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The split-step backward Euler method for linear stochastic delay differential equations. (English) Zbl 1183.65007

The authors consider a scalar linear system of Itô stochastic delay differential equations

dy(t)=(ay(t)+by(t-τ)dt+(cy(t)+dy(t-τ))dW(t),t0,y(t)=ψ(t),t[-τ,0](1)

where W(t) is on dimensional standard Wiener process, τ>0· A split-step backward Euler (SSBE) scheme for solving this system is constructed. The authors constructed the SSBE method by Y k =ψ(kh), when k=-m,-m+1,,0, h=t N and when k0

Y k * =Y k +h[aY k * +bY k-m+1 ],Y k+1 =Y k * +(cY k * +dY k-m+1 )ΔW k

where Y k is the numerical approximation of y(t k ) with t k =kh· The following theorem is the main result of this paper.

Theorem: Assume the condition a<-|b|-1 2(|c|+|d|) 2 · is satisfied.

if ad-bc=0 and 4|b|c 2 +b 2 -a 2 0 then the SSBE method is general mean spare-stable

if ad-bc=0 and 4|b|c 2 +b 2 -a 2 >0 then the SSBE methods is MS-stable and the stepsize satisfies

h(0,h 1 (a,b,c,d)), where

h 1 (a,b,c,d)=-[2a+2|b|+(|c|+|d|) 2 ] 4|b|c 2 +b 2 -a 2 ·

if ad-bc0 then the SSBE methods is MS-stable and the stepsize satisfies h(0,h 2 (a,b,c,d)), where

h 2 (a,b,c,d)=-[2|b|c 2 -2a|cd|+b 2 -2ad 2 +2bcd-a 2 ]+Δ 2(ad-bc) 2 ·

Here,

Δ=[2|b|c 2 -2a|cd|+b 2 -2ad 2 +2bcd-a 2 ] 2 -4(ad-bc) 2 [2a+2|b|+(|c|+|d|) 2 ]·

Several illustrative numerical examples of applying the SSBE method are presented.

MSC:
65C30Stochastic differential and integral equations
60H10Stochastic ordinary differential equations
34F05ODE with randomness
60H35Computational methods for stochastic equations
34K50Stochastic functional-differential equations
65L06Multistep, Runge-Kutta, and extrapolation methods
65L20Stability and convergence of numerical methods for ODE