引言

随机微分方程的稳定性一般是指解对于初值的稳定性，简单来说，稳定性表现了当系统的初始状态或者系统中的元素遇到了细小变化时系统的不灵敏度。对于一个稳定的系统，如果在特定的瞬间轨迹相互之间很接近，那么在后继的瞬间也会持续保持这样接近的状态。在稳定性概念中，指数稳定性是最重要的概念之一，在各领域的问题研究中也有较多的应用^[1-4]。由于很难得到随机微分方程具体的解析解，一般会用各种数值方法求得随机微分方程的近似解，所以研究这些数值解的稳定性是十分必要的。Higham等^[5]研究了随机微分方程经典和向后欧拉方法数值解的矩稳定性和几乎处处稳定性；Wu等^[6]研究了时滞随机微分方程经典欧拉方法数值解的几乎处处稳定性；Hu等^[7]研究了随机微分方程截断EM方法数值解的渐进稳定性。

Milosevic^[8]研究了中立型时滞随机微分方程几种隐式数值方法的收敛性和向后欧拉解的几乎处处渐近稳定性，在稳定性方面将漂移项和扩散项系数分开考虑，对每个系数设定了严格的增长性限制。本文在文献[8]基础上将两个系数放入同一个式子考虑，降低了对每个系数增长性的限制，并且不同于文献[8]的常数滞后项，本文考虑的是滞后项依赖于时间的情况，最后得到了向后欧拉方法和前后欧拉方法的几乎处处渐进稳定性。

1 基本假设与定义

设$ (\mathit{\Omega }, \mathscr{F}, {\{ {\mathscr{F}_t}\} _{t \ge 0}}, P)$是一个完备的概率空间，子流$ {\{ {\mathscr{F}_t}\} _{t \ge 0}}$满足常规条件：递增且右连续，$ \mathscr{F}_0$包含所有的零测集。

已知ω(t)=(ω₁(t), ω₂(t), …ω_n(t))^T是一个$ \mathscr{F}_t$可测的n维标准布朗运动并与$\mathscr{F}_0 $独立。令|·|代表欧氏范数，将矩阵B的迹x_trace(B^TB)简写为|B|²。对于一个给定的τ>0，定义C([-τ, 0];R^d)为从[-τ, 0]到R^d的连续函数φ的函数族，令δ:[0, +∞)→[0, τ]为一个Borel可测的时滞函数。引入中立型随机微分方程如下

$ \begin{array}{l} {\rm{d}}\left[ {\mathit{\boldsymbol{x}}\left( t \right) - u\left( {\mathit{\boldsymbol{x}}\left( {t - \delta \left( t \right)} \right)} \right)} \right] = f\left( {\mathit{\boldsymbol{x}}\left( t \right), \mathit{\boldsymbol{x}}\left( {t - } \right.} \right.\\ \left. {\left. {\delta \left( t \right)} \right)} \right){\rm{d}}t + g\left( {\mathit{\boldsymbol{x}}\left( t \right), \mathit{\boldsymbol{x}}\left( {t - \delta \left( t \right)} \right)} \right){\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right), t \ge 0 \end{array} $

(1)

式(1)满足初值条件x₀=ξ={ξ(θ), θ∈[-τ, 0]}，且$ {\mathit{\boldsymbol{x}}_0} \in C_{{\mathscr{F}_0}}^b(\left[ { - \tau , 0} \right];{{\bf{R}}^d})$；同时函数f、g、u满足

$ \left\{ \begin{array}{l} f:{{\bf{R}}^d} \times {{\bf{R}}^d} \to {{\bf{R}}^d}\\ g:{{\bf{R}}^d} \times {{\bf{R}}^d} \to {{\bf{R}}^{d \times n}}\\ u:{{\bf{R}}^d} \to {{\bf{R}}^d} \end{array} \right. $

为了保证式(1)解的存在唯一性，给出以下4个假设条件。

H₁:对于任意正整数i，存在一个正的常数K_i，使得对于任意的x₁, x₂, y₁, y₂∈R^d，且当|x₁|∨|x₂|∨|y₁|∨|y₂|≤i时，都有

$ \begin{array}{l} {\left| {f\left( {{\mathit{\boldsymbol{x}}_1}, {\mathit{\boldsymbol{y}}_1}} \right) - f\left( {{\mathit{\boldsymbol{x}}_2}, {\mathit{\boldsymbol{y}}_2}} \right)} \right|^2} \vee \left| {g\left( {{\mathit{\boldsymbol{x}}_1}, {\mathit{\boldsymbol{y}}_1}} \right) - g\left( {{\mathit{\boldsymbol{x}}_2}, } \right.} \right.\\ {\left. {\left. {{\mathit{\boldsymbol{y}}_2}} \right)} \right|^2} \le {K_i}\left( {{{\left| {{\mathit{\boldsymbol{x}}_2} - {\mathit{\boldsymbol{x}}_1}} \right|}^2} + {{\left| {{\mathit{\boldsymbol{y}}_2} - {\mathit{\boldsymbol{y}}_1}} \right|}^2}} \right) \end{array} $

