The existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and Lévy jumps

Abstract

In this paper, we consider a class of stochastic pantograph differential equations with Lévy jumps (SPDEwLJs). By using the Burkholder-Davis-Gundy inequality and the Kunita's inequality, we prove the existence and uniqueness of solutions to SPDEwLJs whose coefficients satisfying the Lipschitz conditions and the local Lipschitz conditions. Meantime, we establish the p-th exponential estimations and almost surely asymptotic estimations of solutions to SPDEwLJs.

The Averaging Principle for Stochastic Pantograph Equations with Non-Lipschitz Conditions

“…Also, the averaging principle is applied to study stochastic pantograph equations for the first time. In view of the previous literatures [14,15], the authors have studied the stochastic pantograph equations driven by Lévy noise and G-Brownian motion. To the best of our knowledge, the averaging principle can also use to study stochastic pantograph equations driven by Lévy noise and G-Brownian motion with the non-Lipschitz conditions.…”

Existence and Hyers–Ulam Stability for Random Impulsive Stochastic Pantograph Equations with the Caputo Fractional Derivative

“…Mao et al [9] considered a class of stochastic pantograph equations with Lévy jumps as follows:      dx(t) = f (t, x(t), x(θt))dt + g(t, x(t), x(θt))dB(t), t ∈ [t 0 , T], B(t) = t t 0 U h(x(s), x(θs), u)N P(ds, du), x(t) = φ(t), t ∈ [θt 0 , t 0 ],…”

“…We investigate the existence, uniqueness and Hyers-Ulam stability of a class of random impulsive fractional stochastic pantograph equations under relaxed linear growth conditions. Compared with the previous literature [5][6][7][8][9]14,16], the corresponding conditions are required to satisfy the Lipschitz condition and the linear growth condition. However, in practical cases, the linear growth condition is usually violated.…”

Persistence and Extinction of a Stochastic Modified Bazykin Predator-Prey System with Lévy Jumps

“…Recently, the stochastic differential equation driven by jump has drawn more and more researchers' attention [10][11][12][13][14][15][16][17]. This is mainly according to sudden perturbation of environment, such as toxic contamination of water, torrential flood, and hurricane.…”