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In this paper, we present two methods which generalize static team theory todynamic team theory, in the context of continuous-time stochastic nonlineardifferential decentralized decision systems, with relaxed strategies, which aremeasurable to different noisy information structures. For both methods we applyGirsanov's measure transformation to obtain an equivalent dynamic team problemunder a reference probability measure, so that the observations and informationstructures available for decisions, are not affected by any of the teamdecisions. The first method is based on function space integration with respectto products of Wiener measures, and generalizes Witsenhausen's [1] definitionof equivalence between discrete-time static and dynamic team problems. Thesecond method is based on stochastic Pontryagin's maximum principle. The teamoptimality conditions are given by a "Hamiltonian System" consisting of forwardand backward stochastic differential equations, and a conditional variationalHamiltonian with respect to the information structure of each team member,expressed under the initial and a reference probability space via Girsanov'smeasure transformation. Under global convexity conditions, we show that thatPbP optimality implies team optimality. In addition, we also show existence ofteam and PbP optimal relaxed decentralized strategies (conditionaldistributions), in the weak$^*$ sense, without imposing convexity on the actionspaces of the team members. Moreover, using the embedding of regular strategiesinto relaxed strategies, we also obtain team and PbP optimality conditions forregular team strategies, which are measurable functions of decentralizedinformation structures, and we use the Krein-Millman theorem to showrealizability of relaxed strategies by regular strategies.

Dynamic Team Theory of Stochastic Differential Decision Systems with  Decentralized Noisy Information Structures via Girsanov's Measure  Transformation