Showing papers on "Discrete time and continuous time published in 1985"

Probabilistic Properties of Deterministic Systems

Abstract: 1. Introduction 2. The toolbox 3. Markov and Frobenius-Perron operators 4. Studying chaos with densities 5. The asymptotic properties of densities 6. The behaviour of transformations on intervals and manifolds 7. Continuous time systems: an introduction 8. Discrete time processes embedded in continuous time systems 9. Entropy 10. Stochastic perturbation of discrete time systems 11. Stochastic perturbation of continuous time systems.

Upper and Lower Bounds of Put and Call Option Value: Stochastic Dominance Approach

Abstract: Applying stochastic dominance rules with borrowing and lending at the risk-free interest rate, we derive upper and lower values for an option price for all unconstrained utility functions and alternatively for concave utility functions. The derivation of these bounds is quite general and fits any kind of stock price distribution as long as it is characterized by a "nonnegative beta." Transaction costs and taxes can be easily incorporated in the model presented here since investors are not required to revise their portfolios continuously. The "price" that is paid for this generalization is that a range of values rather than a unique value is obtained. IN THE LAST DECADE, there has been an increasing interest in the options market by both academicians and practitioners. The most well-known valuation models are the Black-Scholes [4] model, the jump stochastic process model (see Cox and Ross [6]), the binomial model (see Cox, Ross, and Rubinstein [7], Rendleman and Bartler [19], and Sharpe [22]), and Rubinstein's discrete model [21]. Recently, Perrakis and Ryan [18], using Rubinstein's approach, have derived upper and lower bounds for option prices.' The relative pricing theory of contingent claims differs in the continuous time and discrete time models. In a continuous time framework, the instantaneous expected rate of return of the option and that of the underlying stock must follow a certain market equilibrium condition. In this case, the equilibrium condition is easily determined by a continuous hedging strategy. In the discrete time case, there is no hedging strategy to provide a simple relationship between the option and the stock price. However, for a given value distribution of the stock, the value distribution of the option is uniquely determined by the option exercise price, and this suggests some possible relationships between the stock and option values.

Discrete Time Stochastic Petri Nets

Abstract: Basic graph models of processes, such as Petri nets, have usually omitted the concept of time as a parameter. Time has been added to the Petri net model in two ways. The timed Petri net (TPN) uses a fixed number of discrete time intervals. The stochastic Petri net (SPN) uses an exponentially distributed random variable. In this paper, a discrete time stochastic Petri model is described. These discrete time SPN's fill the gap between TPN and normal SPN. However, the use of discrete time complicates the SPN model in that more than one transition may fire at a time step. Finally, an example of a live and bounded Petri net which has nonempty, disjoint, recurrent subsets of markings is given.

A diffusion process and its applications to detecting a change in the drift of Brownian motion

Abstract: : The purpose of the present paper is to make a quantitative comparison of the Shiryayev-Roberts and Page procedures We do this in the context of continuous time in order to use the machinery of diffusion processes to perform explicitly certain calculations, which seem impossible in discrete time Although the continuous time results are not especially good approximations to the corresponding quantities in discrete time, they provide very useful comparative information on which to base selection of a stopping rule This paper is organized as follows The Shiryayev-Roberts process is defined and shown to be a novel diffusion process with some surprising properties We also specify more precisely the basis for our comparison of the two procedures and give the results of some elementary calculations These developments contain an asymptotic evaluation We define a modification of our basic procedure and give an asymptotic evaluation of its average run length Numerical comparisons and a discussion of their significance are contained next Our conclusions are roughly these In simple situations where the two procedures can be directly compared, neither seems dramatically superior to the other However, the Shiryayev-Roberts procedure is more easily adapted to complex circumstances and consequently warrants additional study

Multirate and composite control of two-time-scale discrete-time systems

Abstract: The design of stabilizing feedback control of singularly perturbed diserete-time systems is decomposed into the design of slow and fast controllers which are combined to form the composite control. Composite control strategies are developed for the case of single rate measurements (all variables are measured at the same rate) as well as for the case of multirate measurements (slow variables are measured at a rate slower than that of fast variables).

