On Continuous Timed Automata with Input-Determined Guards

Suppose that a road network model is given, together with some given demand for travel by (say) car and that the demand for travel varies with time of day but not from day to day. Suppose that this demand is given in the form of specified total outflow rates from each origin headed towards each destination, for each origin-destination pair and for each time of day, and that some initial time-dependent routeinflow rates, meeting the given demand, are given. Finally, suppose that within-day time is represented by a continuous variable. This paper specifies a natural smooth day-to-day route-swapping procedure wherein drivers swap toward less expensive routes as day succeeds day, and shows that under reasonable conditions there is an equilibrium state of this dynamical system. If such a collection of route-inflows has arisen today, say, then there is no incentive for any route-inflow to change tomorrow, in the sense that at each moment of today each of today's route-inflows isalready on a route which today yielded the smallest travel cost. Such a set of “no-incentive-to-change” route-inflows is called adynamic equilibrium, or adynamic user-equilibrium, and may be regarded as a solution of the dynamic equilibrium traffic assignment problem. Thus, the paper introduces a smooth day-to-day dynamic assignment model and, using this model, shows that there is a dynamic user-equilibrium in a continuous time setting. The paper briefly considers the day-to-day stability of the route-swapping process, also in a continuous setting. Finally, the paper gives a simple dynamical example illustrating the stability of the route-swapping process in a simple two-route network when there is deterministic queueing at bottlenecks.

We consider monadic second order logic (MSO) and the modal μ-calculus (Lμ) over transition systems (Kripke structures). It is well known that every class of transition systems which is definable by a sentence of Lμ is definable by a sentence of MSO as well. It will be shown that the converse is also true for an important fragment of MSO: every class of transition systems which is MSO-definable and which is closed under bisimulation - i.e., the sentence does not distinguish between bisimilar models - is also Lμ-definable. Hence we obtain the following expressive completeness result: the bisimulation invariant fragment of MSO and Lμ are equivalent. The result was proved by David Janin and Igor Walukiewicz. Our presentation is based on their article [91]. The main step is the development of automata-based characterizations of Lμ over arbitrary transition systems and of MSO over transition trees (see also Chapter 16). It turns out that there is a general notion of automaton subsuming both characterizations, so we obtain a common ground to compare these two logics. Moreover we need the notion of the ω-unravelling for a transition system, on the one hand to obtain a bisimilar transition tree and on the other hand to increase the possibilities of choosing successors.

We consider dynamic cooperative games in characteristic function form in the sense that the characteristic function evolves over time in accordance with a difference or differential equation that is influenced not only by the current ("instantaneous") characteristic function but also by the solution concept used to allocate the benefits of cooperation among the players. The latter solution concept can be any one of a number of now standard solution concepts of cooperative game theory but, for demonstration purposes, we focus on the core and the Shapley value. In the process, we introduce some new mechanisms by which players may regard the evolution of cooperative game over time and analyse them with respect to the goal of attaining time consistency either in discrete or in continuous time setting. In discrete time, we illustrate the phenomena that can arise when an allocation according to a given solution concept is used to adapt the values of coalitions at successive time points. In continuous time, we introduce the notion of an "instantaneous" game and its integration over time.

In a continuous time setting, we consider a set of -absolutely continuous probability measures, containing at least one equivalent to , and denote by , the smallest m-stable closed convex set of probability measures containing . We shall prove mainly under the martingale representation property with respect to a continuous vector valued martingale S, that is the set of supermartingale measures for a family , where is a set of predictable vector valued processes. We state as an immediate consequence of Föllmer-Kramkov Theorem in H. Föllmer and D. Kramkov (Optional decompositions under constraints, Probab. Theory Relat. Fields 109 (1997), pp. 1–25), the decomposition of -supermartingales.

Rotordynamics prediction in engineering (second edition)

We study the language-theoretical aspects of parameterized communicating automata (PCAs), in which processes communicate via rendez-vous. A given PCA can be run on any topology of bounded degree such as pipelines, rings, ranked trees, and grids. We show that, under a context bound, which restricts the local behavior of each process, PCAs are effectively complementable. Complementability is considered a key aspect of robust automata models and can, in particular, be exploited for verification. In this paper, we use it to obtain a characterization of context-bounded PCAs in terms of monadic second-order (MSO) logic. As the emptiness problem for context-bounded PCAs is decidable for the classes of pipelines, rings, and trees, their model-checking problem wrt. MSO properties also becomes decidable. While previous work on model checking parameterized systems typically uses temporal logics without next operator, our MSO logic allows one to express several natural next modalities.

Environmental and Resource Management Under Myopia

Automata walking on graphs are a mathematical formalisation of autonomous mobile agents with limited memory operating in discrete environments. This paper establishes a framework in which to model and automatically verify that autonomous mobile agents correctly perform their tasks. The framework consists of a logical language tailored for expressing agent tasks, and an algorithm solving the parameterized verification problem, where the graphs are treated as the parameter. We reduce the parameterised verification problem to classic questions in automata theory and monadic second order logic, i.e., universality and validity problems.We illustrate the framework by instantiating it to a popular model of robot from the distributed computing literature.This work clarifies the border between classes of mobile-agent systems that have decidable parameterised verification problem, and those that do not.

Tail generating functions for extendable branching processes

This article shows how structured financing can be used to solve the asset substitution problem in a dynamic setting. Structuring induces the firm’s owner to optimally choose the first best operating strategy even though the owner’s value function might be locally convex (concave), which would ordinarily lead to overinvestment (underinvestment) in risky projects. This result is demonstrated in two different continuous time settings—one that is based on the risk-shifting framework of Leland (1998) and one that generalizes the scaled return model of Green (1984). It is shown that the contractual nature of the structuring is a key determinant of the issuing firm’s dynamic asset volatility. Furthermore, unlike nonstructured financing, the default (conversion) probability of a structured debt security may be increasing (decreasing) in the firm’s total assets. Structured securities are therefore hedge assets, which potentially explains the popularity of structured securities among investors and third-party issuers.

We define Backward Stochastic Difference Equations on spaces related to discrete time, finite state processes. Solutions exist and are unique under weaker assumptions than are needed in the continuous time setting. A comparison theorem for these solutions is also given. Applications to the theory of nonlinear expectations are explored, including a representation result.

We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain in a random environment under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection and supremum of σ-fields, which is key for the stability of the nonlinear filter and partially resolves a long-standing gap in the proof of a result of Kunita [J. Multivariate Anal. 1 (1971) 365–393]. A similar result is obtained also in the continuous time setting. The proofs are based on an ergodic theorem for Markov chains in random environments in a general state space.

A minimum principle for chaotic dynamical systems

This note indicates two alternative proofs for a result of Galambos and Seneta [1] on the distribution of ratios of successive record times in a sequence of independent and identically distributed random variables. Some consequences of one method of proof are discussed, and its analogues in a continuous time setting are described.

Local Volatility Calibration during Turbulent Periods