成立。

H₂:存在常数β∈(0, 1)，对任意x、y∈R^d，都有|u(x)-u(y)|≤β|x-y|；假设u(0)=0，则有|u(x)|≤β|x|。

H₃:函数δ是连续可微的，且δ′(t)≤δ < 1。

H₄:存在函数V∈C^{2, 1}(R^d×[-τ, +∞]; R₊)、U∈C(R^d×[-τ, +∞]; R₊)和c₁, c₂, c₃, c₄>0，使得$ {c_4} > \frac{{{c_3}}}{{1 - \bar \delta }}$成立；以及对任意x、y∈R^d和t≥0，都有式(2)成立

$ \begin{array}{l} {c_1}{\left| {\mathit{\boldsymbol{x}}\left( t \right)} \right|^p} \le V\left( {\mathit{\boldsymbol{x}}\left( t \right)} \right) \le {c_2}{\left| {\mathit{\boldsymbol{x}}\left( t \right)} \right|^p}{V_t}\left( {\mathit{\boldsymbol{x}}\left( t \right) - u} \right.\\ \left. {\left( {\mathit{\boldsymbol{y}}\left( t \right)} \right)} \right) + {V_\mathit{\boldsymbol{x}}}\left( {\mathit{\boldsymbol{x}}\left( t \right) - u\left( {\mathit{\boldsymbol{y}}\left( t \right)} \right)} \right)f\left( {\mathit{\boldsymbol{x}}\left( t \right), \mathit{\boldsymbol{y}}\left( t \right)} \right) + \frac{1}{2}\\ {x_{{\rm{trace}}}}\left[ {g{{\left( {\mathit{\boldsymbol{x}}\left( t \right), \mathit{\boldsymbol{y}}\left( t \right)} \right)}^{\rm{T}}}{V_{xx}}\left( {\mathit{\boldsymbol{x}}\left( t \right) - u\left( {\mathit{\boldsymbol{y}}\left( t \right)} \right)} \right)g\left( {\mathit{\boldsymbol{x}}\left( t \right), } \right.} \right.\\ \left. {\left. {\mathit{\boldsymbol{y}}\left( t \right)} \right)} \right] \le {c_3}\left[ {1 + V\left( {\mathit{\boldsymbol{x}}\left( t \right)} \right) + V\left( {\mathit{\boldsymbol{y}}\left( {t - \delta \left( t \right)} \right)} \right)} \right] - {c_4}\\ U\left( {\mathit{\boldsymbol{x}}\left( t \right)} \right) \end{array} $

(2)

定理1^[9]  若条件H₁~H₄成立，那么对任意初值$ {\mathit{\boldsymbol{x}}_0} \in C_{{\mathscr{F}_0}}^b(\left[ { - \tau , 0} \right];{{\bf{R}}^d})$，式(1)的整体解x={x(t), t∈[-τ, +∞)}存在且唯一，并满足

$ EV\left( {\mathit{\boldsymbol{x}}\left( t \right)} \right) < \infty , E\int_0^t {U\left( {\mathit{\boldsymbol{x}}\left( s \right)} \right){\rm{d}}s} < \infty , t \ge 0 $

为了得到本文的最终结论，还需要以下假设条件和引理。

H₅:存在一个正的常数η，使得

|δ(t)-δ(s)|≤η|t-s|, t, s≥0成立。

H₆:存在正的常数λ₁、λ₂，满足λ₁>(((1-η)^-1)+1)λ₂，使得2(x-u(y))^Tf(x, y)+|g(x, y)|²≤-λ₁|x|²+ λ₂|y|²成立。

引理1^[10]  在条件H₅成立的情况下，对于任意非负整数i，令i-[δ(iΔ)/Δ]=a，那么

$ * \left\{ {i:i - \left[ {\delta \left( {j\Delta } \right)/\Delta } \right] = a} \right\} \le \left[ {{{\left( {1 - \eta } \right)}^{ - 1}}} \right] + 1 $

(3)

式(3)中，[δ(jΔ)/Δ]表示δ(jΔ)/Δ的整数部分，*{A}表示集合A中元素的个数。

引理2^[8]  令{A_i}、{U_i}(i=1, 2, …)为两列非负随机向量，且A_i和U_i都是$\mathscr{F}_{i-1} $可测的，A₀=U₀=0 a.s.；令M_i为一个实值局部鞅，M₀=0；令{X_i}是一个非负的半鞅且满足X_i=ξ+A_i-U_i+M_i，其中ξ为一个非负$ \mathscr{F}_0$可测的随机变量。若$ \mathop {\lim }\limits_{i \to \infty } {A_i} < \infty \;a.s.$，那么就有$\mathop {\lim }\limits_{i \to \infty } {X_i} < \infty $，且$\mathop {\lim }\limits_{i \to \infty } {U_i} < \infty $。

