Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions

Lebovits, Joachim; Véhel, Jacques Lévy; Herbin, Érick

doi:10.1016/j.spa.2013.09.004

Cited by 12 publications

(8 citation statements)

References 14 publications

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“…ST is normally distributed since the integral of a deterministic function with respect to Brownian is Gaussian (Lebovits (2014), Shahnazi et al (2021; The expectation of ST is given by ( 7):…”

Section: Statistical Properties Of Ornstein Uhlenbeck Modelsmentioning

confidence: 99%

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Modification of the Ornstein Uhlenbeck Process to Incorporate the Influence of Speculation on Volatility in Financial Markets

Hendricks¹,

Ong'ala²,

Ntirampeba³

2022

Journal of Mathematics and Statistics

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Financial market participants often speculate on how markets would behave in the light of certain information at hand. This speculation contributes to volatility within the financial market and consequently, it makes the market unstable. The Ornstein Uhlenbeck (OU) model has intensively been used in modelling volatility, however, the contribution of speculation on volatility has not been studied in the OU model. Therefore, this study focuses on the modification of the OU model by incorporating a time dependent exponential function that caters for the contribution of speculation on volatility. The statistical properties of the Improved OU model are then studied and the results compared with properties of the OU model. NAD/USD exchange rate data is used to compare and validate the Improved model with the OU model. It was found that both the OU and Improved OU model had a similar expected price, while variance of price for the OU model stabilised upwards up to 16 and variance of price for the Improved OU model stabilised downwards up to 0.01. The variance of the Improved model was found to be much lower than that of the OU model. Additionally, it was found that the distribution of the forecasted price changed with different lead times for the OU model whereas, the distribution of the forecasted price for the Improved OU model did not change with different lead times. Thus, the OU model is a time specific model whereas the Improved OU model is an invariant time model. Consequently, the Improved OU model was found to be more efficient than the OU model.

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Section: Statistical Properties Of Ornstein Uhlenbeck Modelsmentioning

confidence: 99%

“…ST is normally distributed since the integral of the brownian motion part is Gaussian (Lebovits (2014), Shahnazi et al (2021). To find the mean of ST, we take the expectation on both sides of ( 23…”

Section: S S E E E E K T T Dw E E K T T Dw R E K T T Dwmentioning

confidence: 99%

Modification of the Ornstein Uhlenbeck Process to Incorporate the Influence of Speculation on Volatility in Financial Markets

Hendricks¹,

Ong'ala²,

Ntirampeba³

2022

Journal of Mathematics and Statistics

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“…Wick product is a significant phenomena in stochastic and fractional stochastic calculus, mostly applied in the solutions of nonlinear differential equations in financial mathematics and fluid mechanics. A detailed treatment to Wick prod- uct and its applications can be seen in Kaligotla et al [34] , Duncan et al [35] , Levajkovic et al [36] , Parczewski [37] , Biagini et al [38] , Lebovits et al [39] , Kim et al [49] . In this paper, we employ the Wick product in the fractional Brownian motion settings to represent fractional white noise.…”

Section: Sirs Model With Fractional Brownian Motionmentioning

confidence: 99%

Solutions of a disease model with fractional white noise

Akinlar

Gómez‐Aguilar

Boutarfa

2020

Chaos, Solitons & Fractals

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Elsevier has created a COVID-19 resource centre with free information in English and Mandarin on the novel coronavirus COVID-19. The COVID-19 resource centre is hosted on Elsevier Connect, the company's public news and information website. Elsevier hereby grants permission to make all its COVID-19-related research that is available on the COVID-19 resource centre-including this research content-immediately available in PubMed Central and other publicly funded repositories, such as the WHO COVID database with rights for unrestricted research re-use and analyses in any form or by any means with acknowledgement of the original source. These permissions are granted for free by Elsevier for as long as the COVID-19 resource centre remains active.

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“…We summarize here the minimum background on White Noise Theory, written e.g. in [LLVH14,. More precisely, let (L 2 ) denote the space L 2 (Ω, G, µ), where G is the σ-field generated by (< ., f >) f ∈L 2 (R) .…”

Section: The Spaces Of Stochastic Test Functions and Stochastic Distrmentioning

confidence: 99%

Stochastic Calculus with Respect to Gaussian Processes

Lebovits

2017

Potential Anal

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The aim of this work is to define and perform a study of local times of all Gaussian processes that have an integral representation over a real interval (that maybe infinite). Very rich, this class of Gaussian processes, contains Volterra processes (and thus fractional Brownian motion), multifractional Brownian motions as well as processes, the regularity of which varies along the time. Using the White Noise-based anticipative stochastic calculus with respect to Gaussian processes developed in [Leb17], we first establish a Tanaka formula. This allows us to define both weighted and non-weighted local times and finally to provide occupation time formulas for both these local times. A complete comparison of the Tanaka formula as well as the results on Gaussian local times we present here, is made with the ones proposed in [MV05, LN12, SV14].

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Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions

Cited by 12 publications

References 14 publications

Modification of the Ornstein Uhlenbeck Process to Incorporate the Influence of Speculation on Volatility in Financial Markets

Modification of the Ornstein Uhlenbeck Process to Incorporate the Influence of Speculation on Volatility in Financial Markets

Solutions of a disease model with fractional white noise

Stochastic Calculus with Respect to Gaussian Processes

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