# Mean square exponential stability of numerical methods for stochastic
differential delay equations 111Supported by Natural Science Foundation of
Beijing Municipality (1192013).

Guangqiang Lan222Corresponding author: Guangqiang Lan<EMAIL_ADDRESS>and
Qi Liu333Qi Liu<EMAIL_ADDRESS>
College of Mathematics and Physics, Beijing University of Chemical Technology,
Beijing 100029, China

###### Abstract

Mean square exponential stability of $\theta$-EM and modified truncated Euler-
Maruyama (MTEM) methods for stochastic differential delay equations (SDDEs)
are investigated in this paper. We present new criterion of mean square
exponential stability of the $\theta$-EM and MTEM methods for SDDEs, which are
different from most existing results under Khasminskii-type conditions. Two
examples are provided to support our conclusions.

MSC 2010: 65C30, 65C20, 65L05, 65L20.

Key words: stochastic differential delay equations, $\theta$-EM method,
modified truncated Euler-Maruyama method, mean square exponential stability.

##  1 Introduction

Consider the nonlinear SDDE

$dx(t)=f(x(t),x(t-\tau(t)))dt+g(x(t),x(t-\tau(t)))dB(t),t\geq 0$ (1.1)

with initial value $\xi$, $B(t):=\left(B_{1}(t),\ldots,B_{m}(t)\right)^{T}$ be
an $m$-dimensional standard Brownian motion,
$f:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d},\quad
g:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d\times m}$ and
$\tau(t):\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ are Borel-measurable
functions and $0<\tau(t)\leq\tau,t\in\mathbb{R}_{+}$, for some $\tau>0$.

Asymptotic stability of numerical approximations for the above SDDE (1.1) or
more general model has been widely investigated in recent years. In general,
to obtain the exponential stability results, the following Khasminskii-type
conditions are usually parts of the sufficient conditions.

$2\langle x,f(x,y)\rangle+||g(x,y)||^{2}\leq-C_{1}|x|^{2}+C_{2}|y|^{2}$ (1.2)

where $C_{1}>C_{2}>0.$

For example, in [10], Chapter 5, Razumikhin type Theorems are presented for
SDDEs, where conditions in Corollary 6.6 implies (1.2) if $k=1$. In [3],
Theorem 4.1 implied that if there is no neutral term, then the MTEM method is
mean square exponentially stable under condition (1.2). Under the same
condition, [14] considered exponential mean-square stability of two classes of
theta Milstein methods for stochastic differential equations with constant
delay. [16] obtained exponential stability of stochastic theta method for
nonlinear stochastic differential equations with piecewise continuous
arguments (in this case, $\tau(t)=t-[t]$). [6] obtained mean square
exponential stability of split-step method under stronger conditions (note
that they need the linearity of both $f$ and $g$), while [7] and [13]
investigated mean square stability of two class of theta method of neutral
stochastic differential delay equations under similar conditions.

Recently, by considering each component separately, [12] presented a different
type of sufficient conditions under which the trivial solution of given SDDE
is mean square exponentially stable.

Motivated by that paper, now suppose that $f$ and $g$ jointly satisfy the
following

$2x_{i}f_{i}(x,y)+\sum_{l=1}^{m}\left(g_{il}(x,y)\right)^{2}\leq\sum_{j=1}^{d}a_{ij}x_{j}^{2}+\sum_{j=1}^{d}b_{ij}y_{j}^{2},\quad
i\in\underline{d}:={1,2,...,d}.$ (1.3)

By the same method in [12], it is not difficult to obtain mean square
exponential stability of $x(t)$ under (1.3) if $a_{ij}$ and $b_{ij}$ satisfy
suitable conditions. Note that $f$ and $g$ in (1.3) do not necessarily satisfy
linear growth condition.

Note also that by (1.3) we can only have

$2\langle
x,f(x,y)\rangle+||g(x,y)||^{2}\leq\sum_{j=1}^{d}\sum_{i=1}^{d}a_{ij}x_{j}^{2}+\sum_{j=1}^{d}\sum_{i=1}^{d}b_{ij}y_{j}^{2}.$

If $\sum_{i=1}^{d}a_{ij}\geq 0$ for some $j\in\underline{d}$, or
$0<\min_{j\in\underline{d}}(-\sum_{i=1}^{d}a_{ij})<\max_{j\in\underline{d}}\sum_{i=1}^{d}b_{ij}$,
then (1.2) can never hold. Therefore, (1.3) is a different type condition from
(1.2).

