow underly constraints described by closed, but not necessarily convex, sets. The final wealths obtained by trading under these constraints are identified as stochastic processes which usually are supermartingales, and even martingales for particular strategies. These strategies are seen to be optimal, and the corresponding value functions determined simply by the initial values of the supermartingales. We separately treat the cases of exponential, power and logarithmic utility.Introduction. In this paper we consider a small trader on an incomplete financial market who can trade in a finite time interval [0, T ] by investing in risky stocks and a riskless bond. He aims at maximizing the utility he draws from his final wealth measured by some utility function. The trading strategies he may choose to attain his wealth underly some restriction formalized by a constraint. For example, he may be forced not to have a negative number of shares or that his investment in risky stocks is not allowed to exceed a certain threshold. We will be interested not only in describing the trader's optimal utility, but also the strategies which he may follow to reach this goal. As opposed to most of the papers dealing so far with the maximization of expected utility under constraints, we essentially relax the hypotheses to be fulfilled by them. They are formulated as usual by the requirement that the strategies take their values in some set, which is supposed to simply

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International audienceIn this paper, we formulate a general time-inconsistent stochastic linear--quadratic (LQ) control problem. The time-inconsistency arises from the presence of a quadratic term of the expected state as well as a state-dependent term in the objective functional. We define an equilibrium, instead of optimal, solution within the class of open-loop controls, and derive a sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we find an explicit equilibrium control. As an application, we then consider a mean-variance portfolio selection model in a complete financial market where the risk-free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes. Applying the general sufficient condition, we obtain explicit equilibrium strategies when the risk premium is both deterministic and stochastic