Robustness of discrete-time direct adaptive controllers

Abstract: The problem of preserving stability of discrete-time adaptive controllers in spite of reduced-order modeling and output disturbances is addressed in this paper. Conditions for global stability (convergence of the tracking error with bounded signals) are derived for a discrete-time pole-zero placement adaptive controller where the parameter estimator is modified in terms of normalized signals. Following an input-output perpective, the overall system is decomposed into two subsystems reflecting the parameter estimation and modeling errors, respectively, and its stability is studied using the sector stability and passivity theorems. First the analysis is carried for the class of disturbances and reference inputs that are either decaying or can be exactly hulled by a linear controller of the chosen structure. In this L 2 -framework, it is shown that the only substantive assumption to assure stability is the existence of a linear controller such that the closed-loop transfer function verifies certain conicity conditions. The convergence speed and alertness properties of various parameter adaptation algorithms regarding this condition are discussed. The results are further extended to a broader class of L_{\infty} disturbances and reference inputs.

Asymptotic recovery for discrete-time systems

Abstract: An asymptotic recovery design procedure is proposed for square, discrete-time, linear, time-invariant multivariable systems, which allows a state-feedback design to be approximately recovered by a dynamic output feedback scheme. Both the case of negligible processing time (compared to the sampling interval) and of significant processing time are discussed. In the former case, it is possible to obtain perfect recovery if the plant is minimum-phase and has the smallest possible number of zeros at infinity. In other cases good recovery is frequently possible. New conditions are found which ensure that the return-ratio being recovered exhibits good robustness properties.

The discrete-time bounded-real lemma in digital filtering

Abstract: The Lossless Bounded-Real lemma is developed in the discrete-time domain, based only on energy balance arguments. The results are used to prove a discrete-time version of the general Bounded-Real lemma, based on a matrix spectral-factorization result that permits a transfer matrix embedding process. Some applications of the results in digital filter theory are finally outlined.

An Investigation of Commodity Futures Prices Using the Consumption‐Based Intertemporal Capital Asset Pricing Model

Abstract: In this paper we extend the multigood futures pricing model of Grauer and Litzenberger [9] to a dynamic discrete time setting. We then test the model using data on futures prices for corn, wheat, and soybeans. The parameter estimates we obtain are similar to those obtained by other researchers using stock return data. The model itself is rejected and we offer some suggestions as to which assumption may be violated. We also give an interpretation to the Hansen-Singleton nonlinear instrumental variables estimation technique used in our empirical work.

A Transfer-Function Approach to Linear Time-Varying Discrete-Time Systems

Abstract: In the first part of the paper a transfer-function approach is developed for the class of linear time-varying discrete-time systems. The theory is specified in terms of skew (noncommutative) rings of polynomials and formal power series, both with coefficients in a ring of time functions. The transfer-function matrix is defined to be a matrix whose entries belong to a skew ring of formal power series. It is shown that various system properties, such as asymptotic stability, can be characterized in terms of the skew-ring framework. In the last part of the paper, the transfer-function framework is applied to the study of feedback control. New results are obtained on assignability of system dynamics by using dynamic output feedback and dynamic state feedback. The results are applied to the control of an armature-controlled do motor with a variable loading.

Recursive limited memory filtering and scattering theory (Corresp.)

Abstract: Redheffer scattering theory is reviewed in a generalized setting as a method to derive recursive solutions of linear two-point boundary value problems (TPBVP) over arbitrarily varying intervals. The results can be used to derive a complete solution for the problem of limited-memory (or sliding-window) estimation, when a usual state-space model for the signal is available. Recursive limited-memory filters are derived for both continuous and discrete time signals.

Numerical Comparisons of Inventory Policies for Periodic Review Systems

Abstract: This paper studies the numerical computation of the two parameters the reorder level s and the order up to level S of inventory policies for discrete time shortage cost systems. Our goal is to obtain approximately optimal policies with little computational effort. The paper introduces three new methods that are designed to achieve this goal. Two of the methods are shortcuts based on the method of Freeland and Porteus and one is a heuristic that makes several modifications to a standard continuous review approximation. The paper provides a fairly detailed survey of other methods for easily computing approximately optimal inventory policies. It then numerically compares all these methods on a reasonably broad range of problems. One of the shortcuts and the new heuristic method performed very well: the percentage error of their average costs was approximately 1%. Some commonly cited competing methods had percentage errors of over 10% and a commonly cited continuous review approximation had a percentage error of over 80%. To study the effect of extreme parameter choices in the test bed, the paper introduces a procedure to determine a subset of the parameter values, called the 1% contiguous test bed, for which each method performed well. The results show that, depending on the range of values that apply in a given practical situation, either i any of a large number of methods will yield good performance or ii a carefully selected method can achieve superior performance.

