  Decentralized optimization of distributed stochastic differential systems has
been an active area of research for over half a century. Its formulation
utilizing static team and person-by-person optimality criteria is well
investigated. However, the results have not been generalized to nonlinear
distributed stochastic differential systems possibly due to technical
difficulties inherent with decentralized decision strategies.
  In this first part of the two-part paper, we derive team optimality and
person-by-person optimality conditions for distributed stochastic differential
systems with different information structures. The optimality conditions are
given in terms of a Hamiltonian system of equations described by a system of
coupled backward and forward stochastic differential equations and a
conditional Hamiltonian, under both regular and relaxed strategies. Our
methodology is based on the semi martingale representation theorem and
variational methods. Throughout the presentation we discuss similarities to
optimality conditions of centralized decision making.
