Rigorous mathematical finance relies strongly on two additional fields: optimal stopping and stochastic analysis. This book is the first one which presents not only main results in the mathematical finance but also these "related topics" with all proofs and in a self-contained form. The book treats both discrete and continuous time mathematical finance. Some topics, such as Israeli (game) contingent claims, and several proofs have not appeared before in a self-contained book form. The book contains exercises with solutions at the end of it and it can be used for a yearlong advanced graduate course for mathematical students.