scholarly journals Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Caishi Wang ◽  
Jinshu Chen
Keyword(s):  
Brownian Motion ◽  
Growth Condition ◽  
Discrete Time ◽  
Continuous Time ◽  
Orthonormal Basis ◽  
Continuous Time Processes ◽  
Characterization Theorems ◽  
Normal Martingale

We aim at characterizing generalized functionals of discrete-time normal martingales. LetM=(Mn)n∈Nbe a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals ofMwith an appropriate orthonormal basis forM’s square integrable functionals. Then we introduce a transform, called the Fock transform, for these functionals and characterize them via the transform. Several characterization theorems are established. Finally we give some applications of these characterization theorems. Our results show that generalized functionals of discrete-time normal martingales can be characterized only by growth condition, which contrasts sharply with the case of some continuous-time processes (e.g., Brownian motion), where both growth condition and analyticity condition are needed to characterize generalized functionals of those continuous-time processes.

Markovian contact processes

1978 ◽  
Vol 10 (01) ◽  
pp. 85-108 ◽  
Author(s):  
Denis Mollison
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Birth Process ◽  
Linear Processes ◽  
Contact Processes ◽  
Spatial Spread ◽  
Non Linear ◽  
And Migration ◽  
Contact Distribution ◽  
Continuous Time Processes

Markovian contact processes (mcp's) include some of the commoner models for the spatial spread of population processes, such as the ‘simple’ and ‘general’ epidemics, percolation processes, and birth, death and migration processes. They are here set in a framework which relates them to each other, and to a particular basic model, the contact birth process (cbp); indeed they are defined as modifications of the cbp. The immediate advantage of this is that bounds for the velocity of the cbp, which can be obtained from the linear equation describing its expected numbers (provided the contact distribution has exponentially bounded tail) apply also to general mcp's. Existence of moments (and for some processes their time-derivatives) of St (θ), the distance to the furthest individual in an arbitrary direction θ, can also be deduced whenever the corresponding moment of the contact distribution exists. An important subclass consists of population-monotone mcp's, for which the distributions of St can be shown to be subconvolutive, so that the work of Hammersley (1974) can be applied to obtain convergence theorems for St /t; and these can be extended to convergence of the convex hull of the set of inhabited points. These results are particularly valuable because they apply to some non-linear processes, e.g. the simple epidemic. Some special results on the cbp (Section 6) emphasize the differences between linear and non-linear spatial stochastic processes. Although the paper is written throughout in terms of continuous-time processes, the results on subconvolutive distributions are actually more easily applied in the discrete-time case. (Conversely, the approach used in the accompanying paper by Biggins (pp. 62–84), which is written in terms of discrete-time processes, can be extended to deal also with continuous-time processes.)

scholarly journals Continuous-time gambling problems

1974 ◽  
Vol 6 (4) ◽  
pp. 651-665 ◽  
Author(s):  
David C. Heath ◽  
William D. Sudderth
Keyword(s):  
Brownian Motion ◽  
Discrete Time ◽  
Continuous Time ◽  
Gambling Problems ◽  
Brownian Motion Process ◽  
Motion Process

An abstract gambler's problem is formulated in a continuous-time setting and analogues are proved for some of the discrete-time results of Dubins and Savage in their book How to Gamble if You Must. Applications are made to problems of controlling a Brownian motion process.

scholarly journals Limiting stochastic processes of shift-periodic dynamical systems

2019 ◽  
Vol 6 (11) ◽  
pp. 191423
Author(s):  
Julia Stadlmann ◽  
Radek Erban
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Discrete Time ◽  
Real Line ◽  
Continuous Time ◽  
Initial Distribution ◽  
Dynamical Behaviour ◽  
One Dimensional

A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences x n +1 = F ( x n ) generated by such maps display rich dynamical behaviour. The integer parts ⌊ x n ⌋ give a discrete-time random walk for a suitable initial distribution of x 0 and converge in certain limits to Brownian motion or more general Lévy processes. Furthermore, for certain shift-periodic maps with small holes on [0,1], convergence of trajectories to a continuous-time random walk is shown in a limit.

scholarly journals A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour

2004 ◽  
Vol 41 (03) ◽  
pp. 601-622 ◽  
Author(s):  
Claudia Klüppelberg ◽  
Alexander Lindner ◽  
Ross Maller
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Lévy Process ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Levy Process ◽  
Volatility Models ◽  
Garch Processes ◽  
Continuous Time Processes ◽  
Necessary And Sufficient

We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.

