Financial markets with infinite assets, quadratic hedging and insider trading.

Summary This thesis consists on several applications of stochastic calculus to mathematical finance. Its structure is as follows. In the first chapter, we study the relation between market completeness and extremality of equivalent martingale measures in the case of infinitely many assets. In the second one, we find equivalent conditions to the existence and uniqueness of an equivalent martingale measure under which the price process follows some given n-dimensional distributions with n fixed. In the third, we extend to a large financial market a characterization of the mean-variance optimal hedging strategy based on a technique combining change of numéraire and artificial extension. Finally, the fourth and last chapter deals with the hedging problem of a given contingent claim in a market with asymmetric information.