Stochastic optimization problems arise in decision-making problems under uncertainty, and find numerous and various applications in economics and finance. On the other hand, problems in finance led recently to new developments in the theory of stochastic control. This volume intends to provide a systematic treatment of stochastic optimization problems applied to finance, by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastic differential equations, and martingale duality methods. The theory is discussed with the recent developments in this field, with complete and detailed proofs, and is illustrated by concrete examples occurring in finance: portfolio allocation, option hedging, real options, optimal investment, etc This volume is directed towards graduate students and researchers in mathematical finance. It will also appeal to applied mathematicians interested in financial applications and practitioners wishing to know more about the use of stochastic optimization methods in finance.