The randomization method in stochastic optimal control

In this paper, we make a survey on the so called randomization method, a recent methodology to study stochastic optimization problems. It allows for the representation of the value function of an optimal control problem by a suitable backward stochastic differential equation (BSDE), by means of an auxiliary optimization problem having the same value as the starting one. This method works for a large class of control problems and provides a BSDE representation to many related PDEs of Hamilton-Jacobi-Bellman type, even in the fully nonlinear case. After a general informal introduction, we explain the method giving full details in a basic case. Then, we try to give a complete picture of the existing applications and we present some related open problems.