令$ \Delta = \frac{\tau }{N}$，其中N为大于τ的整数，则Δ∈(0，1)。

定义1^[8](向后欧拉方法)  定义向后欧拉数值解y_k

$ \left\{ \begin{array}{l} {\mathit{\boldsymbol{y}}_k} = \mathit{\boldsymbol{\xi }}\left( {k\Delta } \right), k = - N, - N + 1, \cdots , 0\\ {\mathit{\boldsymbol{y}}_k} = {\mathit{\boldsymbol{y}}_{k - 1}} + u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right) - \\ \;\;\;\;\;\;\;u\left( {{\mathit{\boldsymbol{y}}_{\left( {k - 1} \right) - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right) + \\ \;\;\;\;\;\;\;f\left( {{\mathit{\boldsymbol{y}}_k}, {\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)\Delta + \\ \;\;\;\;\;\;\;g\left( {{\mathit{\boldsymbol{y}}_{k - 1}}, {\mathit{\boldsymbol{y}}_{k - 1 - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right)\Delta {\mathit{\boldsymbol{\omega }}_{k - 1}}, \\ \;\;\;\;\;\;\;k = 1, 2, \cdots \end{array} \right. $

(4)

式(4)中，Δω_k-1=ω(kΔ)-ω((k-1)Δ)。

定义2^[8](前后欧拉方法)  定义前后欧拉数值解y_k

$ \left\{ \begin{array}{l} {{\mathit{\boldsymbol{\bar y}}}_k} = \mathit{\boldsymbol{\xi }}\left( {k\Delta } \right), k = - N, - N + 1, \cdots , 0\\ {{\mathit{\boldsymbol{\bar y}}}_k} = {{\mathit{\boldsymbol{\bar y}}}_{k - 1}} + u\left( {{{\mathit{\boldsymbol{\bar y}}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right) - \\ \;\;\;\;\;\;\;u\left( {{{\mathit{\boldsymbol{\bar y}}}_{\left( {k - 1} \right) - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right) + \\ \;\;\;\;\;\;\;f\left( {{\mathit{\boldsymbol{y}}_{k - 1}}, {\mathit{\boldsymbol{y}}_{k - 1 - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right)\Delta + \\ \;\;\;\;\;\;\;g\left( {{\mathit{\boldsymbol{y}}_{k - 1}}, {\mathit{\boldsymbol{y}}_{k - 1 - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right)\Delta {\mathit{\boldsymbol{\omega }}_{k - 1}}, \\ \;\;\;\;\;\;\;k = 1, 2, \cdots \end{array} \right. $

(5)

定义3^[6](几乎处处渐近指数稳定)  若存在一个常数ε₀>0，使得数值解q_k满足$ \mathop {\lim \sup }\limits_{k \to + \infty } \frac{{\lg |{\mathit{\boldsymbol{q}}_k}|}}{{k\Delta }} \le - {\varepsilon _0}\;a.s.$，那么q_k是几乎处处渐近指数稳定的。

2 主要结果及证明

定理2  在条件H₁~H₆成立情况下，用向后欧拉方法得到的数值解y_k和用前后欧拉方法得到的数值解y_k是几乎处处渐近指数稳定的。

证明  首先证明y_k的几乎处处渐进稳定性。

由式(4)得到

$ \begin{array}{l} {\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} \le \left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|\\ f\left( {{\mathit{\boldsymbol{y}}_k}, {\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)\Delta + \frac{1}{2}{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} + \frac{1}{2}\\ {\left| {{\mathit{\boldsymbol{y}}_{k - 1}} - u\left( {{\mathit{\boldsymbol{y}}_{\left( {k - 1} \right) - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} + \frac{1}{2}\left| {g\left( {{\mathit{\boldsymbol{y}}_k}, } \right.} \right.\\ {\left. {\left. {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2}\Delta + {m_{k - 1}} \end{array} $

(6)

式(6)中

$ \begin{array}{l} {m_{k - 1}} = \frac{1}{5}{\left| {g\left( {{\mathit{\boldsymbol{y}}_{k - 1}}, {\mathit{\boldsymbol{y}}_{\left( {k - 1} \right) - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right)\Delta {\mathit{\boldsymbol{\omega }}_{k - 1}}} \right|^2} - \\ \frac{1}{2}{\left| {g\left( {{\mathit{\boldsymbol{y}}_k}, {\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2}\Delta + \left\langle {{\mathit{\boldsymbol{y}}_{k - 1}} - } \right.\\ \mathit{\boldsymbol{u}}\left( {{\mathit{\boldsymbol{y}}_{\left( {k - 1} \right) - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right), g\left( {{\mathit{\boldsymbol{y}}_{k - 1}}, {\mathit{\boldsymbol{y}}_{\left( {k - 1} \right) - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right)\\ \left. {\Delta {\mathit{\boldsymbol{\omega }}_{k - 1}}} \right\rangle \end{array} $