To the best of our knowledge, there isn’t any result of stability of numerical
methods under condition (1.3). So the key aim of this paper is to obtain mean
square exponential stability of $\theta$-EM method and MTEM method of the
given stochastic differential delay equations under (1.3) with suitable
$a_{ij}$ and $b_{ij}$.

The rest of the paper is organized as follows. In Section 2, $\theta$-EM
method, MTEM method and some necessary assumptions will be presented. In
Section 3, we will prove that the given $\theta$-EM method is mean square
exponentially stable under (1.3). Section 4 will prove that the MTEM method
will replicate the mean square exponential stability of the exact solution for
the stochastic differential delay equation under the same conditions. Finally,
in Section 5, two examples will be presented to interpret our conclusion.

##  2 Model description and preliminaries

Let $\mathbb{N}$ be the set of all natural numbers. Denote by $\mathbb{R}$ the
set of all real numbers. For given integers $l,q\geq 1,\mathbb{R}^{l}$ is the
$l$-dimensional vector space over $\mathbb{R}$ and $\mathbb{R}^{l\times q}$
stands for the set of all $l\times q$ matrices with entries in $\mathbb{R}$.
For given $m\in\mathbb{N}$, let $\underline{m}:=\\{1,2,\ldots,m\\}$. Let
$\left(\Omega,\mathcal{F},\\{\mathcal{F}\\}_{t\geq 0},P\right)$ be a complete
probability space with a filtration $\left\\{\mathcal{F}_{t}\right\\}_{t\geq
0}$ satisfying the usual conditions (i.e., right continuous and
$\mathcal{F}_{0}$ containing all $P$ null sets). Additionally, $\|\cdot\|$
denotes the Hilbert-Schmidt norm
$\|g\|^{2}:=\sum_{i=1}^{d}\sum_{j=1}^{m}g^{2}_{ij}$ for any $d\times m$ matrix
$g=(g_{ij})\in\mathbb{R}^{d\times m}$. Let
$C\left([-\tau,0],\mathbb{R}^{d}\right)$ be the Banach space of all continuous
functions on $[-\tau,0]$, endowed with the norm $\|\varphi\|=$
$\max_{s\in[-\tau,0]}\|\varphi(s)\|$. Denote by
$C_{\mathcal{F}_{0}}^{b}\left([-\tau,0],\mathbb{R}^{d}\right)$, the family of
$\mathcal{F}_{0}$-measurable bounded
$C\left([-\tau,0],\mathbb{R}^{d}\right)$-valued random variables.

An initial condition for (1.1) is defined by

$x\left(s\right)=\xi(s),\quad s\in[-\tau,0]$ (2.1)

where $\xi\in C_{\mathcal{F}_{0}}^{b}\left([-\tau,0],\mathbb{R}^{d}\right)$.

An $\mathbb{R}^{d}$-valued stochastic process $\\{x(t)\\}_{t\geq-\tau}$ is
called a solution of (1.1) with initial value (2.1), if $\\{x(t)\\}_{t\geq 0}$
is continuous, $\mathcal{F}_{t}$-adapted such that

$\displaystyle
x(t)=\xi(0)+\int_{0}^{t}f(x(s),x(s-\tau(s)))ds+\int_{0}^{t}g(x(s),x(s-\tau(s)))dB(s)$
(2.2)

holds for each $t\geq 0$, with probability 1 and the initial condition (2.1)
is fulfilled.

###### Assumption 2.1

Assume that both the coefficients $f$ and $g$ in (1.1) are locally Lipschitz
continuous, that is, for each $R>0$ there is $L_{R}>0$ $($depending on $R$$)$
such that

$|f(x,y)-f(\bar{x},\bar{y})|\vee|g(x,y)-g(\bar{x},\bar{y})|\leq
L_{R}(|x-\bar{x}|+|y-\bar{y}|)$ (2.3)

for all $|x|\vee|y|\vee|\bar{x}|\vee|\bar{y}|\leq R>0$.

