Our objective in this work is to give a presentation of some basic results of stochastic control. It is thus a text intended to advanced students and researchers willing to learn the theory. Stochastic control covers a broad area of disciplines and problems. is also a field in full development, and some important aspects remain to be cleaned up. That is why in presenting stochastic control a choice is necessary. We have emphasized this choice in the title. of partial differential equations, the semi-group theory, variational and quasi variational inequalities play a very important role in solving problems of stochastic control. We have tried to use them as much as possible, since they bring tools and results which are very important, specially for computational purposes, and which cannot be obtained in an other way, namely regularity results, and weak solution concepts.

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