Periodic Ornstein-Uhlenbeck processes driven by Lévy processes

In this paper, the class of periodic Ornstein-Uhlenbeck processes is defined. It is shown that periodic Ornstein-Uhlenbeck processes are stationary Markov random fields and the class of stationary distributions is characterized. In particular, any self-decomposable distribution is the stationary distribution of some periodic Ornstein-Uhlenbeck process. As examples, gamma periodic Ornstein-Uhlenbeck processes and Gaussian periodic Ornstein-Uhlenbeck processes are considered.

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Markov Chains in Many Dimensions

Advances in Applied Probability ◽

10.2307/1427819 ◽

1994 ◽

Vol 26 (3) ◽

pp. 756-774 ◽

Cited By ~ 9

Author(s):

Dimitris N. Politis

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Maximum Entropy ◽

Stationary Distribution ◽

Random Fields ◽

Special Class ◽

Markov Random Fields ◽

One Dimension ◽

Stationary Markov Chain ◽

Markov Random

A generalization of the notion of a stationary Markov chain in more than one dimension is proposed, and is found to be a special class of homogeneous Markov random fields. Stationary Markov chains in many dimensions are shown to possess a maximum entropy property, analogous to the corresponding property for Markov chains in one dimension. In addition, a representation of Markov chains in many dimensions is provided, together with a method for their generation that converges to their stationary distribution.

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Generalized fractional Lévy processes with fractional Brownian motion limit

Advances in Applied Probability ◽

10.1017/s000186780004903x ◽

2015 ◽

Vol 47 (04) ◽

pp. 1108-1131 ◽

Cited By ~ 1

Author(s):

Claudia Klüppelberg ◽

Muneya Matsui

Keyword(s):

Brownian Motion ◽

Fractional Brownian Motion ◽

Lévy Processes ◽

Sample Path ◽

Levy Processes ◽

Order Structure ◽

Uhlenbeck Process ◽

Sample Paths ◽

Volatility Models ◽

Ornstein Uhlenbeck Process

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernel to a regularly varying function. We call the resulting stochastic processes generalized fractional Lévy processes (GFLPs) and show that they may have short or long memory increments and that their sample paths may have jumps or not. Moreover, we define stochastic integrals with respect to a GFLP and investigate their second-order structure and sample path properties. A specific example is the Ornstein-Uhlenbeck process driven by a time-scaled GFLP. We prove a functional central limit theorem for such scaled processes with a fractional Ornstein-Uhlenbeck process as a limit process. This approximation applies to a wide class of stochastic volatility models, which include models where possibly neither the data nor the latent volatility process are semimartingales.

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Markov Chains in Many Dimensions

Advances in Applied Probability ◽

10.1017/s0001867800026537 ◽

1994 ◽

Vol 26 (03) ◽

pp. 756-774 ◽

Cited By ~ 1

Author(s):

Dimitris N. Politis

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Maximum Entropy ◽

Stationary Distribution ◽

Random Fields ◽

Special Class ◽

Markov Random Fields ◽

One Dimension ◽

Stationary Markov Chain ◽

Markov Random

A generalization of the notion of a stationary Markov chain in more than one dimension is proposed, and is found to be a special class of homogeneous Markov random fields. Stationary Markov chains in many dimensions are shown to possess a maximum entropy property, analogous to the corresponding property for Markov chains in one dimension. In addition, a representation of Markov chains in many dimensions is provided, together with a method for their generation that converges to their stationary distribution.

Download Full-text

On association and other forms of positive dependence for Feller processes

Journal of Applied Probability ◽

10.1017/jpr.2019.36 ◽

2019 ◽

Vol 56 (2) ◽

pp. 624-646

Author(s):

Eddie Tu

Keyword(s):

State Space ◽

Lévy Processes ◽

Levy Processes ◽

Poisson Processes ◽

Jump Processes ◽

Uhlenbeck Process ◽

Positive Dependence ◽

Positive Orthant ◽

Ornstein Uhlenbeck Process ◽

Lévy Measures

AbstractWe characterize various forms of positive dependence, such as association, positive supermodular association and dependence, and positive orthant dependence, for jump-Feller processes. Such jump processes can be studied through their state-space dependent Lévy measures. It is through these Lévy measures that we will provide our characterization. Finally, we present applications of these results to stochastically monotone Feller processes, including Lévy processes, the Ornstein–Uhlenbeck process, pseudo-Poisson processes, and subordinated Feller processes.

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Generalized fractional Lévy processes with fractional Brownian motion limit

Advances in Applied Probability ◽

10.1239/aap/1449859802 ◽

2015 ◽

Vol 47 (4) ◽

pp. 1108-1131 ◽

Cited By ~ 5

Author(s):

Claudia Klüppelberg ◽

Muneya Matsui

Keyword(s):

Brownian Motion ◽

Fractional Brownian Motion ◽

Lévy Processes ◽

Sample Path ◽

Levy Processes ◽

Order Structure ◽

Uhlenbeck Process ◽

Sample Paths ◽

Volatility Models ◽

Ornstein Uhlenbeck Process

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernel to a regularly varying function. We call the resulting stochastic processes generalized fractional Lévy processes (GFLPs) and show that they may have short or long memory increments and that their sample paths may have jumps or not. Moreover, we define stochastic integrals with respect to a GFLP and investigate their second-order structure and sample path properties. A specific example is the Ornstein-Uhlenbeck process driven by a time-scaled GFLP. We prove a functional central limit theorem for such scaled processes with a fractional Ornstein-Uhlenbeck process as a limit process. This approximation applies to a wide class of stochastic volatility models, which include models where possibly neither the data nor the latent volatility process are semimartingales.

Download Full-text