scholarly journals LAMN property for the drift and volatility parameters of a sde driven by a stable Lévy process

2019 ◽  
Vol 23 ◽  
pp. 136-175 ◽  
Author(s):  
Emmanuelle Clément ◽  
Arnaud Gloter ◽  
Huong Nguyen
Keyword(s):  
Stable Process ◽  
Transition Density ◽  
Fixed Time ◽  
Small Time ◽  
Time Interval ◽  
Lévy Measure ◽  
Stable Lévy Process ◽  
Continuous Time Process ◽  
Asymptotic Mixed Normality ◽  
Drift Coefficient

This work focuses on the local asymptotic mixed normality (LAMN) property from high frequency observations, of a continuous time process solution of a stochastic differential equation driven by a truncated α-stable process with index α ∈ (0, 2). The process is observed on the fixed time interval [0,1] and the parameters appear in both the drift coefficient and scale coefficient. This extends the results of Clément and Gloter [Stoch. Process. Appl. 125 (2015) 2316–2352] where the index α ∈ (1, 2) and the parameter appears only in the drift coefficient. We compute the asymptotic Fisher information and find that the rate in the LAMN property depends on the behavior of the Lévy measure near zero. The proof relies on the small time asymptotic behavior of the transition density of the process obtained in Clément et al. [Preprint HAL-01410989v2 (2017)].

scholarly journals Asymptotics in small time for the density of a stochastic differential equation driven by a stable Lévy process

2018 ◽  
Vol 22 ◽  
pp. 58-95
Author(s):  
Emmanuelle Clément ◽  
Arnaud Gloter ◽  
Huong Nguyen
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Asymptotic Behavior ◽  
Stochastic Differential Equation ◽  
Malliavin Calculus ◽  
Stable Process ◽  
Small Time ◽  
Stable Lévy Process ◽  
Asymptotic Mixed Normality ◽  
Drift Coefficient

This work focuses on the asymptotic behavior of the density in small time of a stochastic differential equation driven by a truncated α-stable process with index α ∈ (0, 2). We assume that the process depends on a parameter β = (θ, σ)T and we study the sensitivity of the density with respect to this parameter. This extends the results of [E. Clément and A. Gloter, Local asymptotic mixed normality property for discretely observed stochastic dierential equations driven by stable Lévy processes. Stochastic Process. Appl. 125 (2015) 2316–2352.] which was restricted to the index α ∈ (1, 2) and considered only the sensitivity with respect to the drift coefficient. By using Malliavin calculus, we obtain the representation of the density and its derivative as an expectation and a conditional expectation. This permits to analyze the asymptotic behavior in small time of the density, using the time rescaling property of the stable process.

Linear impulsive control system with impulse time windows

2016 ◽  
Vol 23 (1) ◽  
pp. 111-118 ◽  
Author(s):  
Yuming Feng ◽  
Junzhi Yu ◽  
Chuandong Li ◽  
Tingwen Huang ◽  
Hangjun Che
Keyword(s):  
Impulsive Control ◽  
Time Window ◽  
Time Windows ◽  
Practical Importance ◽  
Fixed Time ◽  
Small Time ◽  
Time Interval ◽  
Impulsive Systems ◽  
Impulse Time Windows ◽  
System States

We formulate the linear impulsive control systems with impulse time windows. Different from the most impulsive systems where the impulses occur at fixed time or when the system states hit a certain hyperplane, the impulse time in the presented systems might be uncertain, but limited to a small time interval, i.e. a time window. Compared with the existing impulsive systems, the systems with impulse time windows is of practical importance. We then study the asymptotic stability of the case of linear systems and obtain several stability criteria. Numerical examples are given to verify the effectiveness of the theoretical results.

Utilization of remote sensing for estimation of scale of cutting and growing of forest zones

2017 ◽  
Vol 920 (2) ◽  
pp. 57-60
Author(s):  
F.E. Guliyeva
Keyword(s):  
Remote Sensing ◽  
Fixed Time ◽  
Time Interval ◽  
Time Points ◽  
Average Value ◽  
Time Period ◽  
Remote Estimation ◽  
The Given ◽  
Forest Cutting

The study of results of relevant works on remote sensing of forests has shown that the known methods of remote estimation of forest cuts and growth don’t allow to calculate the objective average value of forests cut volume during the fixed time period. The existing mathematical estimates are not monotonous and make it possible to estimate primitively the scale of cutting by computing the ratio of data in two fixed time points. In the article the extreme properties of the considered estimates for deforestation and reforestation models are researched. The extreme features of integrated averaged values of given estimates upon limitations applied on variables, characterizing the deforestation and reforestation processes are studied. The integrated parameter, making it possible to calculate the averaged value of estimates of forest cutting, computed for all fixed time period with a fixed step is suggested. It is shown mathematically that the given estimate has a monotonous feature in regard of value of given time interval and make it possible to evaluate objectively the scales of forest cutting.