是一个鞅，且m₀=0。

整理式(6)得

$ \begin{array}{l} {\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} - \left| {{\mathit{\boldsymbol{y}}_{k - 1}} - } \right.\\ {\left. {u\left( {{\mathit{\boldsymbol{y}}_{\left( {k - 1} \right) - \left( {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right)}}} \right)} \right|^2} \le 2\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|f\\ \left( {{\mathit{\boldsymbol{y}}_k}, {\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)\Delta + {\left| {g\left( {{\mathit{\boldsymbol{y}}_k}, {\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2}\Delta + 2{m_{k - 1}} \end{array} $

利用H₆得到

$ \begin{array}{l} {\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} - \left| {{\mathit{\boldsymbol{y}}_{k - 1}} - } \right.\\ {\left. {u\left( {{\mathit{\boldsymbol{y}}_{\left( {k - 1} \right) - \left( {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right)}}} \right)} \right|^2} \le \left( { - {\lambda _1}{{\left| {{\mathit{\boldsymbol{y}}_k}} \right|}^2} + } \right.\\ \left. {{\lambda _2}{{\left| {{y_{k - }}\left( {\delta \left( {k\Delta } \right)/\Delta } \right)} \right|}^2}} \right)\Delta + 2{m_{k - 1}} \end{array} $

(7)

对于任意一个常数C≥1，有

$ \begin{array}{l} {C^{k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} - {C^{\left( {k - 1} \right)\Delta }}\left| {{\mathit{\boldsymbol{y}}_{k - 1}} - u} \right.\\ {\left. {\left( {{\mathit{\boldsymbol{y}}_{\left( {k - 1} \right) - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} = {C^{k\Delta }}\left( {{{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|}^2} - } \right.\\ {\left| {{\mathit{\boldsymbol{y}}_{k - 1}} - u\left( {{\mathit{\boldsymbol{y}}_{\left( {k - 1} \right) - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} + \left( {1 - {C^{ - \Delta }}} \right)\left| {{\mathit{\boldsymbol{y}}_{k - 1}} - } \right.\\ \left. {{{\left. {u\left( {{\mathit{\boldsymbol{y}}_{\left( {k - 1} \right) - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right)} \right|}^2}} \right) \end{array} $

(8)

将式(7)代入式(8)中，再利用H₂得到

$ \begin{array}{l} {C^{k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} - {C^{\left( {k - 1} \right)\Delta }}\left| {{\mathit{\boldsymbol{y}}_{k - 1}} - } \right.\\ {\left. {u\left( {{\mathit{\boldsymbol{y}}_{\left( {k - 1} \right) - \left( {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right)}}} \right)} \right|^2} \le {C^{k\Delta }}\left( {\left( { - {\lambda _1}{{\left| {{\mathit{\boldsymbol{y}}_k}} \right|}^2} + } \right.} \right.\\ \left. {{\lambda _2}{{\left| {{\mathit{\boldsymbol{y}}_{k - \left( {\delta \left( {k\Delta } \right)/\Delta } \right)}}} \right|}^2}} \right)\Delta + 2{m_{k - 1}} + \left( {1 - {C^{ - \Delta }}} \right)\left( {2{{\left| {{\mathit{\boldsymbol{y}}_{k - 1}}} \right|}^2} + } \right.\\ \left. {\left. {2{\beta ^2}{{\left| {{\mathit{\boldsymbol{y}}_{\left( {k - 1} \right) - \left( {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right)}}} \right|}^2}} \right)} \right) \end{array} $

(9)

将式(9)迭代得到

$ \begin{array}{l} {C^{k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} \le {\left| {{\mathit{\boldsymbol{y}}_0} - u\left( {{\mathit{\boldsymbol{y}}_{ - \left[ {\delta \left( 0 \right)/\Delta } \right]}}} \right)} \right|^2} - \\ {\lambda _1}\Delta \sum\limits_{i = 1}^k {{C^{i\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_i}} \right|}^2}} + {\lambda _2}\Delta \sum\limits_{i = 1}^k {{C^{i\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_{i - \left[ {\delta \left( {i\Delta } \right)/\Delta } \right]}}} \right|}^2}} + 2\left( {1 - } \right.\\ \left. {{C^{ - \Delta }}} \right)\sum\limits_{i = 1}^k {{C^{i\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_{i - 1}}} \right|}^2}} + 2{\beta ^2}\left( {1 - {C^{ - \Delta }}} \right)\sum\limits_{i = 1}^k {{C^{i\Delta }}} \\ {\left| {{\mathit{\boldsymbol{y}}_{\left( {i - 1} \right) - \left[ {\delta \left( {\left( {i - 1} \right)\Delta } \right)/\Delta } \right]}}} \right|^2} + {M_k} \end{array} $

(10)

式(10)中$ {M_k} = 2\mathop \sum \limits_{i = 1}^k {C^{i\Delta }}{m_{i - 1}}$为一个鞅, 且M=0。