It is obvious that $L_{R}$ is an increasing function with respect to $R$.

Let $\Delta$ be a stepsize such that $\tau=\bar{m}\Delta$ for some positive
integer $\bar{m}$. Then for any $\theta\in[0,1],$ we can define $\theta$
Euler-Maruyama method ($\theta$-EM for short) $X_{k}$ as the following:

$\displaystyle X_{k}=\xi(k\Delta),\quad k=-\bar{m},-\bar{m}+1,\ldots,0.$ (2.4)
$\displaystyle
X_{k+1}=X_{k}+\left((1-\theta)f\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)+\theta
f\left(X_{k+1},X_{k+1-[\frac{\tau((k+1)\Delta)}{\Delta}]}\right)\right)\Delta$
$\displaystyle\qquad+g\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta
B_{k},\quad k=0,1,2,\ldots,$

Here $\Delta B_{k}=B((k+1)\Delta)-B(k\Delta)$ is the increment of the
$m$-dimensional standard Brownian motion.

Note that if $\theta=0$, it becomes to classical EM method, if $\theta=1$, it
is called Backward Euler method. Moreover, since it is an implicit method for
$\theta\in(0,1]$, then it is necessary to make sure that $\theta$-EM method is
well defined. So we need the following assumption

###### Assumption 2.2

Assume that $f$ is one-sided Lipschitz continuous, that is, there is $L>0$
such that

$\left\langle x-\bar{x},f(x,y)-f(\bar{x},y)\right\rangle\leq L|x-\bar{x}|^{2}$
(2.5)

for all $x,y,\bar{x}\in\mathbb{R}^{d}$.

According to [15], if $L\theta\Delta<1$, then the $\theta$-EM scheme (2.4) is
well defined, see also [2, 11] for details.

Now let us give the modified truncated Euler-Maruyama method for (1.1).

For $\Delta^{*}>0$ , let $h(\Delta)$ be a strictly positive decreasing
function $h:\left(0,\Delta^{*}\right]\rightarrow(0,\infty)$ such that

$\lim_{\Delta\rightarrow 0}h(\Delta)=\infty\text{ and }\lim_{\Delta\rightarrow
0}L_{h(\Delta)}^{2}\Delta=0.$

According to [5], Remark 2.1, such $h$ always exists.

For any $\Delta\in\left(0,\Delta^{*}\right)$, we define the modified truncated
function of $f$ as the following:

$f_{\Delta}(x,y)=\left\\{\begin{array}[]{lc}f(x,y),\quad\quad\quad\quad\quad\quad\quad|x|\vee|y|\leq
h(\Delta),\\\
\frac{|x|\vee|y|}{h(\Delta)}f\left(\frac{h(\Delta)}{|x|\vee|y|}(x,y)\right),\quad|x|\vee|y|>h(\Delta).\end{array}\right.$

$g_{\Delta}$ is defined in the same way as $f_{\Delta}$. Here $f(a(x,y))\equiv
f(ax,ay)$ for any $t\geq 0,a\in(0,1),x,y\in\mathbb{R}^{d}$. It is obvious that
the functions $f_{\Delta}$ and $g_{\Delta}$ defined above are unbounded while
the truncated functions defined in [8, 9] are bound for any fixed $\Delta$.

Then by using $f_{\Delta}$ and $g_{\Delta}$, we can define the modified
truncated Euler-Maruyama method (MTEM for short) $X_{k}$ as the following:

$\displaystyle
X_{k+1}=X_{k}+f_{\Delta}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta+g_{\Delta}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta
B_{k},\quad k=0,1,2,\ldots,$ (2.6) $\displaystyle X_{k}=\xi(k\Delta),\quad
k=-\bar{m},-\bar{m}+1,\ldots,0.$

###### Assumption 2.3

Let $f(x,y):=\left(f_{1}(x,y),\ldots,f_{d}(x,y)\right)^{T}\in\mathbb{R}^{d}$
and $g(x,y):=\left(g_{il}(x,y)\right)\in\mathbb{R}^{d\times m}$. Suppose there
exist constants $a_{ii}\in\mathbb{R};a_{ij}\geq 0,i\neq j;b_{ij}\geqslant 0,$
$i,j\in\underline{d}$, such that

$2x_{i}f_{i}(x,y)+\sum_{l=1}^{m}\left(g_{il}(x,y)\right)^{2}\leq\sum_{j=1}^{d}a_{ij}x_{j}^{2}+\sum_{j=1}^{d}b_{ij}y_{j}^{2}$
(2.7)

holds for any $i\in\underline{d}$.

