A note on record times

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 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 consider a general class of timed automata parameterized by a set of “input-determined” operators, in a continuous time setting. We show that for any such set of operators, we have a monadic second order logic characterization of the class of timed languages accepted by the corresponding class of automata. Further, we consider natural timed temporal logics based on these operators, and show that they are expressively equivalent to the first-order fragment of the corresponding MSO logics. As a corollary of these general results we obtain an expressive completeness result for the continuous version of MTL.

LetY0, Y1, Y2, … be an i.i.d. sequence of random variables with continuous distribution function, and letPbe a simple point process on 0≦t≦∞, independent of theYj's.We assume thatPhas a point att= 0; we associateYjwith thejth point ofj≧0, and we say that theYj'soccur at the arrival times ofP. Y0is considered a ‘reference value'. The firstYj(j≧1) to exceed all previous ones is called the first ‘record value', and the time of its occurrence is the first ‘record time'. Subsequent record values and times are defined analogously. We give an infinite series representation for the joint characteristic function of the firstnrecord times, for generalP; in some cases the series can be summed. We find the intensity of the record process whenPis a general birth process, and whenPis a linear birth process withmimmigration sources we find the distribution of the number of records in (0,t]. Form =0 (the Yule process) we give moments of record times and a compact form for the record process intensity. We show that the records occur according to a homogeneous Poisson process whenm =1, and we display a different model with the same behavior, leading to statistical non-identifiability if only the record times are observed. Form =2, the records occur according to a semi-Markov process; again we display a different model with the same behavior. Finally we give a new derivation of the joint distribution of the interrecord times whenPis an arbitrary Poisson process. We relate this result to existing work and to the classical record model. We also obtain a new characterization of the exponential distribution.

Let Y 0 , Y 1 , Y 2, … be an i.i.d. sequence of random variables with continuous distribution function, and let P be a simple point process on 0≦t≦∞, independent of the Yj 's. We assume that P has a point at t = 0; we associate Yj with the jth point of j≧0, and we say that the Yj's occur at the arrival times of P . Y 0 is considered a ‘reference value'. The first Yj (j≧1) to exceed all previous ones is called the first ‘record value', and the time of its occurrence is the first ‘record time'. Subsequent record values and times are defined analogously. We give an infinite series representation for the joint characteristic function of the first n record times, for general P; in some cases the series can be summed. We find the intensity of the record process when P is a general birth process, and when P is a linear birth process with m immigration sources we find the distribution of the number of records in (0, t]. For m = 0 (the Yule process) we give moments of record times and a compact form for the record process intensity. We show that the records occur according to a homogeneous Poisson process when m = 1, and we display a different model with the same behavior, leading to statistical non-identifiability if only the record times are observed. For m = 2, the records occur according to a semi-Markov process; again we display a different model with the same behavior. Finally we give a new derivation of the joint distribution of the interrecord times when P is an arbitrary Poisson process. We relate this result to existing work and to the classical record model. We also obtain a new characterization of the exponential distribution.

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

Local Volatility Calibration during Turbulent Periods

This paper studies reachability problems in discrete time affine systems on simplices-that is, how to steer the states in a simplex to a target polytope in finite time. Reachability problems in a discrete time setting have several distinct features from those in a continuous time setting and the solvability depends not only on the simplex itself but also on the target polytope. In this paper we first study the case where the union of the simplex and the target polytope is convex. Then we generalize the results by dropping the convex assumption. For theoretical completeness, the problem of making a polytope positive invariant is considered as well. Finally, a temperature control system is presented to illustrate our results.

The random decrement technique (RDT), introduced in the sixties by Cole [NASA CR-2005, 1973], is a very performing method of analysis for vibration signature of a structure under ambient loading. But the real nature of the random decrement signature has been misunderstood until now. Moreover, the various interpretations were made in continuous time setting, while real experimental data are obtained in discrete time. In this paper, the really implemental discrete time algorithms are studied. The asymptotic analysis as the number of triggering points go to infinity is achieved, and a Law of Large Numbers as well as a Central Limit Theorem is proved. Moreover, the limit as the discretization time step goes to zero is computed, giving more tractable formulas to approximate the random decrement. This is a new approach of the famous “Kac-Slepian paradox” [Ann. Math. Stat., 30, 1215–1228 (1959)]. The main point might be that the RDT is a very efficient functional estimator of the correlation function of a stationary ergodic Gaussian process. Very fast, it is to classical estimators what Fast Fourier Transform (FFT) is to ordinary Discrete Fourier Transforms.