Automata over continuous time

On the Consistency of the Lucas Pricing Formula

American Option Pricing with Discrete and Continuous Time Models: An Empirical Comparison

SummaryLinear quadratic regulator (LQR), a popular technique for designing optimal state feedback controller, is used to derive a mapping between continuous and discrete time inverse optimal equivalence of proportional integral derivative (PID) control problem via dominant pole placement. The aim is to derive transformation of the LQR weighting matrix for fixed weighting factor, using the discrete algebraic Riccati equation (DARE) to design a discrete time optimal PID controller producing similar time response to its continuous time counterpart. Continuous time LQR-based PID controller can be transformed to discrete time by establishing a relation between the respective LQR weighting matrices that will produce similar closed loop response, independent of the chosen sampling time. Simulation examples of first/second order and first-order integrating processes exhibiting stable/unstable and marginally stable open loop dynamics are provided, using the transformation of LQR weights. Time responses for set-point and disturbance inputs are compared for different sampling times as fraction of the desired closed loop time constant.

We investigate experimentally the impact of continuous time on a four-player Hotelling location game. The static pure strategy Nash equilibrium (NE) consists of firms paired-up at the first and third quartiles of the linear city. In a repeated simultaneous move (discrete time) treatment, we largely replicate previous findings in which subjects fail to converge to the NE. However, in asynchronous move (continuous time) treatments we see clear convergence towards the NE.

Sain and Massey [1] have obtained necessary and sufficient conditions for the invertibility of continuous and discrete time linear systems and have also found bounds for what they termed the "inherent integration" (continuous time) or the "inherent delay" (discrete time) of an invertible linear system. In this note, we tighten these bounds by modifying an argument used to prove the Sain-Massey result.

Most models of ecological and eco‐evolutionary processes involve creating trajectories of something, be it population densities, average trait values, or environmental states, over time. This requires decision‐making regarding how to represent the flow of time in models. Most ecologists are exposed to continuous‐time models (typically in the form of ordinary differential equations) as part of their training, especially since the famous Lotka‐Volterra predator–prey dynamics are formulated this way. However, few appear sufficiently well trained to produce their own work with continuous‐time models and may lack exposure to the true versatility of available methods. Specifically, knowledge that discrete individuals can be modelled in continuous time using the Gillespie algorithm is not as widespread as it should be. I will illustrate the flexibility of continous‐time modelling methods such that researchers can make informed choices, and not resort to discretizing time as a ‘default’ without a clear biological motivation to do so. I provide three example‐based tutorials: (1) a comparison of deterministic and stochastic dynamics of the Lotka‐Volterra predator–prey model, (2) an evaluation of matelessness in a hypothetical insect population (and of selection to mate more often by either searching more efficiently or by shortening the ‘time out’ after each mating) and (3) within‐season density dependence followed by a birth pulse leading to Beverton‐Holt or Ricker dynamics depending on whether the deaths of conspecifics help reduce the mortality of others or not (compensatory mortality). I highlight properties of the exponential distribution that, while counter‐intuitive, are good to know when deriving expected lifetime reproductive success or other similar quantities. I also give guidance on how to proceed if the so‐called memorylessness assumption does not hold in a given situation, and show how continuous and discrete times can be freely mixed if the biological situation dictates this to be the preferred option. Continuous‐time models can also be empirically fitted to data, and I review briefly the insight this gives into the so‐called ‘do hares eat lynx?’ paradox that has been plaguing the interpretation of the Hudson Bay hare and lynx dataset.

The accumulated wealth from investment in risky assets is a random variable. If investment strategies are to be ordered, so that one is preferred to another, then the ordering of random variables is required. In this paper the levels of stochastic dominance for random variables are used to define bi-criteria problems for determining an efficient investment strategy. The criteria are characterized as growth and security, respectively, and produce an ordering of strategies consistent with stochastic dominance. In the case where the dynamics of asset returns follow geometric Brownian motion in continuous time, the efficient strategies are shown to be proportional to the growth optimum or Kelly strategy. The analogous problem in discrete time requires solving a stochastic program. An example is provided which compares the continuous and discrete time solutions.KeywordsAsset PricePlanning HorizonInvestment StrategyRisky AssetStochastic DominanceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The ORCCAD programming environment for robotic systems allows users to address automatic control laws in continuous time at the lower levels, and aspects of discrete-time logic at the higher lev els. ORCCAD provides tools of specification, formal verification, simulation, and real-time code generation integrated within a set of dedicated graphical interfaces. Basic robot actions, which are in trinsically hybrid entities, are handled by the ROBOT-TASK struc ture, which smartly interfaces aspects of continuous and discrete time. ROBOT-TASKS are further logically composed into more complex actions, ROBOT-PROCEDURES, through a dedicated lan guage. While system performance can be checked using simula tions, crucial properties such as deadlock avoidance, safety, and liveness can be formally verified at both levels. The approach is illustrated with an underwater inspection mission.

Software reliability assessment models based on cumulative bernoulli trial processes

To reduce simulation times, continuous time (CT) sigma-delta modulators can be modeled as equivalent discrete time (DT) systems so that DT simulators can be used. This method reviews two popular methods of converting from CT to DT and applies them to a second order CT complex ADC. The ADC being designed is for use with a low IF receiver and achieves 80dB SNDR in simulation.

Optimal Hedging in Discrete and Continuous Time

Continuous approximation of collective system behaviour: A tutorial

Network theory is a useful framework for studying interconnected systems of interacting entities. Many networked systems evolve continuously in time, but most existing methods for the analysis of time-dependent networks rely on discrete or discretized time. In this paper, we propose an approach for studying networks that evolve in continuous time by distinguishing between interactions , which we model as discrete contacts, and ties , which encode the strengths of relationships over time. To illustrate our tie-decay network formalism, we adapt the well-known PageRank centrality score to our tie-decay framework in a mathematically tractable and computationally efficient way. We apply this framework to a synthetic example and then use it to study a network of retweets during the 2012 National Health Service controversy in the United Kingdom. Our work also provides guidance for similar generalizations of other tools from network theory to continuous-time networks with tie decay, including for applications to streaming data.

We conduct an experiment on a minimum effort coordination game in a (quasi-)continuous time-frame, where effort choices can be switched freely during a 60-s period. The cooperation levels of the continuous time treatments are not significantly different from the discrete time treatments. Providing subjects with the information on the effort choices of all group members increases the average effort level in continuous time only. The minimum effort level in continuous time with full information feedback is also substantially higher than that with limited information feedback, but the difference is statistically insignificant. With limited information feedback, subjects rarely coordinate to increase their efforts simultaneously to change the group minimum within a period. Our findings imply that continuous time games are not behaviorally equivalent to infinitely repeated discrete time games.

We study lattice QCD in the limit of infinite gauge coupling on a discrete spatial yet continuous Euclidean time lattice at finite baryon chemical potential $\mu_B$. The continuous time framework is based on sending $N_\tau\rightarrow \infty$ and the bare anisotropy to infinity while fixing the temperature in a non-perturbative setup. This leads to a sign problem free algorithm that allows us to study the whole $\mu_B$-$T$ plane. We construct Taylor coefficients required for a Taylor expansion in the pressure at zero chemical potential and in the chiral limit. On that account, cumulants in the baryon number density are measured in various fashions to improve on accuracy. These calculations are based on our worm type Monte Carlo algorithm featuring a polymer resummation scheme and a histogram method.