因为

$ \begin{array}{l} 2\left( {1 - {C^{ - \Delta }}} \right)\sum\limits_{i = 1}^k {{C^{i\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_{i - 1}}} \right|}^2}} = 2\left( {{C^\Delta } - 1} \right)\sum\limits_{i = 1}^k {{C^{i\Delta }}} \\ {\left| {{\mathit{\boldsymbol{y}}_i}} \right|^2} + 2\left( {{C^\Delta } - 1} \right)\left( {{{\left| {{\mathit{\boldsymbol{y}}_0}} \right|}^2} - {C^{k\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_k}} \right|}^2}} \right) \end{array} $

$ \begin{array}{l} 2{\beta ^2}\left( {1 - {C^{ - \Delta }}} \right)\sum\limits_{i = 1}^k {{C^{i\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_{\left( {i - 1} \right) - \left[ {\delta \left( {\left( {i - 1} \right)\Delta } \right)/\Delta } \right]}}} \right|}^2}} = 2{\beta ^2}\\ \left( {{C^\Delta } - 1} \right)\sum\limits_{i = 1}^k {{C^{i\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_{i - \left[ {\delta \left( {i\Delta } \right)/\Delta } \right]}}} \right|}^2}} = 2\left( {{C^\Delta } - 1} \right)\\ \left( {{{\left| {{\mathit{\boldsymbol{y}}_{ - \left[ {\delta \left( 0 \right)/\Delta } \right]}}} \right|}^2} - {C^{k\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right|}^2}} \right) \end{array} $

所以式(10)变为

$ \begin{array}{l} {C^{k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} \le {\left| {{\mathit{\boldsymbol{y}}_0} - u\left( {{\mathit{\boldsymbol{y}}_{ - \left[ {\delta \left( 0 \right)/\Delta } \right]}}} \right)} \right|^2} + \\ 2\left( {{C^\Delta } - 1} \right){\left| {{\mathit{\boldsymbol{y}}_0}} \right|^2} + 2\left( {{C^\Delta } - 1} \right){\left| {{\mathit{\boldsymbol{y}}_{ - \left[ {\delta \left( 0 \right)/\Delta } \right]}}} \right|^2} + \left( { - {\lambda _1}\Delta + } \right.\\ \left. {2\left( {{C^\Delta } - 1} \right)} \right)\sum\limits_{i = 1}^k {{C^{i\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_i}} \right|}^2}} + \left( {{\lambda _2}\Delta + 2{\beta ^2}\left( {{C^\Delta } - } \right.} \right.\\ \left. {\left. 1 \right)} \right)\sum\limits_{i = 1}^k {{C^{i\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_{i - \left[ {\delta \left( {i\Delta } \right)/\Delta } \right]}}} \right|}^2}} + {M_k} \end{array} $

(11)

由引理1可得

$ \begin{array}{l} \sum\limits_{i = 1}^k {{C^{i\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_{i - \left[ {\delta \left( {i\Delta } \right)/\Delta } \right]}}} \right|}^2}} \le \left( {\left[ {{{\left( {1 - \eta } \right)}^{ - 1}}} \right] + 1} \right)\\ {C^\tau }\sum\limits_{i = - N}^k {{C^{i\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_i}} \right|}^2}} \end{array} $

因此式(11)变为

$ \begin{array}{l} {C^{k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} \le h\left( {C, \Delta } \right)\sum\limits_{i = 1}^k {{C^{i\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_i}} \right|}^2}} + \\ {M_k} + X \end{array} $

(12)

式(12)中

$ \begin{array}{l} X = {\left| {{\mathit{\boldsymbol{y}}_0} - u\left( {{\mathit{\boldsymbol{y}}_{ - \left[ {\delta \left( 0 \right)/\Delta } \right]}}} \right)} \right|^2} + 2\left( {{C^\Delta } - 1} \right){\left| {{\mathit{\boldsymbol{y}}_0}} \right|^2} + \\ 2\left( {{C^\Delta } - 1} \right){\left| {{\mathit{\boldsymbol{y}}_{ - \left[ {\delta \left( 0 \right)/\Delta } \right]}}} \right|^2} + \left( {{\lambda _2}\Delta + 2{\beta ^2}\left( {{C^\Delta } - 1} \right)} \right)\\ \left( {\left( {{{\left( {1 - \eta } \right)}^{ - 1}}} \right) + 1} \right){C^\tau }\sum\limits_{i = - N}^0 {{C^{i\Delta }}{{\left| {\xi \left( {i\Delta } \right)} \right|}^2}} < \infty \end{array} $

令h(C, Δ)=-λ₁Δ+2(C^Δ-1)+(λ₂Δ+2β²(C^Δ-1))(((1-η)^-1)+1)C^τ，可以求得

$ \begin{array}{l} \frac{{\rm{d}}}{{{\rm{d}}C}}h\left( {C, \Delta } \right) = 2\Delta {C^{\Delta - 1}} + \left( {\left( {{{\left( {1 - \eta } \right)}^{ - 1}}} \right) + 1} \right)\\ \left( {{C^\tau }2{\beta ^2}\Delta {C^{\Delta - 1}} + \left( {{\lambda _2}\Delta + 2{\beta ^2}\left( {{C^\Delta } - 1} \right)} \right)\tau {C^{\tau - 1}}} \right) > 0 \end{array} $

