A Split-step $\theta$-Milstein Method for Linear Stochastic Delay Differential Equations

Guo, Qian; Tao, Xueyin; Xie, Wenwen

doi:10.12733/jics20101622

Cited by 1 publication

(2 citation statements)

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“…The authors in [28] studied the strong convergence and the mean-square stability of the split-step backward Euler method to linear SDDEs with constant lag and took the stepsize as a multiplier of that. This type of stepsize selection by a semi-implicit split-step θ -Milstein method was developed in [7], too. But Wang et al in [22] proposed a new improved split-step backward Euler method for SDDEs with time-dependent delay where a piecewise linear interpolation is used to approximate the solution at the delayed points.…”

Section: Introductionmentioning

confidence: 99%

“…But Wang et al in [22] proposed a new improved split-step backward Euler method for SDDEs with time-dependent delay where a piecewise linear interpolation is used to approximate the solution at the delayed points. Also, in contrast to [7,28], in [22], the restriction of stepsize is removed and the unconditional stability property is extended as well. In our proposed method, this restriction on stepsize is dropped, too.…”

Section: Introductionmentioning

confidence: 99%

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Numerical solution of stochastic state-dependent delay differential equations: convergence and stability

Akhtari

2019

Adv Differ Equ

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Numerical analysis of stochastic delay differential equations has been widely developed but frequently for the cases where the delay term has a simple feature. In this paper, we aim to study a more general case of delay term which has not been much discussed so far. We mean the case where the delay term takes random values. For this purpose, a new continuous split-step scheme is introduced to approximate the solution and then convergence in the mean-square sense is investigated. Moreover, given a test equation, the mean-square asymptotic stability of the scheme is presented. Numerical examples are provided to further illustrate the obtained theoretical results.Assumption 1 The functions a : (R d ) 2 → R d and b : (R d ) 2 → R d×m are globally Lipschitz continuous, i.e. there is a positive constant K 1 such that a(Assumption 2 In case (L2), let τ : [t 0 , T] → (0, r] be of Lipschitz continuous aswhere K 2 is a positive constant. In case (L3), let τ : [t 0 , T] × R d → (0, r] be of Lipschitz continuous, i.e. there exist two positive constants K 2 and K 3 such thatfor all t, s ∈ [t 0 , T] and X, X 1 , X 2 ∈ R d . Note that in each of the two cases above, the fact that τ is positive guarantees the measurability and then the existence of the Itô integral.

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Section: Introductionmentioning

confidence: 99%