It follows easily from Assumption 2.3 that

$2\left\langle x,f(x,y)\right\rangle+|g(x,y)|^{2}\leq K(1+|x|^{2}+|y|^{2})$
(2.8)

where
$K=\max\limits_{j}\left\\{\sum\limits_{i=1}^{d}|a_{ij}|,\sum\limits_{i=1}^{d}b_{ij}\right\\}\vee
1$.

Similar to Theorem 3.5 in[1] or Theorem 2.1 in [4], it is easy to see that
Assumptions 2.1 and 2.3 guarantee strong convergence of MTEM method. If
Assumption 2.2 also holds, then, by [17], the $\theta$-EM method is also
strongly convergent.

The following Theorem provides explicit criteria for the mean square
exponential stability of (1.1).

###### Theorem 2.4

Suppose Assumptions 2.1 and 2.3 hold. If there exist constants
$p_{j}>0,j\in\underline{d}$ such that

$\sum_{j=1}^{d}(a_{ij}+b_{ij})p_{j}<0,$ (2.9)

then the initial value problem (1.1), (2.1) always has a unique solution
$x\left(t,\xi\right)$ and the trivial solution of (1.1) is exponentially
stable in mean square sense.

Note that (2.9) implies $\max_{i\in\underline{d}}a_{ii}<0$ since $a_{ij}\geq
0,i\neq j;b_{ij}\geqslant 0,$ $i,j\in\underline{d}$.

Proof of Theorem 2.4. Firstly, local Lipschitz continuity of $f$ and $g$ and
(2.8) implies that (1.1) always has a unique local maximum solution. Moreover,
since Assumption 2.3 implies (2.8), then the solution must be non explosive,
that is, (1.1) has a unique global solution $x\left(t,\xi\right)$.

Let $p:=$ $\left(p_{1},p_{2},\ldots,p_{d}\right)^{T}\in\mathbb{R}^{d}$. By
continuity, (2.9) implies that there exists $\beta>0$ small enough such that

$\sum_{j=1}^{d}\left(a_{ij}+e^{\beta\tau}b_{ij}\right)p_{j}\leq-\beta p_{i}$
(2.10)

for any $i\in\underline{d}$.

Then by the same method of the proof of Theorem II.2, [12], it follows that
the trivial solution of (1.1) is exponentially stable in mean square sense.
$\square$

##  3 Mean exponential stability of $\theta$-EM method

In this Section, we consider mean exponential stability of $\theta$-EM method
for all $\theta\in[0,1]$.

###### Theorem 3.1

Suppose Assumption 2.1, Assumption 2.2 and Assumption 2.3 hold. Moreover,
there exist constants
$0<\varepsilon<\frac{\min_{i}|a_{ii}|}{d\max_{i}|a_{ii}|}$ and
$p_{j}>0,j\in\underline{d}$ such that

$\varepsilon a_{ii}p_{i}+\sum_{j\neq
i}a_{ij}p_{j}+\sum_{j=1}^{d}b_{ij}p_{j}<0.$ (3.1)

Then for any fixed $\theta\in(\frac{1}{2},1]$, the $\theta$-EM method (2.4) is
mean quare and almost surely exponentially stable.

###### Remark 3.2

Notice that (3.1) implies

$\sum_{j=1}^{d}a_{ij}p_{j}+\sum_{j=1}^{d}b_{ij}p_{j}<0.$ (3.2)

Then similar to the proof of Theorem II.2 [12], it follows that the Assumption
2.3 implies that the solution $x\left(t,\xi\right)$ to (1.1) is exponentially
stable in mean square sense if (2.9) holds. Theorem 3.1 assures that
$\theta$-EM method (2.4) replicates mean square exponential stability of the
exact solution under given conditions.

