Lagrange lemma and the optimal control of diffusion. 1. Differentiable multipliers

The authors review and further develop the approach to optimal control problems based on the Lagrange lemma they previously introduced (1991). The focus is on linear Lagrange functionals. It is shown that it is possible to solve in an unified framework a large class of deterministic and stochastic control problems, as well as calculus of variations problems, by absolutely elementary mathematics. In particular, the multipliers are pathwise differentiable. The typical problem of dealing with a stochastic differential equation with a terminal condition is therefore completely avoided. The method is illustrated by solving various deterministic and stochastic, convex and nonconvex, smooth and nonsmooth problems