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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

$\left\{\begin{array}{cc}dy\left(t\right)\hfill & =\left(ay\left(t\right)+by\left(t-\tau \right)dt+\left(cy\left(t\right)+dy\left(t-\tau \right)\right)dW\left(t\right),\phantom{\rule{1.em}{0ex}}t\ge 0,\hfill \\ y\left(t\right)\hfill & =\psi \left(t\right),\phantom{\rule{1.em}{0ex}}t\in \left[-\tau ,0\right]\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $W\left(t\right)$ is on dimensional standard Wiener process, $\tau >0·$ A split-step backward Euler (SSBE) scheme for solving this system is constructed. The authors constructed the SSBE method by ${Y}_{k}=\psi \left(kh\right),$ when $k=-m,-m+1,\cdots ,0$, $\frac{h=t}{N}$ and when $k\ge 0$

$\left\{\begin{array}{cc}{Y}_{k}^{*}\hfill & ={Y}_{k}+h\left[a{Y}_{k}^{*}+b{Y}_{k-m+1}\right],\hfill \\ {Y}_{k+1}\hfill & ={Y}_{k}^{*}+\left(c{Y}_{k}^{*}+d{Y}_{k-m+1}\right){\Delta }{W}_{k}\hfill \end{array}\right\$

where ${Y}_{k}$ is the numerical approximation of $y\left({t}_{k}\right)$ with ${t}_{k}=kh·$ The following theorem is the main result of this paper.

Theorem: Assume the condition $a<-|b|-\frac{1}{2}\left(|c|+{|d|\right)}^{2}·$ is satisfied.

if $ad-bc=0$ and $4|b|{c}^{2}+{b}^{2}-{a}^{2}\le 0$ 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

$h\in \left(0,{h}_{1}\left(a,b,c,d\right)\right),$ where

${h}_{1}\left(a,b,c,d\right)=\frac{{-\left[2a+2|b|+\left(|c|+|d|\right)}^{2}\right]}{4|b|{c}^{2}+{b}^{2}-{a}^{2}}·$

if $ad-bc\ne 0$ then the SSBE methods is MS-stable and the stepsize satisfies $h\in \left(0,{h}_{2}\left(a,b,c,d\right)\right),$ where

${h}_{2}\left(a,b,c,d\right)=\frac{-\left[2|b|{c}^{2}-2a|cd|+{b}^{2}-2a{d}^{2}+2bcd-{a}^{2}\right]+\sqrt{{\Delta }}}{2{\left(ad-bc\right)}^{2}}·$

Here,

${\Delta }=\left[2|b|{c}^{2}-2a|cd|+{b}^{2}-2a{d}^{2}+2bcd-{a}^{2}{\right]}^{2}-4{\left(ad-bc\right)}^{2}{\left[2a+2|b|+\left(|c|+|d|\right)}^{2}\right]·$

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

##### MSC:
 65C30 Stochastic differential and integral equations 60H10 Stochastic ordinary differential equations 34F05 ODE with randomness 60H35 Computational methods for stochastic equations 34K50 Stochastic functional-differential equations 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L20 Stability and convergence of numerical methods for ODE