Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise

Applebaum, David; Siakalli, Michailina

doi:10.1239/jap/1261670692

Cited by 101 publications

(16 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Define τ N = inf t : |X n (t)| p + t 0 X n (s) p ds > N as the stopping time. We have to find the p th moment estimates for the above system (for similar formulation see Theorem 4.4 of [2]). For this, let us take the function f (x) = |x| p and apply the Itô's formula (see Theorem 5.1, chapter II of [16], Theorem 4.4.7 of [1], Theorem 4.4 of [31]) to the process X n (t) to obtain,…”

Section: Hypothesis and Localmentioning

confidence: 99%

See 1 more Smart Citation

Two-dimensional magneto-hydrodynamic system with jump processes: well posedness and invariant measures

Manna¹,

Mohan²

2013

COSA

View full text Add to dashboard Cite

In this work we prove the existence and uniqueness of the strong solution to the two-dimensional stochastic magneto-hydrodynamic system perturbed by Lévy noise. The local monotonicity arguments have been exploited in the proofs. The existence of a unique invariant measures has been proved using the exponential stability of solutions.2000 Mathematics Subject Classification. Primary 60H15; Secondary 35Q35, 76D03, 76D06.

show abstract

Section: Hypothesis and Localmentioning

confidence: 99%

“…A simple calculation gives the extension of the result for the càdlàg local martingale. (σ k (t), X(s)) 2 (ε + s) 2 ds…”

Section: Invariant Measuresmentioning

confidence: 99%

Two-dimensional magneto-hydrodynamic system with jump processes: well posedness and invariant measures

Manna¹,

Mohan²

2013

COSA

View full text Add to dashboard Cite

show abstract

“…It has been a long time since stochastic differential equations (SDEs) started being applied in various areas, including biology [6], physics [4], engineering [16], finance [5]. SDEs are taken as important tools in modeling and simulating real phenomena, the stability of SDEs has been studied widely by mathematicians in different senses, such as stochastically stable, stochastically asymptotically stable, moment exponentially stable, almost surely stable, mean square polynomial stable, see [1,9,15,18]. A systematic introduction of stabilities is provided by Mao in [11].…”

Section: Introductionmentioning

confidence: 99%

“…Kobayashi also studied Itô formula driven by time-changed SDE which is provided under certain conditions as below, where f : R → R is a C 2 function. 1 In light of time-changed Itô formula, recent paper [17] analyzes the SDE driven by time-changed Brownian motion (1.4) dX(t) = k(t, E t , X(t−))dt + f (t, E t , X(t−))dE t + g(t, E t , X(t−))dB Et ,…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stability of the solution of stochastic differential equation driven by time-changed Lévy noise

Nane

2017

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

This paper studies stabilities of stochastic differential equation (SDE) driven by timechanged Lévy noise in both probability and moment sense. This provides more flexibility in modeling schemes in application areas including physics, biology, engineering, finance and hydrology. Necessary conditions for solution of time-changed SDE to be stable in different senses will be established. Connection between stability of solution to time-changed SDE and that to corresponding original SDE will be disclosed. Examples related to different stabilities will be given. We study SDEs with time-changed Lévy noise, where the time-change processes are inverse of general Lévy subordinators. These results are important improvements of the results in Wu [17].

show abstract

Stability equivalence between regime‐switching jump diffusion delayed systems and corresponding systems with piecewise continuous arguments and application to discrete‐time feedback control

Liu,

Cheng,

Cai

2024

Asian Journal of Control

View full text Add to dashboard Cite

In this paper, we mainly study the equivalence of exponential stability for regime‐switching jump diffusion delayed systems (RSJDDSs) and RSJDDSs with piecewise continuous arguments (RSJDDSs‐PCA). Our results show that if one of the RSJDDS and the RSJDDS‐PCA is th moment exponentially stable, then another system is also th moment exponentially stable when time delay and segment step size have a common upper bound, while both equations are almost surely exponentially stable, and we also provided a method to calculate this upper bound. In addition, as an application of the stability equivalence theorem, we design discrete‐time state and mode observations feedback control to stabilize unstable RSJDDSs and investigate that controllers of the drift, diffusion, and jump terms are all able to play a stabilizing effect on the controlled system.

show abstract

Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise

Cited by 101 publications

References 18 publications

Two-dimensional magneto-hydrodynamic system with jump processes: well posedness and invariant measures

Two-dimensional magneto-hydrodynamic system with jump processes: well posedness and invariant measures

Stability of the solution of stochastic differential equation driven by time-changed Lévy noise

Stability equivalence between regime‐switching jump diffusion delayed systems and corresponding systems with piecewise continuous arguments and application to discrete‐time feedback control

Contact Info

Product

Resources

About