Proof of Theorem 3.1. By (2.4), for any $k\geqslant 0$ and $i=1,2,...,d$, we
have

$\displaystyle F_{k+1}^{i}$
$\displaystyle=F_{k}^{i}+f_{i}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta+\sum_{l=1}^{m}g_{il}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta
B_{k}$ (3.3)

where $F_{k}^{i}=X_{k}^{i}-\theta
f_{i}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta$.

So,

$\displaystyle|F_{k+1}^{i}|^{2}$
$\displaystyle=|F_{k}^{i}|^{2}+\left(2X_{k}^{i}f_{i}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)+\sum_{l=1}^{m}g_{il}^{2}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\right)\Delta$
(3.4)
$\displaystyle\quad+(1-2\theta)f^{2}_{i}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta^{2}+M^{i}_{k}$

where

$\displaystyle M^{i}_{k}$
$\displaystyle=2\left(F_{k}^{i}+f_{i}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta\right)\sum_{l=1}^{m}g_{il}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta
B^{l}_{k}\rangle$
$\displaystyle\quad+\left(\left|\sum_{l=1}^{m}g_{il}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta
B^{l}_{k}\right|^{2}-\sum_{l=1}^{m}g^{2}_{il}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta\right).$

It is easy to see that $M^{i}_{k}$ is a $\mathscr{F}_{k\Delta}$ martingale and
$EM^{i}_{k}=0$. Moreover, we claim that

$\displaystyle\quad
2X_{k}^{i}f_{i}(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]})+\sum_{l=1}^{m}g^{2}_{il}(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]})+(1-2\theta)f^{2}_{i}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta$
(3.5) $\displaystyle\leq-C|F_{k}^{i}|^{2}+\sum_{j\neq
i}^{d}a_{ij}|X_{k}^{j}|^{2}+\sum_{j=1}^{d}b_{ij}|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}^{j}|^{2}$

if $C<\min_{i}|a_{ii}|.$

In fact, denote
$f_{k}^{i}:=f_{i}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right),g_{k}^{il}:=g_{il}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)$
it follows that

$\displaystyle(2\theta-1)|f_{k}^{i}|^{2}\Delta-C|F_{k}^{i}|^{2}$
$\displaystyle=((2\theta-1)\Delta-C\theta^{2}\Delta^{2})|f_{k}^{i}|^{2}+2C\theta\Delta
X_{k}^{i}f^{i}_{k}-C|X_{k}^{i}|^{2}$
$\displaystyle=a(f_{k}^{i}+bX_{k}^{i})^{2}-(ab^{2}+C)|X_{k}^{i}|^{2}$

where $a=(2\theta-1)\Delta-C\theta^{2}\Delta^{2},$
$b=\frac{C\theta\Delta}{a}.$

Now we can choose $\Delta$ small enough such that $a>0$ since
$\theta>\frac{1}{2}$. Moreover, $ab^{2}+C\leq-a_{ii},\forall i=1,2,...,d$, and
$a_{ii}<0$ by (2.9). Then by Assumption 2.3, we have

$\displaystyle(2\theta-1)|f_{k}^{i}|^{2}\Delta-C|F_{k}^{i}|^{2}$
$\displaystyle\geq-\frac{ab^{2}+C}{a_{ii}}a_{ii}|X_{k}^{i}|^{2}$
$\displaystyle\geq a_{ii}|X_{k}^{i}|^{2}$ $\displaystyle\geq
2X_{k}^{i}f_{k}^{i}+\sum_{l=1}^{m}(g_{k}^{il})^{2}-(\sum_{j\neq
i}^{d}a_{ij}|X_{k}^{j}|^{2}+\sum_{j=1}^{d}b_{ij}|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}^{j}|^{2})$

which implies (3.5).

