In this talk, I will discuss distributional robustness in a dynamic setting, where the deviation is measured by the causal transport distance between stochastic processes. Such a choice accounts for information evolution, making it hedge against a plausible family of data processes. I will present a recursive reformulation to evaluate the worst-case risk of any given random sequence and explore the intricacies of time consistency in dynamic risk measures. Furthermore, I will present dynamic programming reformulations for finding the optimal robust policy.

Markov decision process (MDP) is a decision making framework where a decision maker is interested in maximizing the expected discounted value of a stream of rewards received at future stages at various states which are visited according to a controlled Markov chain. Many algorithms including linear programming methods are available in the literature to compute an optimal policy when the rewards and transition probabilities are deterministic.
In many practical situations, it is often the case that only a partial information about the underlying distribution is available based on the historical data. In that case, a distributionally robust approach, is used to model the uncertainties, which assumes that the true distribution belongs to an uncertainty set based on its partially available information.
In this paper, we consider an MDP problem where the reward vector is known and the transition probability vector is a random vector which follows a discrete distribution whose information is not completely known. We formulate the MDP problem using distributionally robust chance-constrained optimization framework under various types of moments based uncertainty sets, and statistical-distance based uncertainty sets defined using $\phi$-divergence and Wasserstein distance metric. For each uncertainty set, we propose an equivalent mix-integer bilinear programming problem or a mix-integer semidefinite programming problem with bilinear constraints. As an application, we study a machine replacement problem and perform numerical experiments on randomly generated instances.

We discuss risk evaluation and risk-averse optimization of complex distributed systems with general risk functionals.
We postulate a set of axioms for the functionals evaluating the total risk of the system and show the dual representation for the systemic risk measures. Furthermore, we propose two new families of measures constructed by using either collections of linear scalarizations or non-linear risk aggregation. The new framework facilitates risk-averse sequential decision making by distributed methods. The proposed approach is compared theoretically and numerically to some of the systemic risk measurements in the existing literature.
We also formulated a multi-stage decision problem arising in communication networks when a team of robots explores an area and each robot reports relevant information. The goal is to determine a few reporting points so that the communication is conducted most efficiently while managing the risk of losing information. A distributed numerical method is developed for solving the problem. In this context, we compare the new risk measures to other methods of systemic risk evaluation. We show that the proposed framework is less conservative and results in a substantially better solution of the problem at hand as compared to a linear aggregation of the risk of individual agents as well as other methods.