This thesis is divided into two chapters: the first one deals with some optimal control problems in one dimension and the second one with these problems in two or more dimensions. Note that, in this thesis, the time variable is not taken into account. In Chapter 1, at first we compute the dynamic programming equation for the minimal expected value F of the cost function considered. Next, we apply Whittle's theorem if the condition between the noise v and the functions b and q associated with the control variable is satisfied. Otherwise, we proceed differently. Indeed, if we make a change of variable, we obtain a Riccati equation for G= F', but without initial conditions. In some cases, from the symmetry of the infinitesimal parameters and of the function q, we can deduce the point x' where G(x')=0. If this is not possible, we limit ourselves to good approximations. The same approach is still possible if we are in specific situations, for example, when we have only one barrier. In Chapter 2, we discuss problems in dimension two or more. Since the condition in Whittle's theorem is difficult to satisfy in this case, we try to generalize the results obtained in Chapter 1. We then use in some examples the method of similarity solutions, which enables us to transform the problem into a one-dimensional one. Next, we propose a new resolution method. This method linearises the dynamic programming equation, which is a non-linear partial differential equation. Finally, we must find initial conditions for the new function, and also verify that the n expressions for F are equivalent.

Let p be a polynomial in the variable z. There exist several inequalities between the coefficents of p and its maximum modulus. In this work, we shall mainly study known proofs of the Visser inquality together with some extensions. We shall finally apply the inequality of Visser to obtain extensions of the Chebyshev inequality.