An existence theorem for discrete-time infinite-horizon optimal control problems

Abstract: An existence theorem is given for a general class of deterministic, infinite-horizon, discrete-time optimal control problems The hypotheses of the theorem are weak and can be easily verified The objective function may be either a summation or supremum over time

Discrete-time linear periodic systems: gramian and modal criteria for reachability and controllability†

Abstract: This paper deals with the reachability and controllability of periodic discrete-time systems. First, we supply two necessary and sufficient complete reachability conditions, which apply to reversible and non-reversible systems, respectively. Then, a necessary and sufficient complete controllability condition is provided. This condition, as well as the complete reachability criteria, is given in terms of the reachability gramian matrix. Equivalent modal criteria for reachability and controllability are established in the second part of the paper.

Statistical Identification of Storage Models with Application to Stochastic Hydrology

Abstract: This paper describes a method for the statistical identification of storage models for daily riverflow time series, together with numerical results. The first step in the identification process is to obtain a discrete time version of a storage model using a local linearization approach. It is shown that the discrete time version thus obtained may be utilized in the identification of the original storage model. A statistical method for the identification of daily rainfall time series models used in simulation is also presented. This identification procedure employs the maximum likelihood method for point process data analysis and is illustrated by means of numerical examples.

A new nonlinear filtering formula for discrete time measurements

Abstract: An exact formula for computing the conditional mean of a random variable is derived for discrete time observations. This formula is analogous to the well-known result of Fujisaki-Kallianpur-Kunita for continuous time observations, and it is similar to the discrete time formula recently derived by Takeuchi and Akashi. The derivation of the new formula is extremely elementary, and it is based on the judicious choice of a certain homotopy function.

Markov Decision Drift Processes; Conditions for Optimality Obtained by Discretization

Abstract: In a recent paper (Hordijk, A., F. A. van der Duyn Schouten. 1983. Discretization and weak convergence in Markov decision drift processes. Math. Oper. Res. 8 112–141.) the authors gave sufficient conditions for the weak convergence of a sequence of discrete time Markov decision drift processes to a related continuous time Markov decision drift process. The goal of this analysis was to obtain for the continuous time model conditions for optimality of a limit point of a sequence of discounted discrete time optimal policies. However, the general conditions in (Hordijk, A., F. A. van der Duyn Schouten. 1983. Discretization and weak convergence in Markov decision drift processes. Math. Oper. Res. 8 112–141.) can only be applied to a very restrictive class of models. To obtain more widely applicable conditions we are concerned in this paper not with the weak convergence of the stochastic processes induced by policies, but rather with the convergence of the total expected discounted costs. Special attention is p...

Stability of Converter Control for Multiterminal HVDC Systems

Abstract: The stability analysis of converter control, particularly for multiterminal HVDC systems, is complex and requires a systematic approach. This paper presents a new discrete converter model based on average system quantities and the development of the state space model of the AC/DC system. The linearized, discrete time, state space model of the overall system is derived utilizing the linearized component models and identifying the interconnections between them. Both frequency and time domain techniques are employed for stability investigations. The system model is demonstrated through the stability analysis of some sample systems and the results are validated using digital simulation.

Tracking and disturbance rejection of MIMO nonlinear systems with a PI or PS controller

Abstract: We study tracking and disturbance rejection for a class of MIMO nonlinear systems, with a Linear proportional plus integral (PI) compensator, in the continuous time Case, and a linear proportional plus sum (PS) compensator in the discrete time case. We show that if the nonlinear plant is exponentially stable and has a strictly increasing dc steady state I/O map then a simple PI or PS compensator can be used to yield a stable unity feedback closed loop system which asymptotically tracks reference inputs that tend to constant vectors and asymptotically rejects disturbances that tend to constant vectors. This extends earlier work of Desoer and Lin

Convergence of an algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states

Abstract: In [1] an algorithm was presented to find an approximant to the maximal state constraint set for a linear discrete time dynamical system with polyhedral state and input bounds. Here it is shown that the algorithm will yield an approximant arbitrarily close to the maximal state constraint set and the number of iterations is given as a function of the prescribed precision of the approximant.