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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
\[ \begin{cases} dy(t) & = (a y(t) + b y(t -\tau) dt + (c y (t) + dy(t-\tau))dW(t), \quad t \geq 0,\\ y(t) & = \psi (t), \quad t\in [-\tau, 0]\end{cases}\tag{1} \]
where \(W(t)\) 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 \( Y_k = \psi (kh),\) when \(k=-m, -m+1, \dots, 0\), \(h=t \over N\) and when \(k \geq 0\)
\[ \begin{cases} Y_{k}^* & = Y_k + h[a Y_k^* +b Y_{k-m+1}],\\ Y_{k+1} & = Y_k^* + (c Y_k^* +d Y_{k-m+1}) \Delta W_k \end{cases} \]
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 \(a< -|b| - \frac12 ( |c| + |d|)^2.\) is satisfied. 7mm
(i)
if \(ad -bc =0\) and \(4|b| c^2 + b^2 -a^2 \leq 0\) then the SSBE method is general mean spare-stable
(ii)
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
\(h \in (0, h_1 (a, b, c, d)),\) where
\[ h_1(a,b ,c,d)= \frac{-[2a +2|b| + (|c| +|d|)^2]}{4|b| c^2 + b^2 - a^2}. \]
(iii)
if \(ad -bc \neq 0\) then the SSBE methods is MS-stable and the stepsize satisfies \(h \in (0, h_2 (a, b, c, d)),\) where
\[ h_2(a,b ,c,d)= \frac{-[2|b|c^2 -2a |cd| + b^2 - 2ad^2 + 2bcd -a^2] +\sqrt\Delta}{2(ad - bc)^2}. \]
Here,
\[ \Delta = [2|b|c^2 - 2a|cd| +b^2 -2ad^2 +2bcd -a^2]^2 - 4(ad-bc)^2 [2a + 2|b| +( |c| +|d|)^2]. \]
Several illustrative numerical examples of applying the SSBE method are presented.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
34K50 Stochastic functional-differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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References:

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