arXiv subjects

Eduardo L. Pasiliao

Publications and source records attributed to Eduardo L. Pasiliao.

6 recordsLinked to original sources

arXiv2018

On the hierarchical risk-averse control problems for diffusion processes

In this paper, we consider a risk-averse control problem for diffusion processes, in which there is a partition of the admissible control strategy into two decision-making groups (namely, the {\it leader} and {\it follower}) with different cost functionals and risk-averse satisfactions. Our approach, based on a hierarchical optimization framework, requires that a certain level of risk-averse satisfaction be achieved for the {\it leader} as a priority over that of the {\it follower's} risk-averseness. In particular, we formulate such a risk-averse control problem involving a family of time-consistent dynamic convex risk measures induced by conditional $g$-expectations (i.e., filtration-consistent nonlinear expectations associated with the generators of certain backward stochastic differential equations). Moreover, under suitable conditions, we establish the existence of optimal risk-averse solutions, in the sense of viscosity solutions, for the corresponding risk-averse dynamic programming equations. Finally, we briefly comment on the implication of our results.

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arXiv2016

Acceptable risks and related decision problems with multiple risk-averse agents

In this paper, we consider a risk-averse decision problem for controlled-diffusion processes, with dynamic risk measures, in which multiple risk-averse agents choose their decisions in such a way to minimize their individual accumulated risk-costs over a finite-time horizon. In particular, we introduce multi-structure dynamic risk measures induced from conditional $g$-expectations, where the latter are associated with the generator functionals of certain BSDEs that implicitly take into account the risk-cost functionals of the risk-averse agents. Here, we also require that such solutions of the BSDEs to satisfy a stochastic viability property with respect to a given closed convex set. Moreover, using a result similar to that of the Arrow-Barankin-Blackwell theorem, we establish the existence of consistent optimal decisions for the risk-averse agents, when the set of all Pareto optimal solutions, in the sense of viscosity, for the associated dynamic programming equations is dense in the given closed convex set. Finally, we briefly comment on the characteristics of acceptable risks vis-á-vis some uncertain future costs or outcomes, in which results from the dynamic risk analysis constitute part of the information used in the risk-averse decision criteria.

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arXiv2016

On the dynamic consistency of hierarchical risk-averse decision problems

In this paper, we consider a risk-averse decision problem for controlled-diffusion processes, with dynamic risk measures, in which there are two risk-averse decision makers (i.e., {\it leader} and {\it follower}) with different risk-averse related responsibilities and information. Moreover, we assume that there are two objectives that these decision makers are expected to achieve. That is, the first objective being of {\it stochastic controllability} type that describes an acceptable risk-exposure set vis-á-vis some uncertain future payoff, and while the {\it second one} is making sure the solution of a certain risk-related system equation has to stay always above a given continuous stochastic process, namely {\it obstacle}. In particular, we introduce multi-structure, time-consistent, dynamic risk measures induced from conditional $g$-expectations, where the latter are associated with the generator functionals of two backward-SDEs that implicitly take into account the above two objectives along with the given continuous obstacle process. Moreover, under certain conditions, we establish the existence of optimal hierarchical risk-averse solutions, in the sense of viscosity solutions, to the associated risk-averse dynamic programming equations that formalize the way in which both the {\it leader} and {\it follower} consistently choose their respective risk-averse decisions. Finally, we remark on the implication of our result in assessing the influence of the {\it leader'}s decisions on the risk-averseness of the {\it follower} in relation to the direction of {\it leader-follower} information flow.

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arXiv2015

Remarks on the hierarchical control problems with model uncertainty

In this paper, we consider a hierarchical control problem with model uncertainty. Specifically, we consider the following objectives that we would like to accomplish. The first one being of a controllability-type that consists of guaranteeing the terminal state to reach a target set starting from an initial condition, while the second one is keeping the state trajectory of the system close to a given reference trajectory over a finite time interval. We introduce the following framework. First, we partition the control subdomain into two disjoint open subdomains, with smooth boundaries, that are compatible with the strategy subspaces of the {\it leader} (which is responsible for the controllability-type criterion) and that of the {\it follower} (which is associated with the second criterion), respectively. Moreover, we account at the optimization stage for model uncertainty by allowing the {\it leader} to choose its control strategy based on a class of alternative models about the system, whereas the {\it follower} makes use of an approximate model about the system. Using the notion of Stackelberg's optimization, we provide conditions on the existence of optimal control strategies for such a hierarchical control problem, under which the {\it follower} is required to respond optimally to the strategy of the {\it leader} so as to achieve the overall objectives. Apart from the issue of modeling and uncertainty, this paper is a companion to our previous work.

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arXiv2015

On the attainable distributions of controlled-diffusion processes pertaining to a chain of distributed systems

We consider a controlled-diffusion process pertaining to a chain of distributed systems with random perturbations that satisfies a weak Hörmander type condition. In particular, we consider a stochastic control problem with the following objectives that we would like to achieve. The first one being of a reachability-type that consists of determining a set of attainable distributions at a given time starting from an initial distribution, while the second one involves minimizing the relative entropy subject to the initial and desired final attainable distributions. Using the logarithmic transformations approach from Fleming, we provide a sufficient condition on the existence of an optimal admissible control for such a stochastic control problem which is amounted to changing the drift by a certain perturbation suggested by Jamison in the context of reciprocal processes. Moreover, such a perturbation coincides with a minimum energy control among all admissible controls forcing the controlled-diffusion process to the desired final attainable distribution starting from the initial distribution. Finally, we briefly remark on the invariance property of the path-space measure for such a controlled-diffusion process pertaining to the chain of distributed systems.

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arXiv2015

On the hierarchical optimal control of a chain of distributed systems

In this paper, we consider a chain of distributed systems governed by a degenerate parabolic equation, which satisfies a weak Hörmander type condition, with a control distributed over an open subdomain. In particular, we consider two objectives that we would like to accomplish. The first one being of a controllability type that consists of guaranteeing the terminal state to reach a target set starting from an initial condition; while the second one is keeping the state trajectory of the overall system close to a given reference trajectory on a finite, compact time intervals. We introduce the following framework. First, we partition the control subdomain into two disjoint open subdomains that are compatible with the strategy subspaces of the {\it leader} and that of the {\it follower}, respectively. Then, using the notion of Stackelberg's optimization (which is a hierarchical optimization framework), we provide a new result on the existence of optimal strategies for such an optimization problem -- where the {\it follower} (which corresponds to the second criterion) is required to respond optimally, in the sense of {\it best-response correspondence} to the strategy of the {\it leader} (that is associated to the controllability-type criterion) so as to achieve the overall objectives. Finally, we remark on the implication of our result in assessing the influence of the reachable target set on the optimal strategy of the {\it follower} in relation to the direction of {\it leader-follower} and {\it follower-leader} information flows.

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