scholarly journals Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

2014 ◽  
Vol 51 (03) ◽  
pp. 699-712 ◽  
Author(s):  
Lingjiong Zhu
Keyword(s):  
Stochastic Process ◽  
Large Deviations ◽  
Central Limit ◽  
Limit Theorems ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Laplace Transforms ◽  
Hawkes Process ◽  
Large Numbers ◽  
Special Case

In this paper we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. We obtain Laplace transforms and limit theorems, including the law of large numbers, central limit theorems, and large deviations.

scholarly journals Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

2014 ◽  
Vol 51 (3) ◽  
pp. 699-712 ◽  
Author(s):  
Lingjiong Zhu
Keyword(s):  
Stochastic Process ◽  
Large Deviations ◽  
Central Limit ◽  
Limit Theorems ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Laplace Transforms ◽  
Hawkes Process ◽  
Large Numbers ◽  
Special Case

In this paper we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. We obtain Laplace transforms and limit theorems, including the law of large numbers, central limit theorems, and large deviations.

scholarly journals A Note on Rényi's ‘Record’ Problem and Engel's Series

2018 ◽  
Vol 61 (2) ◽  
pp. 363-369 ◽  
Author(s):  
Lulu Fang ◽  
Min Wu
Keyword(s):  
Number Theory ◽  
Large Deviations ◽  
Markov Processes ◽  
Classical Limit ◽  
Limit Theorems ◽  
London Math ◽  
Law Of Large Numbers ◽  
Record Times ◽  
Iterated Logarithm ◽  
Large Numbers

AbstractIn 1973, Williams [D. Williams, On Rényi's ‘record’ problem and Engel's series, Bull. London Math. Soc.5 (1973), 235–237] introduced two interesting discrete Markov processes, namely C-processes and A-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between C-processes and A-processes in the context of large deviations.

scholarly journals Order Book Dynamics with Liquidity Fluctuations: Limit Theorems and Large Deviations

2021 ◽  
Author(s):  
Helder Rojas ◽  
Anatoly Yambartsev ◽  
Artem Logachov
Keyword(s):  
Large Deviations ◽  
Stochastic Models ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Limit Order Book ◽  
Order Book ◽  
Limit Order ◽  
Bid Ask Spread ◽  
Large Numbers ◽  
The One

We propose a class of stochastic models for a dynamics of limit order book with different type of liquidities. Within this class of models we study the one where a spread decreases uniformly, belonging to the class of processes known as a population processes with uniform catastrophes. The law of large numbers (LLN), central limit theorem (CLT) and large deviations (LD) are proved for our model with uniform catastrophes. Our results allow us to satisfactorily explain the volatility and local trends in the prices, relevant empirical characteristics that are observed in this type of markets. Furthermore, it shows us how these local trends and volatility are determined by the typical values of the bid-ask spread. In addition, we use our model to show how large deviations occur in the spread and prices, such as those observed in flash crashes.

The Hawkes Process with Different Exciting Functions and its Asymptotic Behavior

2015 ◽  
Vol 52 (01) ◽  
pp. 37-54 ◽  
Author(s):  
Raúl Fierro ◽  
Víctor Leiva ◽  
Jesper Møller
Keyword(s):  
Asymptotic Behavior ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Hawkes Process ◽  
Homogeneous Poisson Process ◽  
Large Numbers ◽  
Main Shocks

The standard Hawkes process is constructed from a homogeneous Poisson process and uses the same exciting function for different generations of offspring. We propose an extension of this process by considering different exciting functions. This consideration may be important in a number of fields; e.g. in seismology, where main shocks produce aftershocks with possibly different intensities. The main results are devoted to the asymptotic behavior of this extension of the Hawkes process. Indeed, a law of large numbers and a central limit theorem are stated. These results allow us to analyze the asymptotic behavior of the process when unpredictable marks are considered.

scholarly journals The Hawkes Process with Different Exciting Functions and its Asymptotic Behavior

2015 ◽  
Vol 52 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Raúl Fierro ◽  
Víctor Leiva ◽  
Jesper Møller
Keyword(s):  
Asymptotic Behavior ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Hawkes Process ◽  
Homogeneous Poisson Process ◽  
Large Numbers ◽  
Main Shocks

The standard Hawkes process is constructed from a homogeneous Poisson process and uses the same exciting function for different generations of offspring. We propose an extension of this process by considering different exciting functions. This consideration may be important in a number of fields; e.g. in seismology, where main shocks produce aftershocks with possibly different intensities. The main results are devoted to the asymptotic behavior of this extension of the Hawkes process. Indeed, a law of large numbers and a central limit theorem are stated. These results allow us to analyze the asymptotic behavior of the process when unpredictable marks are considered.

Limit theorems for a supercritical Poisson random indexed branching process

2016 ◽  
Vol 53 (1) ◽  
pp. 307-314 ◽  
Author(s):  
Zhenlong Gao ◽  
Yanhua Zhang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Limit Theorems ◽  
Branching Process ◽  
Law Of Large Numbers ◽  
Moderate Deviation ◽  
Large Numbers

Abstract Let {Zn, n = 0, 1, 2, . . .} be a supercritical branching process, {Nt, t ≥ 0} be a Poisson process independent of {Zn, n = 0, 1, 2, . . .}, then {ZNt, t ≥ 0} is a supercritical Poisson random indexed branching process. We show a law of large numbers, central limit theorem, and large and moderate deviation principles for log ZNt.

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