A PDE Approach to the Problems of Optimality of Expectations

Let (X, Z) be a bivariate random vector. A predictor of X based on Z is just Borel function g(Z). The problem "least squares prediction" given the observation to find global minimum point functional E[(X − g(Z))2] with respect all variables g(Z), where g function. It well known that solution this conditional expectation E(X|Z). We also know that, if for nonnegative smooth F: R×R → R, arg ming(Z)E[F(X, g(Z))] = E[X|Z], and Z, then F(x, y) Bregmann loss interest, fixed ϕ F (x, y), satisfying, ϕ(E[X|Z]), Z. In more general setting, stronger satisfying miny∈RE[F y)] ϕ(E[X]), ∀X. study develop partial differential equation (PDE) approach these problems.