A stochastic variational approach to study economic equilibrium problems under uncertainty

On the study of multistage stochastic vector quasi-variational problems

“…To study multicriteria multistage problems under uncertainty and partial information, we preliminarily introduce a suitable multistage-functional framework in which we operate; for further details, the interested reader can refer to [22,27].…”

“…In a stochastic framework, in [32], the authors proposed a Sample Average Approximation method for solving a class of stochastic vector variational inequalities. Moreover, for the scalar case, the well-known Progressive Hedging algorithm is used to study multistage stochastic variational inequalities; see, e.g., [15,22,28,29]. In particular, it works on the extensive formulation of the multistage stochastic variational inequality by solving in parallel a problem for each possible scenario.…”

“…Indeed, to capture the dynamics that are essential to stochastic decision processes in response to an increasing level of information, Rockafellar and Wets [27] introduced a multistage stochastic variational problem. This formulation provides innovative and flexible tools to study real-life problems complicated by time, uncertainty, risk and allows us to capture the role of information in the recursive decision processes; see, e.g., [15,20,22,29]. The key concept of this new formulation turns out to be the nonanticipativity: some constraints have to be included in the formulation of the problem to take into account the partial information structure progressively revealed.…”

On a Class of Multistage Stochastic Hierarchical Problems

“…To this goal, we firstly introduce the multistage framework and the functional space in which we operate; for further details, the interested reader can refer to [17,18] and the references therein.…”

“…Remark 1. Condition (i) means that at stage t = 0 no uncertainty is resolved; condition (ii) means that information is progressively revealed, that is, one has only partial information; (iii) indicates that all information is revealed at stage T. In order to link the multistage stochastic and the information partition, we can use a particular oriented graph, called event-tree; see, e.g., [18]. In this way, for each pair (ω, t) identified in P corresponds a node ξ j t t and at each node ξ j t t of the oriented graph, we associate the elementary event F j t t , that is, F…”