由H₆得到

$ h\left( {1, \Delta } \right) = - {\lambda _1}\Delta + {\lambda _2}\Delta \left( {\left( {{{\left( {1 - \eta } \right)}^{ - 1}}} \right) + 1} \right) < 0 $

因此存在一个C_*>1，使得h(C_*, Δ)=0，因C_*的值与Δ有关，所以式(10)变为

$ C_ * ^{k\Delta }{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} \le X + {M_k} $

(13)

由引理2可得

$ \begin{array}{l} \mathop {\limsup }\limits_{k \to \infty } \left( {C_ * ^{k\Delta }{{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|}^2}} \right) \le \mathop {\limsup }\limits_{k \to \infty } \\ \left( {X + {M_k}} \right) \end{array} $

令$\mu = \ln {C_*}, \mathop {\lim }\limits_{\Delta \to 0} \mu = \varepsilon $, 则由式(13)得到

$ \mathop {\limsup }\limits_{k \to \infty } {{\rm{e}}^{\mu k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} < \infty $

所以对于任意$\sigma \in \left( {0, \frac{\varepsilon }{2}} \right) $，存在常数α，使得

$ \mathop {\limsup }\limits_{k \to \infty } {{\rm{e}}^{\left( {\varepsilon - 2\sigma } \right)k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} = \alpha $

(14)

对于任意$ \gamma \in \left( {0, - \frac{1}{\tau }\ln \beta \wedge \left( {\varepsilon - 2\sigma } \right)} \right)$，存在整数k₁，使得式(15)成立。

$ {{\rm{e}}^{\gamma k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} \le \alpha + \gamma , k \ge {k_1} $

(15)

因为

$ {\left( {a + b} \right)^p} \le {\left( {1 + c} \right)^{p - 1}}\left( {{a^p} + {c^{1 - p}}{b^p}} \right), a, b > 0, p > \mathit{1}, c > 0 $

令$p = 2, c = \frac{\beta }{{1 - \beta }} $，则得到

$ \begin{array}{l} {{\rm{e}}^{\gamma k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k}} \right|^2} \le {\left( {1 - \beta } \right)^{ - 1}}{{\rm{e}}^{\gamma k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} + \\ \beta {{\rm{e}}^{\gamma k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right|^2} \end{array} $

(16)

因此对任意整数k₂≥k₁，有

$ \begin{array}{l} \mathop {\sup }\limits_{{k_1} \le k \le {k_2}} {{\rm{e}}^{\gamma k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k}} \right|^2} \le {\left( {1 - \beta } \right)^{ - 1}}\left( {\alpha + \gamma } \right) + \beta {{\rm{e}}^{\gamma \tau }}\\ \mathop {\sup }\limits_{{k_1} - N \le k \le {k_1}} {{\rm{e}}^{\gamma k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k}} \right|^2} + \beta {{\rm{e}}^{\gamma \tau }}\mathop {\sup }\limits_{{k_1} \le k \le {k_2}} {{\rm{e}}^{\gamma k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k}} \right|^2} \end{array} $

令k₂→+∞，则得到

$ \begin{array}{l} \mathop {\sup }\limits_{{k_1} \le k < + \infty } {{\rm{e}}^{\gamma k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k}} \right|^2} \le {\left( {1 - \beta {{\rm{e}}^{\gamma \tau }}} \right)^{ - 1}}\left( {{{\left( {1 - \beta } \right)}^{ - 1}}\left( {\alpha + } \right.} \right.\\ \left. {\left. \gamma \right) + \beta {{\rm{e}}^{\gamma \tau }}\mathop {\sup }\limits_{{k_1} - N \le k \le {k_1}} {{\rm{e}}^{\gamma k\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_k}} \right|}^2}} \right) < \infty \end{array} $

即$ \mathop {\lim \sup }\limits_{k \to + \infty } {e^{\gamma k\Delta }}|{\mathit{\boldsymbol{y}}_k}{|^2} < \infty $，可得y_k有界。

因此式(16)可以变为

$ \begin{array}{l} \mathop {\lim \sup }\limits_{k \to + \infty } {{\rm{e}}^{\gamma k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k}} \right|^2} \le {\left( {1 - \beta } \right)^{ - 1}}\alpha + \beta {{\rm{e}}^{\gamma \tau }}\mathop {\lim \sup }\limits_{k \to + \infty } \\ {{\rm{e}}^{\gamma \left( {k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]} \right)\Delta }}{\left| {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right|^2} \end{array} $

