scholarly journals Predicting IBNYR Events and Delays II. Discrete Time

1990 ◽  
Vol 20 (1) ◽  
pp. 93-111 ◽  
Author(s):  
William S. Jewell
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Bayesian Method ◽  
Predictive Accuracy ◽  
Random Delay ◽  
Continuous Time Model ◽  
Practical Situation ◽  
Time Model ◽  
Numerical Example ◽  
Exposure Interval

AbstractAn IBNYR event is one that occurs randomly during some fixed exposure interval and incurs a random delay before it is reported. A previous paper developed a continuous-time model of the IBNYR process in which both the Poisson rate at which events occur and the parameters of the delay distribution are unknown random quantities; a full-distributional Bayesian method was then developed to predict the number of unreported events. Using a numerical example, the success of this approach was shown to depend upon whether or not the occurrence dates were available in addition to the reporting dates. This paper considers the more usual practical situation in which only discretized epoch information is available; this leads to a loss of predictive accuracy, which is investigated by considering various levels of quantization for the same numerical example.

DISCRETE AND CONTINUOUS TIME POPULATION MODELS, A COMPARISON CONCERNING PROLIFERATION BY FISSION

1995 ◽  
Vol 03 (02) ◽  
pp. 543-558 ◽  
Author(s):  
B.W. KOOI ◽  
M.P. BOER
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Matrix Theory ◽  
Dimensional Torus ◽  
Continuous Time Model ◽  
Time Model ◽  
Discrete Time Model ◽  
Fixed Part ◽  
Number Of Individuals ◽  
Insight Into

We present two approaches, discrete time and continuous time models, for individuals which propagate through binary fission. The volumes of the two daughters are a fixed part of that of the mother, not necessarily the half, and their growth rates may differ. The discrete time approach gives more insight into the results obtained with the continuous time model. We define classes in the continuous time model such that the total number of individuals in these classes at specific moments in time is equal to the unknown number in a discrete time model. Then the discrete time model is homologous to the continuous one in the sense of having the same solutions at specific moments. Population matrix theory applies when the ratio of the inter-division times of the two daughters is rational. There is inter-class convergence but no intra-class convergence. The latter feature implies that there is no convergence of the size distribution in the continuous time model either. When the ratio is irrational the continuous time model holds and there is convergence but the rate of convergence can become infinitesimally small. This phenomenon is linked with quasi-periodicity on a 2-dimensional torus.

scholarly journals Asymptotic synthesis of contingent claims with controlled risk in a sequence of discrete‐time markets

2021 ◽  
Vol 16 (1) ◽  
pp. 25-47
Author(s):  
David M. Kreps ◽  
Walter Schachermayer
Keyword(s):  
Financial Markets ◽  
Discrete Time ◽  
Continuous Time ◽  
Contingent Claims ◽  
Contingent Claim ◽  
Complete Markets ◽  
Continuous Time Model ◽  
Time Model ◽  
Black Scholes ◽  
Discrete Time Models

We examine the connection between discrete‐time models of financial markets and the celebrated Black–Scholes–Merton (BSM) continuous‐time model in which “markets are complete.” Suppose that (a) the probability law of a sequence of discrete‐time models converges to the law of the BSM model and (b) the largest possible one‐period step in the discrete‐time models converges to zero. We prove that, under these assumptions, every bounded and continuous contingent claim can be asymptotically synthesized, controlling for the risks taken in a manner that implies, for instance, that an expected‐utility‐maximizing consumer can asymptotically obtain as much utility in the (possibly incomplete) discrete‐time economies as she can at the continuous‐time limit. Hence, in economically significant ways, many discrete‐time models with frequent trading resemble the complete‐markets model of BSM.

A Continuous-Time Model of Income Dynamics

2011 ◽  
Author(s):  
Mark Matthias Trede ◽  
Thorsten Heimann
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Income Dynamics ◽  
Continuous Time Model ◽  
Time Intervals ◽  
Time Model ◽  
Varying Length ◽  
Accounting Period ◽  
Stochastic Properties ◽  
Social Security Agency

Most models of income dynamics are set in a discrete-time framework with an arbitrarily chosen accounting period. This article introduces a continuous-time stochastic model of income flows, without the need to define an accounting period. Our model can be estimated using unbalanced panel data with arbitrarily spaced observations. Although our model describes the stochastic properties of income flows, estimation is based on observed incomes accruing during time intervals of possibly varying length. Our model of income dynamics is close in spirit to the discrete-time two-stage models prevalent in the literature. We impose a parsimoniously parameterized continuous-time stochastic process (possibly containing a unit root) to model the deviation from a traditional earnings function. We illustrate our approach by estimating a simplified model using microeconomic data from the German social security agency from 1975 to 1995.

