A multi-dimensional martingale for Markov additive processes and its applications

2000 ◽  
Vol 32 (2) ◽  
pp. 376-393 ◽  
Author(s):  
Søren Asmussen ◽  
Offer Kella
Keyword(s):  
Brownian Motion ◽  
Lévy Processes ◽  
Levy Processes ◽  
One Dimensional ◽  
Additive Processes ◽  
Motion Models ◽  
And Brownian Motion ◽  
Storage Processes ◽  
Reflecting Barriers ◽  
Markov Additive Processes

We establish new multidimensional martingales for Markov additive processes and certain modifications of such processes (e.g., such processes with reflecting barriers). These results generalize corresponding one-dimensional martingale results for Lévy processes. This martingale is then applied to various storage processes, queues and Brownian motion models.

A multi-dimensional martingale for Markov additive processes and its applications

2000 ◽  
Vol 32 (02) ◽  
pp. 376-393 ◽  
Author(s):  
Søren Asmussen ◽  
Offer Kella
Keyword(s):  
Brownian Motion ◽  
Lévy Processes ◽  
Levy Processes ◽  
One Dimensional ◽  
Additive Processes ◽  
Motion Models ◽  
And Brownian Motion ◽  
Storage Processes ◽  
Reflecting Barriers ◽  
Markov Additive Processes

We establish new multidimensional martingales for Markov additive processes and certain modifications of such processes (e.g., such processes with reflecting barriers). These results generalize corresponding one-dimensional martingale results for Lévy processes. This martingale is then applied to various storage processes, queues and Brownian motion models.

scholarly journals Potential Measures of One-Sided Markov Additive Processes with Reflecting and Terminating Barriers

2014 ◽  
Vol 51 (04) ◽  
pp. 1154-1170 ◽  
Author(s):  
Jevgenijs Ivanovs
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Additive Process ◽  
Markov Additive Process ◽  
Additive Processes ◽  
Quasistationary Distributions ◽  
Markov Additive Processes

Consider a one-sided Markov additive process with an upper and a lower barrier, where each can be either reflecting or terminating. For both defective and nondefective processes, and all possible scenarios, we identify the corresponding potential measures, which help to generalize a number of results for one-sided Lévy processes. The resulting rather neat formulae have various applications in risk and queueing theories, and, in particular, they lead to quasistationary distributions of the corresponding processes.

scholarly journals Potential Measures of One-Sided Markov Additive Processes with Reflecting and Terminating Barriers

2014 ◽  
Vol 51 (04) ◽  
pp. 1154-1170 ◽  
Author(s):  
Jevgenijs Ivanovs
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Additive Process ◽  
Markov Additive Process ◽  
Additive Processes ◽  
Quasistationary Distributions ◽  
Markov Additive Processes

Consider a one-sided Markov additive process with an upper and a lower barrier, where each can be either reflecting or terminating. For both defective and nondefective processes, and all possible scenarios, we identify the corresponding potential measures, which help to generalize a number of results for one-sided Lévy processes. The resulting rather neat formulae have various applications in risk and queueing theories, and, in particular, they lead to quasistationary distributions of the corresponding processes.

scholarly journals Potential Measures of One-Sided Markov Additive Processes with Reflecting and Terminating Barriers

2014 ◽  
Vol 51 (4) ◽  
pp. 1154-1170 ◽  
Author(s):  
Jevgenijs Ivanovs
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Additive Process ◽  
Markov Additive Process ◽  
Additive Processes ◽  
Quasistationary Distributions ◽  
Markov Additive Processes

Consider a one-sided Markov additive process with an upper and a lower barrier, where each can be either reflecting or terminating. For both defective and nondefective processes, and all possible scenarios, we identify the corresponding potential measures, which help to generalize a number of results for one-sided Lévy processes. The resulting rather neat formulae have various applications in risk and queueing theories, and, in particular, they lead to quasistationary distributions of the corresponding processes.

Generalized fractional Lévy processes with fractional Brownian motion limit

2015 ◽  
Vol 47 (04) ◽  
pp. 1108-1131 ◽  
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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