Arithmetic Brownian motion subordinated by tempered stable and inverse tempered stable processes

Wyłomańska, Agnieszka

doi:10.1016/j.physa.2012.05.072

Cited by 30 publications

(31 citation statements)

References 53 publications

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“…4) indicates that the waiting times can be described by the tempered stable distribution. Therefore we assume that the sequence {T n } constitutes a sample of independent random variables from the tempered stable distribution, i.e., for each n the random variable T n has the following Laplace transform [17]:…”

Section: Continuous Time Random Walkmentioning

confidence: 99%

“…As we observe, for λ = 0 the tempered stable distribution defined by the Laplace transform in (3) becomes the α-stable random variable U . The tempered stable distribution of waiting times was considered in [17][18][19] from both theoretical and practical point of view. Moreover the rich class of tempered stable distributions was examined in [13,[20][21][22][23].…”

Section: Continuous Time Random Walkmentioning

confidence: 99%

“…Some properties of the inverse tempered stable process one can find in [27] but also in [17], where the inverse tempered stable subordinator was analyzed in comparison with the tempered stable Lévy process. Below, we recall a formula for the pdf of the inverse tempered stable subordinator [17], namely…”

Section: The Subordinated Process As a Limit Of The Ctrw Modelmentioning

confidence: 99%

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Stochastic Modeling of Indoor Air Temperature

et al. 2013

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Temperature is one of the main parameters describing thermal comfort and indoor air quality. In this paper we propose an approach, based on a modification of the continuous time random walk, to model the indoor air temperature. We perform a statistical analysis of the recorded time series, that allows us to point out the main statistical properties of the recorded variable. The obtained conclusions about the nature of the process lead to a continuous time random walk, that in contrast to the classical approach, models time dependence of the jumps distribution. Moreover, we show that the waiting times can be modeled by a tempered stable distribution, which yields a subdiffusive behavior in short times and diffusive behavior in longer times. Finally, by conducting a simulation study we illustrate possible applications of the presented approach in the thermal comfort monitoring and forecasting.

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Section: Continuous Time Random Walkmentioning

confidence: 99%

Section: Continuous Time Random Walkmentioning

confidence: 99%

Section: The Subordinated Process As a Limit Of The Ctrw Modelmentioning

confidence: 99%

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Stochastic Modeling of Indoor Air Temperature

et al. 2013

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“…It is a random variable, in contrast to the ensemble average, which is deterministic [20]. If the sample comes from an H -self-similar process with stationary increments belonging to the domain of attraction of the Lévy α-stable law, then for large n…”

Section: Methodsmentioning

confidence: 99%

Diffusive and subdiffusive dynamics of indoor microclimate: A time series modeling

Maciejewska

Szczurek²,

Sikora³

et al. 2012

Phys. Rev. E

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The indoor microclimate is an issue in modern society, where people spend about 90% of their time indoors. Temperature and relative humidity are commonly used for its evaluation. In this context, the two parameters are usually considered as behaving in the same manner, just inversely correlated. This opinion comes from observation of the deterministic components of temperature and humidity time series. We focus on the dynamics and the dependency structure of the time series of these parameters, without deterministic components. Here we apply the mean square displacement, the autoregressive integrated moving average (ARIMA), and the methodology for studying anomalous diffusion. The analyzed data originated from five monitoring locations inside a modern office building, covering a period of nearly one week. It was found that the temperature data exhibited a transition between diffusive and subdiffusive behavior, when the building occupancy pattern changed from the weekday to the weekend pattern. At the same time the relative humidity consistently showed diffusive character. Also the structures of the dependencies of the temperature and humidity data sets were different, as shown by the different structures of the ARIMA models which were found appropriate. In the space domain, the dynamics and dependency structure of the particular parameter were preserved. This work proposes an approach to describe the very complex conditions of indoor air and it contributes to the improvement of the representative character of microclimate monitoring.

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“…To model the anomalous behavior one of ways is change the real time in Brownian diffusion by inverse subordinators. The time changed process exhibits properties of anomalous diffusion [19]. Subordination of fractional Brownian motion consists of timechanging the paths of FBM by an independent subordinator.…”

Section: Introductionmentioning

confidence: 99%

Time-changed generalized mixed fractional Brownian motion and application to arithmetic average Asian option pricing

Önalan

2017

IJAMR

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In this paper we present a novel model to analyze the behavior of random asset price process under the assumption that the stock price process is governed by time-changed generalized mixed fractional Brownian motion with an inverse gamma subordinator. This model is constructed by introducing random time changes into generalized mixed fractional Brownian motion process. In practice it has been observed that many different time series have long-range dependence property and constant time periods. Fractional Brownian motion provides a very general model for long-term dependent and anomalous diffusion regimes. Motivated by this facts in this paper we investigated the long-range dependence structure and trapping events (periods of prices stay motionless) of CSCO stock price return series. The constant time periods phenomena are modeled using an inverse gamma process as a subordinator. Proposed model include the jump behavior of price process because the gamma process is a pure jump Levy process and hence the subordinated process also has jumps so our model can be capture the random variations in volatility. To show the effectiveness of proposed model, we applied the model to calculate the price of an average arithmetic Asian call option that is written on Cisco stock. In this empirical study first the statistical properties of real financial time series is investigated and then the estimated model parameters from an observed data. The results of empirical study which is performed based on the real data indicated that the results of our model are more accuracy than the results based on traditional models.

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Arithmetic Brownian motion subordinated by tempered stable and inverse tempered stable processes

Cited by 30 publications

References 53 publications

Stochastic Modeling of Indoor Air Temperature

Stochastic Modeling of Indoor Air Temperature

Diffusive and subdiffusive dynamics of indoor microclimate: A time series modeling

Time-changed generalized mixed fractional Brownian motion and application to arithmetic average Asian option pricing

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