Asymptotic expansion of the transition density of the semigroup associated to a SDE driven by Lévy noise

2020 ◽  
pp. 1-18
Author(s):  
Boubaker Smii
Keyword(s):  
Asymptotic Expansion ◽  
Series Expansion ◽  
Transition Probability ◽  
Power Series Expansion ◽  
Transition Density ◽  
Lévy Noise ◽  
Transition Probability Density ◽  
Finite Dimensional ◽  
Feynman Graphs ◽  
Levy Noise

In this work we consider a finite dimensional stochastic differential equation(SDE) driven by a Lévy noise L ( t ) = L t , t > 0. The transition probability density p t , t > 0 of the semigroup associated to the solution u t , t ⩾ 0 of the SDE is given by a power series expansion. The series expansion of p t can be re-expressed in terms of Feynman graphs and rules. We will also prove that p t , t > 0 has an asymptotic expansion in power of a parameter β > 0, and it can be given by a convergent integral. A remark on some applications will be given in this work.

Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation

1983 ◽  
Vol 20 (04) ◽  
pp. 754-765 ◽  
Author(s):  
Etsuo Isobe ◽  
Shunsuke Sato
Keyword(s):  
Differential Equation ◽  
Probability Density Function ◽  
Stochastic Differential Equation ◽  
Probability Density ◽  
Density Function ◽  
Series Expansion ◽  
Transition Probability ◽  
Transition Probability Density ◽  
Hermite Expansion ◽  
Functional Series

In this paper we deal with the Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation. The so-called Wiener kernels which appear in the functional series expansion are expressed in terms of the transition probability density function of the process.

Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation

1983 ◽  
Vol 20 (4) ◽  
pp. 754-765 ◽  
Author(s):  
Etsuo Isobe ◽  
Shunsuke Sato
Keyword(s):  
Differential Equation ◽  
Probability Density Function ◽  
Stochastic Differential Equation ◽  
Probability Density ◽  
Density Function ◽  
Series Expansion ◽  
Transition Probability ◽  
Transition Probability Density ◽  
Hermite Expansion ◽  
Functional Series

In this paper we deal with the Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation. The so-called Wiener kernels which appear in the functional series expansion are expressed in terms of the transition probability density function of the process.

Martingale solution and Markov selection of stochastic non-Newtonian fluid driven by Lévy noise

2014 ◽  
Vol 14 (02) ◽  
pp. 1350017 ◽  
Author(s):  
Jianhua Huang ◽  
Yan Zheng ◽  
Jin Li
Keyword(s):  
Newtonian Fluid ◽  
Embedding Theorem ◽  
Lévy Noise ◽  
Martingale Solution ◽  
Finite Dimensional ◽  
Skorohod Embedding ◽  
Markov Selection ◽  
Selection For ◽  
Levy Noise ◽  
Selection Of

This paper is devoted to stochastic non-Newtonian fluid driven by Lévy noise. By the tight compactness of distribution of the solution for finite-dimensional approximate in a Hilbert space, Skorohod embedding theorem and representation of martingale, the existence of the martingale solution is shown. Moreover, the procedure of the proof for the Markov selection in [J. Differential Equations250 (2011) 2737–2778] are simplified to show the existence of Markov selection for the martingale solution.

Stochastic PDEs in 𝒮' for SDEs driven by Lévy noise

2020 ◽  
Vol 28 (3) ◽  
pp. 217-226
Author(s):  
Suprio Bhar ◽  
Rajeev Bhaskaran ◽  
Barun Sarkar
Keyword(s):  
Differential Equation ◽  
Stochastic Partial Differential Equation ◽  
Lévy Noise ◽  
Invariance Property ◽  
Stochastic Pdes ◽  
Translation Invariance ◽  
Finite Dimensional ◽  
Monotonicity Inequality ◽  
Translation Operators ◽  
Levy Noise

AbstractIn this article we show that a finite-dimensional stochastic differential equation driven by a Lévy noise can be formulated as a stochastic partial differential equation (SPDE) driven by the same Lévy noise. We prove the existence result for such an SPDE by Itô’s formula for translation operators, and the uniqueness by an adapted form of “Monotonicity inequality”, proved earlier in the diffusion case. As a consequence, the solutions that we construct have the “translation invariance” property.

Metastability in a class of hyperbolic dynamical systems perturbed by heavy-tailed Lévy type noise

2015 ◽  
Vol 15 (03) ◽  
pp. 1550019 ◽  
Author(s):  
Michael Högele ◽  
Ilya Pavlyukevich
Keyword(s):  
Dynamical Systems ◽  
Limit Cycles ◽  
Lévy Noise ◽  
Random System ◽  
Finite Dimensional ◽  
Local Attractors ◽  
Heavy Tailed ◽  
Non Gaussian ◽  
Deterministic Dynamical System ◽  
Levy Noise

We consider a finite dimensional deterministic dynamical system with finitely many local attractors Kι, each of which supports a unique ergodic probability measure Pι, perturbed by a multiplicative non-Gaussian heavy-tailed Lévy noise of small intensity ε > 0. We show that the random system exhibits a metastable behavior: there exists a unique ε-dependent time scale on which the system reminds of a continuous time Markov chain on the set of the invariant measures Pι. In particular our approach covers the case of dynamical systems of Morse–Smale type, whose attractors consist of points and limit cycles, perturbed by multiplicative α-stable Lévy noise in the Itô, Stratonovich and Marcus sense. As examples we consider α-stable Lévy perturbations of the Duffing equation and Pareto perturbations of a biochemical birhythmic system with two nested limit cycles.

scholarly journals An Enhanced Intrinsic Time-Scale Decomposition Method Based on Adaptive Lévy Noise and Its Application in Bearing Fault Diagnosis

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 617
Author(s):  
Jianpeng Ma ◽  
Shi Zhuo ◽  
Chengwei Li ◽  
Liwei Zhan ◽  
Guangzhu Zhang
Keyword(s):  
Fault Diagnosis ◽  
Time Scale ◽  
Lévy Noise ◽  
Model Parameters ◽  
Rolling Bearings ◽  
Intrinsic Time ◽  
Time Scale Decomposition ◽  
Weak Fault ◽  
Levy Noise ◽  
The Impact

When early failures in rolling bearings occur, we need to be able to extract weak fault characteristic frequencies under the influence of strong noise and then perform fault diagnosis. Therefore, a new method is proposed: complete ensemble intrinsic time-scale decomposition with adaptive Lévy noise (CEITDALN). This method solves the problem of the traditional complete ensemble intrinsic time-scale decomposition with adaptive noise (CEITDAN) method not being able to filter nonwhite noise in measured vibration signal noise. Therefore, in the method proposed in this paper, a noise model in the form of parameter-adjusted noise is used to replace traditional white noise. We used an optimization algorithm to adaptively adjust the model parameters, reducing the impact of nonwhite noise on the feature frequency extraction. The experimental results for the simulation and vibration signals of rolling bearings showed that the CEITDALN method could extract weak fault features more effectively than traditional methods.

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