Infinite-dimensional Lagrange problem and application to stochastic processes

We consider an infinite-dimensional analog of classical Lagrange problem. The main result of the work is the necessary conditions of the extremum. This result is new (the theory of surface integration in abstract locally convex space is the basic instrument of the investigation, therefore the indicated problem was not and just could not be investigated before). All our considerations and formulas are dimensional-invariant, and in conformity to finite-dimensional case they turn into classical ones.