Mean square asymptotic stability in nonlinear stochastic neutral Volterra-Levin equations with Poisson jumps and variable delays

Benhadri, Mimia; Zeghdoudi, Halim

doi:10.7169/facm/1657

Cited by 4 publications

(5 citation statements)

References 18 publications

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“…Finally, we bound e b1,t−τ −s ∞ ≤ e k(T +1) , we conclude that e b1,t−τ −s τ ∞ ≤ C T,k • τ + e k(T +1) , by the triangular inequality. Thus, all terms inside the integral from 18 By combining both (17) and (19), we conclude that lim τ →0 x τ (t) = x 0 (t) in L p , uniformly on [0, T ].…”

Section: Theorem 32 (Existence and Uniqueness)mentioning

confidence: 88%

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$$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay

2019

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In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay τ > 0: x (t) = ax(t) + bx(t − τ), t ≥ 0, with initial condition x(t) = g(t), −τ ≤ t ≤ 0. The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. By using L p-calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an L p-solution too. An analysis of L p-convergence when the delay τ tends to 0 is also performed in detail.

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Section: Theorem 32 (Existence and Uniqueness)mentioning

confidence: 88%

“…1 and Ch. 10] and in the articles [14][15][16][17][18], for example. Stochastic delay differential equations have also been successfully applied to model real problems in different settings.…”

Section: Introductionmentioning

confidence: 99%

$$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay

2019

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“…A new tool is required to tackle equations of this type, called Itô calculus [23]. Studies on stochastic delay differential equations can be read in [24][25][26][27][28], for example. On the other hand, in fuzzy delay differential equations, uncertainty is driven by fuzzy processes; see [29] for instance.…”

Section: Introductionmentioning

confidence: 99%

Lp-Solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term

López

Jornet

2020

Mathematics

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This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay τ > 0 , by adding a random forcing term f ( t ) that varies with time: x ′ ( t ) = a x ( t ) + b x ( t − τ ) + f ( t ) , t ≥ 0 , with initial condition x ( t ) = g ( t ) , − τ ≤ t ≤ 0 . The coefficients a and b are assumed to be random variables, while the forcing term f ( t ) and the initial condition g ( t ) are stochastic processes on their respective time domains. The equation is regarded in the Lebesgue space L p of random variables with finite p-th moment. The deterministic solution constructed with the method of steps and the method of variation of constants, which involves the delayed exponential function, is proved to be an L p -solution, under certain assumptions on the random data. This proof requires the extension of the deterministic Leibniz’s integral rule for differentiation to the random scenario. Finally, we also prove that, when the delay τ tends to 0, the random delay equation tends in L p to a random equation with no delay. Numerical experiments illustrate how our methodology permits determining the main statistics of the solution process, thereby allowing for uncertainty quantification.

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“…Since X 0 , X 1 ∈ L p (Ω) for all 1 ≤ p < ∞ and A is bounded in [6,14], Theorem 2.4 shows that the stochastic process X(t) defined by (2.3)-(2.5) is the unique L p (Ω) solution to (2.1) on (−1, 1), for each 1 ≤ p < ∞. Analogously to the previous two examples, Table 2.5 and Table 2.6 show the results.…”

Section: Example 29mentioning

confidence: 99%

“…1 and Ch. 10] and in the articles [6,71,126,148,149], for example. Stochastic delay differential equations have also been successfully applied to model real problems in different settings.…”

Section: Introductionmentioning

confidence: 99%

Mean square solutions of random linear models and computation of their probability density function

Sanz¹

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This thesis concerns the analysis of differential equations with uncertain input parameters, in the form of random variables or stochastic processes with any type of probability distributions. In modeling, the input coefficients are set from experimental data, which often involve uncertainties from measurement errors. Moreover, the behavior of the physical phenomenon under study does not follow strict deterministic laws. It is thus more realistic to consider mathematical models with randomness in their formulation. The solution, considered in the sample-path or the mean square sense, is a smooth stochastic process, whose uncertainty has to be quantified. Uncertainty quantification is usually performed by computing the main statistics (expectation and variance) and, if possible, the probability density function.

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Mean square asymptotic stability in nonlinear stochastic neutral Volterra-Levin equations with Poisson jumps and variable delays

Cited by 4 publications

References 18 publications

$$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay

$$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay

Lp-Solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term

Mean square solutions of random linear models and computation of their probability density function

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