Discretization of deflated bond prices

2000 ◽  
Vol 32 (2) ◽  
pp. 540-563 ◽  
Author(s):  
Paul Glasserman ◽  
Hui Wang
Keyword(s):  
Interest Rates ◽  
Term Structure ◽  
Discrete Time ◽  
Continuous Time ◽  
Unit Interval ◽  
Continuous Time Model ◽  
Time Model ◽  
Numerical Work ◽  
Continuous Time Process ◽  
Continuous Time Processes

This paper proposes and analyzes discrete-time approximations to a class of diffusions, with an emphasis on preserving certain important features of the continuous-time processes in the approximations. We start with multivariate diffusions having three features in particular: they are martingales, each of their components evolves within the unit interval, and the components are almost surely ordered. In the models of the term structure of interest rates that motivate our investigation, these properties have the important implications that the model is arbitrage-free and that interest rates remain positive. In practice, numerical work with such models often requires Monte Carlo simulation and thus entails replacing the original continuous-time model with a discrete-time approximation. It is desirable that the approximating processes preserve the three features of the original model just noted, though standard discretization methods do not. We introduce new discretizations based on first applying nonlinear transformations from the unit interval to the real line (in particular, the inverse normal and inverse logit), then using an Euler discretization, and finally applying a small adjustment to the drift in the Euler scheme. We verify that these methods enforce important features in the discretization with no loss in the order of convergence (weak or strong). Numerical results suggest that these methods can also yield a better approximation to the law of the continuous-time process than does a more standard discretization.

ESTIMATING CONTINUOUS-TIME MODELS ON THE BASIS OF DISCRETE DATA VIA AN EXACT DISCRETE ANALOG

2009 ◽  
Vol 25 (4) ◽  
pp. 1120-1137 ◽  
Author(s):  
J. Roderick McCrorie
Keyword(s):  
Time Series ◽  
Discrete Time ◽  
Continuous Time ◽  
Financial Time Series ◽  
Discrete Data ◽  
Discrete Analog ◽  
Continuous Time Model ◽  
Time Model ◽  
Financial Time ◽  
Time Modeling

This paper offers a perspective on A.R. Bergstrom’s contribution to continuous-time modeling, focusing on his preferred method of estimating the parameters of a structural continuous-time model using an exact discrete-time analog. Some inherent difficulties in this approach are discussed, which help to explain why, in spite of his prescience, the methods around his time were not universally adopted as he had hoped. Even so, it is argued that Bergstrom’s contribution and legacy is secure and retains some relevance today for the analysis of macroeconomic and financial time series.

Discretization of deflated bond prices

2000 ◽  
Vol 32 (02) ◽  
pp. 540-563 ◽  
Author(s):  
Paul Glasserman ◽  
Hui Wang
Keyword(s):  
Interest Rates ◽  
Term Structure ◽  
Discrete Time ◽  
Continuous Time ◽  
Unit Interval ◽  
Continuous Time Model ◽  
Time Model ◽  
Numerical Work ◽  
Continuous Time Process ◽  
Continuous Time Processes

This paper proposes and analyzes discrete-time approximations to a class of diffusions, with an emphasis on preserving certain important features of the continuous-time processes in the approximations. We start with multivariate diffusions having three features in particular: they are martingales, each of their components evolves within the unit interval, and the components are almost surely ordered. In the models of the term structure of interest rates that motivate our investigation, these properties have the important implications that the model is arbitrage-free and that interest rates remain positive. In practice, numerical work with such models often requires Monte Carlo simulation and thus entails replacing the original continuous-time model with a discrete-time approximation. It is desirable that the approximating processes preserve the three features of the original model just noted, though standard discretization methods do not. We introduce new discretizations based on first applying nonlinear transformations from the unit interval to the real line (in particular, the inverse normal and inverse logit), then using an Euler discretization, and finally applying a small adjustment to the drift in the Euler scheme. We verify that these methods enforce important features in the discretization with no loss in the order of convergence (weak or strong). Numerical results suggest that these methods can also yield a better approximation to the law of the continuous-time process than does a more standard discretization.

Absence of Cycles in Symmetric Neural Networks

1998 ◽  
Vol 10 (5) ◽  
pp. 1235-1249 ◽  
Author(s):  
Xin Wang ◽  
Arun Jagota ◽  
Fernanda Botelho ◽  
Max Garzon
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Necessary Conditions ◽  
Continuous Time Model ◽  
Time Model ◽  
Discrete Time Model ◽  
Asymptotic Dynamics ◽  
The One ◽  
Discretized Model ◽  
Symmetric Networks

For a given recurrent neural network, a discrete-time model may have asymptotic dynamics different from the one of a related continuous-time model. In this article, we consider a discrete-time model that discretizes the continuous-time leaky integrat or model and study its parallel, sequential, block-sequential, and distributed dynamics for symmetric networks. We provide sufficient (and in many cases necessary) conditions for the discretized model to have the same cycle-free dynamics of the corresponding continuous-time model in symmetric networks.

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