进而得到

$ \mathop {\lim \sup }\limits_{k \to + \infty } {{\rm{e}}^{\gamma k\Delta }}{\left| {{\mathit{\boldsymbol{y}}_k}} \right|^2} \le {\left( {1 - \beta {{\rm{e}}^{\gamma \tau }}} \right)^{ - 1}}{\left( {1 - \beta } \right)^{ - 1}}\alpha $

因此得到

$ \mathop {\lim \sup }\limits_{k \to + \infty } \frac{{\ln \left( {{{\rm{e}}^{\gamma k\Delta }}{{\left| {{\mathit{\boldsymbol{y}}_k}} \right|}^2}} \right)}}{{k\Delta }} = \gamma + 2\mathop {\lim \sup }\limits_{k \to + \infty } \frac{{\ln \left| {{\mathit{\boldsymbol{y}}_k}} \right|}}{{k\Delta }} < 0 $

(17)

由式(17)可知$ \mathop {\lim \sup }\limits_{k \to + \infty } \frac{{\ln |{\mathit{\boldsymbol{y}}_k}|}}{{k\Delta }} < - \frac{\gamma }{2}, {\mathit{\boldsymbol{y}}_k}$的几乎处处渐进稳定性得证。

其次证明y_k的几乎处处渐进稳定性。

将式(4)、(5)联立得到

$ \begin{array}{l} {{\mathit{\boldsymbol{\bar y}}}_k} - {\mathit{\boldsymbol{y}}_k} = {{\mathit{\boldsymbol{\bar y}}}_{k - 1}} - {\mathit{\boldsymbol{y}}_{k - 1}} + u\left( {{{\mathit{\boldsymbol{\bar y}}}_{k - \left( {\delta \left( {k\Delta } \right)/\Delta } \right)}}} \right) - \\ u\left( {{\mathit{\boldsymbol{y}}_{k - \left( {\delta \left( {k\Delta } \right)/\Delta } \right)}}} \right) - u\left( {{{\mathit{\boldsymbol{\bar y}}}_{\left( {k - 1} \right) - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right) + \\ u\left( {{\mathit{\boldsymbol{y}}_{\left( {k - 1} \right) - \left( {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right)}}} \right) + \left( {f\left( {{\mathit{\boldsymbol{y}}_{k - 1}}, {\mathit{\boldsymbol{y}}_{k - 1 - \left[ {\delta \left( {\left( {k - 1} \right)\Delta } \right)/\Delta } \right]}}} \right) - } \right.\\ \left. {f\left( {{\mathit{\boldsymbol{y}}_k}, {\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right)\Delta \end{array} $

(18)

迭代式(18)得到

$ \begin{array}{l} {{\mathit{\boldsymbol{\bar y}}}_k} - u\left( {{{\mathit{\boldsymbol{\bar y}}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right) = {\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right) + \\ \left( {f\left( {{\mathit{\boldsymbol{y}}_0}, {\mathit{\boldsymbol{y}}_{ - \left[ {\delta \left( 0 \right)/\Delta } \right]}}} \right) - f\left( {{\mathit{\boldsymbol{y}}_k}, {\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right)\Delta \end{array} $

$ \begin{array}{l} {\left| {{{\mathit{\boldsymbol{\bar y}}}_k} - u\left( {{{\mathit{\boldsymbol{\bar y}}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} \le 2{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} + \\ 2{\Delta ^2}{\left| {f\left( {{\mathit{\boldsymbol{y}}_0}, {\mathit{\boldsymbol{y}}_{ - \left[ {\delta \left( 0 \right)/\Delta } \right]}}} \right) - f\left( {{\mathit{\boldsymbol{y}}_k}, {\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} \end{array} $

因为y_k有界，故由H₁可得到

$ \begin{array}{l} {\left| {{{\mathit{\boldsymbol{\bar y}}}_k} - u\left( {{{\mathit{\boldsymbol{\bar y}}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} \le 2{\left| {{\mathit{\boldsymbol{y}}_k} - u\left( {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} + \\ 2{\Delta ^2}{K_i}\left( {{{\left| {{\mathit{\boldsymbol{y}}_k} - {\mathit{\boldsymbol{y}}_0}} \right|}^2} + {{\left| {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}} - {\mathit{\boldsymbol{y}}_{ - \left[ {\delta \left( 0 \right)/\Delta } \right]}}} \right|}^2}} \right) \end{array} $

(19)

对于|y_k-y₀|²+|y_{k-[δ(kΔ)/Δ]}-y_-[δ(0)/Δ]|²，有

$ \begin{array}{l} \mathop {\lim \sup }\limits_{k \to + \infty } {\left| {{\mathit{\boldsymbol{y}}_k} - {\mathit{\boldsymbol{y}}_0}} \right|^2} + \mathop {\lim \sup }\limits_{k \to + \infty } {\left| {{\mathit{\boldsymbol{y}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}} - {\mathit{\boldsymbol{y}}_{ - \left[ {\delta \left( 0 \right)/\Delta } \right]}}} \right|^2} = \\ 2\mathop {\lim \sup }\limits_{k \to + \infty } {\left| {{\mathit{\boldsymbol{y}}_k} - {\mathit{\boldsymbol{y}}_0}} \right|^2} \le 4\mathop {\lim \sup }\limits_{k \to + \infty } \left( {{{\left| {{\mathit{\boldsymbol{y}}_k}} \right|}^2} + {{\left| {{\mathit{\boldsymbol{y}}_0}} \right|}^2}} \right) \end{array} $

