Contribution to the mathematical study of arbitrage and market imperfections.

Summary This paper consists of five parts and is particularly interested in the characterization of the no-arbitrage hypothesis in imperfect markets. This hypothesis imposes that one cannot make money out of nothing. The first part introduces the problem that concerns us and positions it in relation to the literature. In the second part, we establish a non-arbitrage theorem in a general framework for finite and discrete models. By applying it to different cases of imperfect markets, we recover the results of the literature and new properties on taxes. To do so, we use an adapted version of the Farkas lemma, results on optimal stopping, and properties of martingales. The next two parts are devoted to the study of a model such that each investment flow can be initiated at any date and in any state of the world under the same (stationary) conditions. The associated model is then necessarily infinite horizon. In the third part, we place ourselves in a deterministic framework. Thanks to the use of radon measurements, we study both the case of discrete and continuous dates. We demonstrate a non-arbitrage result. To do so, we use the properties of radon measurements. Then, in order to be able to 'separate' (c. F. Bourbaki e. V. T) the positive orthant and the set of payments, we show that the latter is locally compact as a compact sole cone. We also use properties of the Laplace transform. In the fourth part, we prove a non-arbitrage result in a stochastic but discrete setting, thanks to our farkas lemma and the brouwer theorem. Then, we apply it to the case of a financial market with transaction costs, which turns out to be stationary. We then obtain stronger results than those already existing in the literature. In the fifth and last part, we give a characterization of the absence of arbitrage when trading strategies are constrained to belong to a convex set. To this end. We show a closure result and then use a separation theorem from Yan (1980). Finally, we give a dual representation of the over-replication cost for a contingent asset.