  In this paper, we present two methods which generalize static team theory to
dynamic team theory, in the context of continuous-time stochastic nonlinear
differential decentralized decision systems, with relaxed strategies, which are
measurable to different noisy information structures. For both methods we apply
Girsanov's measure transformation to obtain an equivalent dynamic team problem
under a reference probability measure, so that the observations and information
structures available for decisions, are not affected by any of the team
decisions. The first method is based on function space integration with respect
to products of Wiener measures, and generalizes Witsenhausen's [1] definition
of equivalence between discrete-time static and dynamic team problems. The
second method is based on stochastic Pontryagin's maximum principle. The team
optimality conditions are given by a "Hamiltonian System" consisting of forward
and backward stochastic differential equations, and a conditional variational
Hamiltonian with respect to the information structure of each team member,
expressed under the initial and a reference probability space via Girsanov's
measure transformation. Under global convexity conditions, we show that that
PbP optimality implies team optimality. In addition, we also show existence of
team and PbP optimal relaxed decentralized strategies (conditional
distributions), in the weak$^*$ sense, without imposing convexity on the action
spaces of the team members. Moreover, using the embedding of regular strategies
into relaxed strategies, we also obtain team and PbP optimality conditions for
regular team strategies, which are measurable functions of decentralized
information structures, and we use the Krein-Millman theorem to show
realizability of relaxed strategies by regular strategies.
