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

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 decision system 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. It generalizes Witsenhausen’s [1] definition of equivalence between discrete-time static and dynamic team problems, and relates Girsanov’s theorem to the so-called “Common Denominator Condition and Change of Variables”. 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 conditional variational Hamiltonians with respect to the information structure of each team member. Under global convexity conditions, we show that PbP optimality implies team optimality. We also obtain team and PbP optimality conditions for regular team strategies, which are measurable functions of decentralized information structures. 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, and their realization by regular team strategies.