Particle methods in stochastic optimal control

When dealing with numerical solution for stochastic optimal control problems, stochastic dynamic programming appears to be the natural framework. Due to the curse of dimensionality the stochastic optimization techniques usually focus on scenarios trees. But this latter method faces an important difficulty: at the first time stages (close to the tree root) we have few Monte-Carlo particles and a small variance of the feedback estimator; on the contrary, at the final time stages (near the tree leaves), we have a large number of Monte-Carlo particles with a huge variance. To tackle these difficulties we present some variational approach for numerical solution of stochastic optimal control problems. We consider two different interpretations of the control problem, an algebraic and a functional one from which we derive optimality conditions. An adaptative mesh discretization method will be used to achieve tractable solution algorithm. An application to a hydro-electric dam production management problem will be presented. (author)