NONPARAMETRIC NONSTATIONARITY TESTS

2013 ◽  
Vol 30 (1) ◽  
pp. 127-149 ◽  
Author(s):  
Federico M. Bandi ◽  
Valentina Corradi
Keyword(s):  
Time Series ◽  
Discrete Time ◽  
Continuous Time ◽  
Diffusion Processes ◽  
Nonlinear Process ◽  
Additive Functional ◽  
Finite Sample ◽  
Occupation Times ◽  
Discrete Time Case ◽  
Continuous Time Processes

We propose additive functional-based nonstationarity tests that exploit the different divergence rates of the occupation times of a (possibly nonlinear) process under the null of nonstationarity (stationarity) versus the alternative of stationarity (nonstationarity). We consider both discrete-time series and continuous-time processes. The discrete-time case covers Harris recurrent Markov chains and integrated processes. The continuous-time case focuses on Harris recurrent diffusion processes. Notwithstanding finite-sample adjustments discussed in the paper, the proposed tests are simple to implement and rely on tabulated critical values. Simulations show that their size and power properties are satisfactory. Our robustness to nonlinear dynamics provides a solution to the typical inconsistency problem between assumed linearity of a time series for the purpose of nonstationarity testing and subsequent nonlinear inference.

Markovian contact processes

1978 ◽  
Vol 10 (1) ◽  
pp. 85-108 ◽  
Author(s):  
Denis Mollison
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Birth Process ◽  
Linear Processes ◽  
Contact Processes ◽  
Spatial Spread ◽  
Non Linear ◽  
And Migration ◽  
Contact Distribution ◽  
Continuous Time Processes

Markovian contact processes (mcp's) include some of the commoner models for the spatial spread of population processes, such as the ‘simple’ and ‘general’ epidemics, percolation processes, and birth, death and migration processes. They are here set in a framework which relates them to each other, and to a particular basic model, the contact birth process (cbp); indeed they are defined as modifications of the cbp. The immediate advantage of this is that bounds for the velocity of the cbp, which can be obtained from the linear equation describing its expected numbers (provided the contact distribution has exponentially bounded tail) apply also to general mcp's. Existence of moments (and for some processes their time-derivatives) of St(θ), the distance to the furthest individual in an arbitrary direction θ, can also be deduced whenever the corresponding moment of the contact distribution exists.An important subclass consists of population-monotone mcp's, for which the distributions of St can be shown to be subconvolutive, so that the work of Hammersley (1974) can be applied to obtain convergence theorems for St/t; and these can be extended to convergence of the convex hull of the set of inhabited points. These results are particularly valuable because they apply to some non-linear processes, e.g. the simple epidemic. Some special results on the cbp (Section 6) emphasize the differences between linear and non-linear spatial stochastic processes.Although the paper is written throughout in terms of continuous-time processes, the results on subconvolutive distributions are actually more easily applied in the discrete-time case. (Conversely, the approach used in the accompanying paper by Biggins (pp. 62–84), which is written in terms of discrete-time processes, can be extended to deal also with continuous-time processes.)

On Wald's equations in continuous time

1970 ◽  
Vol 7 (1) ◽  
pp. 59-68 ◽  
Author(s):  
W. J. Hall
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Stopping Time ◽  
Wiener Processes ◽  
Independent Increments ◽  
Randomly Stopped Sums ◽  
Continuous Time Processes ◽  
Wald's Identity ◽  
Stopped Sums ◽  
Inequality Theorem

Various formulas of Wald relating to randomly stopped sums have well known continuous-time analogs, holding in particular for Wiener processes. However, sufficiently general forms of most of these do not appear explicitly in the literature. Recent papers by Robbins and Samuel (1966) and by Brown (1969) provide general results on Wald's equations in discrete time and these are here extended (Theorems 2 and 3) to Wiener processes and other homogeneous additive processes, that is, continuous-time processes with stationary independent increments. We also give an inequality (Theorem 1) related to Wald's identity in continuous time, and we derive, as corollaries of Wald's equations, bounds on the variance of an arbitrary stopping time. The Wiener process versions of these results find application in a variety of stopping problems. Specifically, all are used in Hall ((1968), (1969)); see also Bechhofer, Kiefer, and Sobel (1968), Root (1969), and Shepp (1967).

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.

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