Then by (3.4) and (3.5), it follows that

$|F_{k+1}^{i}|^{2}\leq(1-C\Delta)|F_{k}^{i}|^{2}+(\sum_{j\neq
i}^{d}a_{ij}|X_{k}^{j}|^{2}+\sum_{j=1}^{d}b_{ij}|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}^{j}|^{2})\Delta+M^{i}_{k}.$

Therefore for any $\alpha>0$,

$\displaystyle|F_{k+1}^{i}|^{2}e^{\alpha(k+1)\Delta}-|F_{k}^{i}|^{2}e^{\alpha
k\Delta}$
$\displaystyle\leq(1-C\Delta-e^{-\alpha\Delta})|F_{k}^{i}|^{2}e^{\alpha(k+1)\Delta}+(\sum_{j\neq
i}^{d}a_{ij}e^{\alpha(k+1)\Delta}|X_{k}^{j}|^{2}$
$\displaystyle\quad+\sum_{j=1}^{d}b_{ij}e^{\alpha(k+1)\Delta}|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}^{j}|^{2})\Delta+M^{i}_{k}e^{\alpha(k+1)\Delta}.$

Then

$\displaystyle e^{\alpha k\Delta}E|F_{k}^{i}|^{2}$ $\displaystyle\leq
E|F_{0}^{i}|^{2}+R(\Delta)\sum_{l=0}^{k-1}e^{\alpha(l+1)\Delta}E|F_{l}^{i}|^{2}$
$\displaystyle\quad+(\sum_{j\neq
i}^{d}a_{ij}\sum_{l=1}^{k-1}e^{\alpha(l+1)\Delta}E|X_{l}^{j}|^{2}+\sum_{j=1}^{d}b_{ij}\sum_{l=1}^{k-1}e^{\alpha(l+1)\Delta}E|X_{l-[\frac{\tau(l\Delta)}{\Delta}]}^{j}|^{2})\Delta$

where $R(\Delta)=1-C\Delta-e^{-\alpha\Delta}.$

Since $R(0)=0$ and

$R^{\prime}(\Delta)=-C+\alpha e^{\alpha\Delta}\leq 0$

for all $\Delta>0$ if $\alpha\leq C$, then $R(\Delta)\leq 0.$

It is obvious that if we choose
$C^{\prime}>E||\xi||^{2}\max_{i}\frac{1}{p_{i}}\vee\max_{i}\frac{E|F_{0}^{i}|^{2}}{\varepsilon^{\prime}p_{i}}$,
where $\varepsilon^{\prime}$ will be given in the following, then we have

$E|X_{k}^{i}|^{2}\leq C^{\prime}p_{i}e^{-\beta k\Delta},\quad
k=-\bar{m},-\bar{m}+1,\ldots,0,i=1,2,...,d.$ (3.6)

Now we will prove that (3.6) also holds for any $k>0$ by using second
principle of mathematical induction.

Assume that

$E|X_{l}^{i}|^{2}\leq C^{\prime}p_{i}e^{-\beta l\Delta},\forall-\bar{m}\leq
l\leq k-1,i=1,2,...,d.$ (3.7)

Then

$\displaystyle e^{\alpha k\Delta}E|F_{k}^{i}|^{2}$ $\displaystyle\leq
E|F_{0}^{i}|^{2}+\left(\sum_{j\neq
i}^{d}a_{ij}\sum_{l=1}^{k-1}e^{\alpha(l+1)\Delta}E|X_{l}^{j}|^{2}+\sum_{j=1}^{d}b_{ij}\sum_{l=1}^{k-1}e^{\alpha(l+1)\Delta}E|X_{l-[\frac{\tau(l\Delta)}{\Delta}]}^{j}|^{2}\right)\Delta$
$\displaystyle\leq|F_{0}^{i}|^{2}+\Delta\sum_{j\neq
i}^{d}a_{ij}\sum_{l=1}^{k-1}e^{\alpha(l+1)\Delta}C^{\prime}p_{j}e^{-\beta
l\Delta}$
$\displaystyle\quad+\Delta\sum_{j=1}^{d}b_{ij}\sum_{l=1}^{k-1}e^{\alpha(l+1)\Delta}C^{\prime}p_{j}e^{-\beta(l-[\frac{\tau(l\Delta)}{\Delta}])\Delta}$
$\displaystyle\leq|F_{0}^{i}|^{2}+C^{\prime}\Delta\left(\sum_{j\neq
i}^{d}a_{ij}p_{j}+\sum_{j=1}^{d}b_{ij}p_{j}e^{\beta\tau}\right)e^{\alpha\Delta}\frac{e^{(\alpha-\beta)k\Delta}-1}{e^{(\alpha-\beta)\Delta}-1}$
$\displaystyle=|F_{0}^{i}|^{2}+C^{\prime}\Delta\left(\varepsilon
a_{ii}p_{i}+\sum_{j\neq
i}^{d}a_{ij}p_{j}+\sum_{j=1}^{d}b_{ij}p_{j}e^{\beta\tau}\right)e^{\alpha\Delta}\frac{e^{(\alpha-\beta)k\Delta}-1}{e^{(\alpha-\beta)\Delta}-1}$
$\displaystyle\quad-C^{\prime}\Delta\varepsilon
a_{ii}p_{i}e^{\alpha\Delta}\frac{e^{(\alpha-\beta)k\Delta}-1}{e^{(\alpha-\beta)\Delta}-1}.$

