Effects of Lévy noise on the Fitzhugh–Nagumo model: A perspective on the maximal likely trajectories

Cai, Rui; He, Ziying; Liu, Yancai; Duan, Jinqiao; Kurths, Jürgen; Li, Xiaofan

doi:10.1016/j.jtbi.2019.08.010

Cited by 16 publications

(7 citation statements)

References 43 publications

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“…Recently, the fractional models have aroused numerous research interests in various fields, ranging from finance, 28,29 neuroscience, 30,31 physics, 32,33 and so on. [34][35][36][37][38][39] We here focus on the fractional model in option pricing.…”

Section: Monte Carlo and Finite Difference Of Fpdementioning

confidence: 99%

Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

Yuan¹,

Ding²,

Duan³

et al. 2021

Preprint

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During the COVID-19 pandemic, many institutions have announced that their counterparties are struggling to fulfill contracts. Therefore, it is necessary to consider the counterparty default risk when pricing options. After the 2008 financial crisis, a variety of value adjustments have been emphasized in the financial industry. The total value adjustment (XVA) is the sum of multiple value adjustments, which is also investigated in many stochastic models such as Heston 1 and Bates 2 models. In this work, a widely used pure jump Lévy process, the CGMY process has been considered for pricing a Bermudan option with various value adjustments. Under a pure jump Lévy process, the value of derivatives satisfies a fractional partial differential equation (FPDE). Therefore, we construct a method which combines Monte Carlo with finite difference of FPDE (MC-FF) to find the numerical approximation of exposure, and compare it with the benchmark Monte Carlo-COS (MC-COS) method. We use the discrete energy estimate method, which is different with the existing works, to derive the convergence of the numerical scheme. Based on the numerical results, the XVA is computed by the financial exposure of the derivative value.

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Section: Monte Carlo and Finite Difference Of Fpdementioning

confidence: 99%

Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

Yuan¹,

Ding²,

Duan³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Indeed, some researchers and we have recently taken these stochastic differential equations as phenomenological models for various random phenomena. Cai et al investigated the escaping phenomena of a neuron system driven by the non-Gaussian  -stable Lé vy noise to detect its excitation behaviors (22,23). Together with co-authors, we analyzed the transitions between the vegetative and the competence regions in a gene network under Lé vy noise by quantitatively computing the exit time and maximal likely transition trajectory (6,24,25).…”

Section: Introductionmentioning

confidence: 99%

A data-driven approach for discovering stochastic dynamical systems with non-Gaussian Lévy noise

Yang

Duan

2021

Physica D: Nonlinear Phenomena

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“…Based on this assertion, Zheng et al developed a probabilistic framework to investigate the maximum likelihood climate change for an energy balance system under the combined influence of greenhouse effect and non-Gaussian α-stable Lévy motions [20]. Some researchers and we studied the escape phenomena [21,22] and stochastic resonance [23,24] of a neuron model driven by the non-Gaussian noise to detect its excitation behaviors. The non-Gaussian Lévy motions are also used to characterize random fluctuations in gene networks [25][26][27], current-biased long Josephson junctions [28] and other scientific fields [29,30].…”

Section: Introductionmentioning

confidence: 99%

Extracting Governing Laws from Sample Path Data of Non-Gaussian Stochastic Dynamical Systems

Li,

Duan

2021

Preprint

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Advances in data science are leading to new progresses in the analysis and understanding of complex dynamics for systems with experimental and observational data. With numerous physical phenomena exhibiting bursting, flights, hopping, and intermittent features, stochastic differential equations with non-Gaussian Lévy noise are suitable to model these systems. Thus it is desirable and essential to infer such equations from available data to reasonably predict dynamical behaviors. In this work, we consider a data-driven method to extract stochastic dynamical systems with non-Gaussian asymmetric (rather than the symmetric) Lévy process, as well as Gaussian Brownian motion. We establish a theoretical framework and design a numerical algorithm to compute the asymmetric Lévy jump measure, drift and diffusion (i.e., nonlocal Kramers-Moyal formulas), hence obtaining the stochastic governing law, from noisy data. Numerical experiments on several prototypical examples confirm the efficacy and accuracy of this method. This method will become an effective tool in discovering the governing laws from available data sets and in understanding the mechanisms underlying complex random phenomena.

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Effects of Lévy noise on the Fitzhugh–Nagumo model: A perspective on the maximal likely trajectories

Cited by 16 publications

References 43 publications

Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

Total value adjustment of Bermudan option valuation under pure jump Lévy fluctuations

A data-driven approach for discovering stochastic dynamical systems with non-Gaussian Lévy noise

Extracting Governing Laws from Sample Path Data of Non-Gaussian Stochastic Dynamical Systems

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