Discretization of distributions in the maximum domain of attraction

Extremes ◽  
2011 ◽  
Vol 15 (3) ◽  
pp. 299-317 ◽  
Author(s):  
Takaaki Shimura
Keyword(s):  
Domain Of Attraction ◽  
Maximum Domain Of Attraction

scholarly journals Maxima of stochastic processes driven by fractional Brownian motion

2005 ◽  
Vol 37 (03) ◽  
pp. 743-764 ◽  
Author(s):  
Boris Buchmann ◽  
Claudia Klüppelberg
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
State Space ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Maximum Domain Of Attraction

We study stationary processes given as solutions to stochastic differential equations driven by fractional Brownian motion. This rich class includes the fractional Ornstein-Uhlenbeck process and those processes that can be obtained from it by state space transformations. An explicit formula in terms of Euler's Γ-function describes the asymptotic behaviour of the covariance function of the fractional Ornstein-Uhlenbeck process near zero, which, by an application of Berman's condition, guarantees that this process is in the maximum domain of attraction of the Gumbel distribution. Necessary and sufficient conditions on the state space transforms are stated to classify the maximum domain of attraction of solutions to stochastic differential equations driven by fractional Brownian motion.

scholarly journals The Finite-Time Ruin Probability with Dependent Insurance and Financial Risks

2011 ◽  
Vol 48 (04) ◽  
pp. 1035-1048 ◽  
Author(s):  
Yiqing Chen
Keyword(s):  
Finite Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Domain Of Attraction ◽  
Random Variable ◽  
Distribution Functions ◽  
Financial Risks ◽  
Finite Time Ruin Probability ◽  
Special Cases ◽  
Maximum Domain Of Attraction

Consider a discrete-time insurance risk model. Within periodi, the net insurance loss is denoted by a real-valued random variableXi. The insurer makes both risk-free and risky investments, leading to an overall stochastic discount factorYifrom timeito timei− 1. Assume that (Xi,Yi),i∈N, form a sequence of independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution with marginal distribution functionsFandG. WhenFis subexponential andGfulfills some constraints in order for the product convolution ofFandGto be subexponential too, we derive a general asymptotic formula for the finite-time ruin probability. Then, for special cases in whichFbelongs to the Fréchet or Weibull maximum domain of attraction, we improve this general formula to be transparent.

scholarly journals The Finite-Time Ruin Probability with Dependent Insurance and Financial Risks

2011 ◽  
Vol 48 (4) ◽  
pp. 1035-1048 ◽  
Author(s):  
Yiqing Chen
Keyword(s):  
Finite Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Domain Of Attraction ◽  
Random Variable ◽  
Distribution Functions ◽  
Financial Risks ◽  
Finite Time Ruin Probability ◽  
Special Cases ◽  
Maximum Domain Of Attraction

Consider a discrete-time insurance risk model. Within period i, the net insurance loss is denoted by a real-valued random variable Xi. The insurer makes both risk-free and risky investments, leading to an overall stochastic discount factor Yi from time i to time i − 1. Assume that (Xi, Yi), i ∈ N, form a sequence of independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution with marginal distribution functions F and G. When F is subexponential and G fulfills some constraints in order for the product convolution of F and G to be subexponential too, we derive a general asymptotic formula for the finite-time ruin probability. Then, for special cases in which F belongs to the Fréchet or Weibull maximum domain of attraction, we improve this general formula to be transparent.

scholarly journals Maxima of stochastic processes driven by fractional Brownian motion

2005 ◽  
Vol 37 (3) ◽  
pp. 743-764 ◽  
Author(s):  
Boris Buchmann ◽  
Claudia Klüppelberg
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
State Space ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Maximum Domain Of Attraction

We study stationary processes given as solutions to stochastic differential equations driven by fractional Brownian motion. This rich class includes the fractional Ornstein-Uhlenbeck process and those processes that can be obtained from it by state space transformations. An explicit formula in terms of Euler's Γ-function describes the asymptotic behaviour of the covariance function of the fractional Ornstein-Uhlenbeck process near zero, which, by an application of Berman's condition, guarantees that this process is in the maximum domain of attraction of the Gumbel distribution. Necessary and sufficient conditions on the state space transforms are stated to classify the maximum domain of attraction of solutions to stochastic differential equations driven by fractional Brownian motion.

scholarly journals MIXED CAUSAL-NONCAUSAL AR PROCESSES AND THE MODELLING OF EXPLOSIVE BUBBLES

2019 ◽  
Vol 35 (6) ◽  
pp. 1234-1270 ◽  
Author(s):  
Sébastien Fries ◽  
Jean-Michel Zakoian
Keyword(s):  
Conditional Distribution ◽  
Stable Distribution ◽  
Financial Time Series ◽  
Domain Of Attraction ◽  
Real Data ◽  
Autoregressive Processes ◽  
Financial Time ◽  
Causal Representation ◽  
Heavy Tailed ◽  
Non Gaussian

Noncausal autoregressive models with heavy-tailed errors generate locally explosive processes and, therefore, provide a convenient framework for modelling bubbles in economic and financial time series. We investigate the probability properties of mixed causal-noncausal autoregressive processes, assuming the errors follow a stable non-Gaussian distribution. Extending the study of the noncausal AR(1) model by Gouriéroux and Zakoian (2017), we show that the conditional distribution in direct time is lighter-tailed than the errors distribution, and we emphasize the presence of ARCH effects in a causal representation of the process. Under the assumption that the errors belong to the domain of attraction of a stable distribution, we show that a causal AR representation with non-i.i.d. errors can be consistently estimated by classical least-squares. We derive a portmanteau test to check the validity of the estimated AR representation and propose a method based on extreme residuals clustering to determine whether the AR generating process is causal, noncausal, or mixed. An empirical study on simulated and real data illustrates the potential usefulness of the results.

Regional stabilization and H∞ congestion control with input saturation

2021 ◽  
pp. 014233122199273
Author(s):  
Sadek Belamfedel Alaoui ◽  
El Houssaine Tissir ◽  
Noreddine Chaibi ◽  
Fatima El Haoussi
Keyword(s):  
Linear Systems ◽  
Controller Design ◽  
Input Saturation ◽  
Domain Of Attraction ◽  
Queue Management ◽  
Variable Delay ◽  
Challenging Problem ◽  
Regional Stabilization ◽  
Small Gain ◽  
Improved Performance

Designing robust active queue management subjected to network imperfections is a challenging problem. Motivated by this topic, we addressed the problem of controller design for linear systems with variable delay and unsymmetrical constraints by the scaled small gain theorem. We designed two mechanisms: robust enhanced proportional derivative; and robust enhanced proportional derivative subjected to input saturation. Discussion of their practical implementations along with extensive comparisons by MATLAB and NS3 illustrate the improved performance and the enlargement of the domain of attraction regarding some literature results.

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