Existence, uniqueness, almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the Euler–Maruyama approximation

Analysis on exponential stability of hybrid pantograph stochastic differential equations with highly nonlinear coefficients

“…However we will replace the linear growth condition by a more general condition, a Khasminskii-Type condition as applied in [18], [19], [22], [23], to guarantee the existence of a unique global solution.…”

“…[18] proposed a Khasminskii-type condition for a nonlinear hybrid PSDE, under which the polynomial stability of the solution could be derived. [22] extended the condition of [18] to the case that different types of functions or polynomials with different orders occured in the Lyapunov operator. [23] investigated the exponential stability of a class of hybrid PSDE, where the coefficients were dominated by polynomials with high orders.…”

“…where r(t) is a right-continuous Markov chain taking value in S = {1, 2} with the generator Γ =  The coefficients aref (x, t, 1) = −x−2x 3 , g(y, t, 1) = 0.5y 2 exp(−0.45t), f (x, t, 2) = x−x 3 ,and g(y, t, 2) = 0.1y exp(−0.45t).Set p = 2, q = 4. It is easy to see xf (x, t, 1) + 1 2 g 2 (y, t, 1) = −x 2 − 2x 4 + 1 8 y 4 exp(−0.9t), and xf (x, t, 2) + 1 2 g 2 (y, t, 2) = x 2 − x 4 + 1 200 y 2 exp(−0.9t).The quantities appeared in(22) are β 11 = 0, β 12 = −1, β 13 = 0, β 14 = 2, β 15 = 1.25; β 21 = 0, β 22 = 1, β 23 = 1/20, β 24 = 1, and β 25 = 0. We see that the matrix A in (23) is A = shows that A is a nonsingular M-matrix.…”

Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation

“…Li et al [15] and Yuan et al [43] established the convergence in probability of the EM approximate solution to the solution of SDDEs with Markovian switching under the Khasminskii-type conditions. While Milosevic [25] showed the convergence in probability of the EM solution for a class of highly nonlinear pantograph stochastic differential equations under the nonlinear growth conditions. However, there is little known on the convergence of numerical solution in probability for hybrid SDEs with jumps under nonlinear growth condition.…”

Polynomial stability of exact solution and a numerical method for stochastic differential equations with time-dependent delay

“…By using the well known discrete semimartingale convergence theorem, sufficient conditions are obtained for both bounded and unbounded delay δ to ensure the polynomial stability of the corresponding numerical approximation. Examples are presented to illustrate the conclusion.MSC 2010: 60H10, 65C30.there are few papers concerning about the polynomial stability of the numerical solution for the underlying stochastic differential equations with unbounded delay except [15]. Recently, Mao [12] introduced truncated EM method for stochastic differential equation without delay, and then he obtained sufficient conditions for the strong convergence rate of it in [13].…”