因此式(16)可变为

$ \begin{array}{l} \mathop {\lim \sup }\limits_{k \to + \infty } {{\rm{e}}^{\left( {\varepsilon - 2\sigma } \right)k\Delta }}{\left| {{{\mathit{\boldsymbol{\bar y}}}_k} - u\left( {{{\mathit{\boldsymbol{\bar y}}}_{k - \left[ {\delta \left( {k\Delta } \right)/\Delta } \right]}}} \right)} \right|^2} \le 2\alpha + \\ 8{\Delta ^2}{K_i}\mathop {\lim \sup }\limits_{k \to + \infty } {{\rm{e}}^{\left( {\varepsilon - 2\sigma } \right)k\Delta }}\left( {{{\left| {{\mathit{\boldsymbol{y}}_k}} \right|}^2} + {{\left| {{\mathit{\boldsymbol{y}}_0}} \right|}^2}} \right) = \bar \alpha < \infty \end{array} $

之后方法与证明y_k几乎处处渐进稳定性的方法完全相同，可以证得y_k也具有几乎处处渐近指数稳定性。定理2得证。证毕。

3 实例分析

考虑如下一维中立型时滞随机微分方程

$ \begin{array}{l} {\rm{d}}\left[ {\mathit{\boldsymbol{x}}\left( t \right) - \frac{1}{9}\mathit{\boldsymbol{x}}\left( {t - \delta \left( t \right)} \right)} \right] = \left( { - 6\mathit{\boldsymbol{x}}\left( t \right) + } \right.\\ \left. {\frac{{\mathit{\boldsymbol{x}}\left( {t - \delta \left( t \right)} \right)}}{{1 + {{\left| {\mathit{\boldsymbol{x}}\left( {t - \delta \left( t \right)} \right)} \right|}^2}}}} \right){\rm{d}}t + \mathit{\boldsymbol{x}}\left( t \right)\sin {\left| {\mathit{\boldsymbol{x}}\left( {t - \delta \left( t \right)} \right)} \right|^2}\\ {\rm{d}}\mathit{\boldsymbol{\omega }}\left( t \right), t \ge 0 \end{array} $

(20)

式(20)中，初值x₀=ξ(θ)=1, θ∈[-τ, 0], τ=2, δ(t)=1-sin t, t≥0。显然漂移项和扩散项系数满足H₁，中立项$ u(\mathit{\boldsymbol{x}}) = \frac{1}{9}\mathit{\boldsymbol{x}}$满足H₂，且$ \delta '\left( t \right) = - \frac{1}{4}\cos t \le \frac{1}{4} = \bar \delta $和$\eta = \frac{1}{4} $满足H₃、H₅；同时对于任意x, y∈R^d，都有

$ \begin{array}{l} 2{\left( {\mathit{\boldsymbol{x}} - u\left( \mathit{\boldsymbol{y}} \right)} \right)^f}\left( {\mathit{\boldsymbol{x}}, \mathit{\boldsymbol{y}}} \right) + {\left| {g\left( {\mathit{\boldsymbol{x}}, \mathit{\boldsymbol{y}}} \right)} \right|^2} = 2\left( {\mathit{\boldsymbol{x}} - \frac{1}{9}\mathit{\boldsymbol{y}}} \right)\\ \left( { - 6\mathit{\boldsymbol{x}} + \frac{\mathit{\boldsymbol{y}}}{{1 + {{\left| \mathit{\boldsymbol{y}} \right|}^2}}}} \right) \le - \frac{{31}}{3}{\left| \mathit{\boldsymbol{x}} \right|^2} + \frac{{17}}{9}{\left| \mathit{\boldsymbol{y}} \right|^2} \end{array} $

即满足H₆。

令$\Delta = 0.1, {Y_k} = \frac{{\ln |{\mathit{\boldsymbol{y}}_k}|}}{{k\Delta }}, {\bar Y_k} = \frac{{\ln |{{\mathit{\boldsymbol{\bar y}}}_k}|}}{{k\Delta }} $, 利用Matlab作图如图 1所示。可以看出，随着k的增大，Y_k、Y_k都是逐渐趋于稳定的，符合几乎处处渐近指数稳定的定义。

4 结束语

通过研究中立型时滞随机微分方程的向后欧拉数值解和前后欧拉数值解的几乎处处渐近指数稳定性，结果表明，与文献[8]相比，本文的漂移项和扩散项系数更加具有普遍性；并通过构造合理的假设条件，将文献[8]中的结果推广到时滞项依赖于时间的情况；最后通过具体的实例分析进一步证明了该结果的有效性。