Tail asymptotics of an infinitely divisible space-time model with convolution equivalent Lévy measure

2021 ◽  
Vol 58 (1) ◽  
pp. 42-67 ◽  
Author(s):  
Mads Stehr ◽  
Anders Rønn-Nielsen
Keyword(s):  
Space Time ◽  
Time Interval ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Time Model ◽  
Spatial Object ◽  
Tail Behaviour ◽  
Fixed Radius ◽  
The Right ◽  
Space Time Model

AbstractWe consider a space-time random field on ${{\mathbb{R}^d} \times {\mathbb{R}}}$ given as an integral of a kernel function with respect to a Lévy basis with a convolution equivalent Lévy measure. The field obeys causality in time and is thereby not continuous along the time axis. For a large class of such random fields we study the tail behaviour of certain functionals of the field. It turns out that the tail is asymptotically equivalent to the right tail of the underlying Lévy measure. Particular examples are the asymptotic probability that there is a time point and a rotation of a spatial object with fixed radius, in which the field exceeds the level x, and that there is a time interval and a rotation of a spatial object with fixed radius, in which the average of the field exceeds the level x.

Bayesian Nonparametric Estimation of Failure Rates with Censored Data

1983 ◽  
Vol 32 (1-2) ◽  
pp. 79-90 ◽  
Author(s):  
J. S. Rao ◽  
R. C. Tiwari
Keyword(s):  
Failure Time ◽  
Dirichlet Distribution ◽  
Survival Function ◽  
Fixed Time ◽  
Bayes Estimation ◽  
Time Interval ◽  
Failure Rates ◽  
Failure Time Distribution ◽  
Accelerated Life ◽  
Bayesian Nonparametric Estimation

The failure time distribution is estimated in the nonparametric context when some of tbe observations arc censored. The time interval is partitioned into fixed class intervals, and number of failures and number censored in each of these intervals are observed. Using a Dirichlet distribution as the prior, the resulting estimates of the survival function and the failure rate have a nice and simple form. If instead of the fixed time intervals, one uses the “natural” intervals formed by the observed failure times, this gives essentially the same results as in Ferauson IUld Phadia (1977), Susarla and Van Ryzin (1976), but in a much simpler way. Bayes estimation under the increasins and decreasing failure rates is also considered, and applications to accelerated life testing are discussed.

scholarly journals Improved bounds for the availability and unavailability in a fixed time interval for systems of maintained, interdependent components

1980 ◽  
Vol 12 (01) ◽  
pp. 200-221 ◽  
Author(s):  
B. Natvig
Keyword(s):  
Lower Bound ◽  
System Reliability ◽  
Nuclear Reactors ◽  
Random Variables ◽  
Fixed Time ◽  
Time Interval ◽  
Coherent System ◽  
Exact Results ◽  
Fixed Time Interval ◽  
Special Case

In this paper we arrive at a series of bounds for the availability and unavailability in the time interval I = [t A , t B ] ⊂ [0, ∞), for a coherent system of maintained, interdependent components. These generalize the minimal cut lower bound for the availability in [0, t] given in Esary and Proschan (1970) and also most bounds for the reliability at time t given in Bodin (1970) and Barlow and Proschan (1975). In the latter special case also some new improved bounds are given. The bounds arrived at are of great interest when trying to predict the performance process of the system. In particular, Lewis et al. (1978) have revealed the great need for adequate tools to treat the dependence between the random variables of interest when considering the safety of nuclear reactors. Satyanarayana and Prabhakar (1978) give a rapid algorithm for computing exact system reliability at time t. This can also be used in cases where some simpler assumptions on the dependence between the components are made. It seems, however, impossible to extend their approach to obtain exact results for the cases treated in the present paper.

A PROPOSED FORMULA FOR HORIZONTAL REVENUE ALLOCATION IN NIGERIA

2012 ◽  
Vol 29 (06) ◽  
pp. 1250033
Author(s):  
VIRTUE U. EKHOSUEHI ◽  
AUGUSTINE A. OSAGIEDE
Keyword(s):  
Optimal Control ◽  
Federal Government ◽  
Optimal Control Theory ◽  
Fixed Time ◽  
Tax Revenue ◽  
Time Interval ◽  
Fixed Time Interval ◽  
Hypothetical Data ◽  
Revenue Allocation ◽  
The Individual

In this study, we have applied optimal control theory to determine the optimum value of tax revenues accruing to a state given the range of budgeted expenditure on enforcing tax laws and awareness creation on the payment of the correct tax. This is achieved by maximizing the state's net tax revenue over a fixed time interval subject to certain constraints. By assuming that the satisfaction derived by the Federal Government of Nigeria on the ability of the individual states to generate tax revenue which is as near as the optimum tax revenue (via the state's control problem) is described by the logarithmic form of the Cobb–Douglas utility function, a formula for horizontal revenue allocation in Nigeria in its raw form is derived. Afterwards, we illustrate the use of the proposed horizontal revenue allocation formula using hypothetical data.

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