Since (3.1) holds for
$\varepsilon<\frac{\min_{i}|a_{ii}|}{d\max_{i}|a_{ii}|}$, then there exist
$\beta>0$ small enough and $\varepsilon^{\prime}>0$ such that
$\varepsilon\frac{\max_{i}|a_{ii}|}{\min_{i}|a_{ii}|}<\varepsilon^{\prime}<\frac{1}{d}$
and

$\varepsilon a_{ii}p_{i}+\sum_{j\neq
i}a_{ij}p_{j}+\sum_{j=1}^{d}b_{ij}p_{j}e^{\beta\tau}\leq-\varepsilon^{\prime}\beta
p_{i}$ (3.8)

for any $i\in\underline{d}$.

It follows that

$\displaystyle e^{\alpha k\Delta}E|F_{k}^{i}|^{2}$
$\displaystyle\leq|F_{0}^{i}|^{2}+C^{\prime}\Delta(-\varepsilon^{\prime}\beta-\varepsilon
a_{ii})p_{i}e^{\alpha\Delta}\frac{e^{(\alpha-\beta)k\Delta}-1}{e^{(\alpha-\beta)\Delta}-1}$
$\displaystyle\leq|F_{0}^{i}|^{2}+C^{\prime}\Delta\varepsilon^{\prime}(\alpha-\beta)p_{i}\frac{e^{(\alpha-\beta)k\Delta}-1}{e^{(\alpha-\beta)\Delta}-1}.$

We have used in the last inequality the fact that
$(-\varepsilon^{\prime}\beta-\varepsilon
a_{ii})e^{\alpha\Delta}\leq\varepsilon^{\prime}(\alpha-\beta)$ if
$\alpha\in(\frac{\varepsilon}{\varepsilon^{\prime}}\max_{i}|a_{ii}|,\min_{i}|a_{ii}|)$
and $\Delta$ is sufficiently small.

Moreover, by the fact that $x\leq e^{x}-1,\forall x>0$, we have

$\displaystyle e^{\alpha k\Delta}E|F_{k}^{i}|^{2}$
$\displaystyle<|F_{0}^{i}|^{2}+C^{\prime}\varepsilon^{\prime}p_{i}(e^{(\alpha-\beta)k\Delta}-1)$
$\displaystyle=|F_{0}^{i}|^{2}-C^{\prime}\varepsilon^{\prime}p_{i}+C^{\prime}\varepsilon^{\prime}p_{i}e^{(\alpha-\beta)k\Delta}$
$\displaystyle\leq
C^{\prime}\varepsilon^{\prime}p_{i}e^{(\alpha-\beta)k\Delta}.$

Thus

$E|F_{k}^{i}|^{2}\leq C^{\prime}\varepsilon^{\prime}p_{i}e^{-\beta k\Delta}.$
(3.9)

On the other hand, since

$\displaystyle|F_{k}^{i}|^{2}$ $\displaystyle=|X_{k}^{i}|^{2}-2\theta\Delta
X_{k}^{i}f_{k}^{i}+\theta^{2}\Delta^{2}|f_{k}^{i}|^{2}$
$\displaystyle\geq|X_{k}^{i}|^{2}-\theta\Delta\left(\sum_{j=1}^{d}a_{ij}|X_{k}^{j}|^{2}+\sum_{j=1}^{d}b_{ij}|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}^{j}|^{2}\right),$

then

$(1-\theta\Delta
a_{ii})|X_{k}^{i}|^{2}\leq|F_{k}^{i}|^{2}+\theta\Delta\left(\sum_{j\neq
i}a_{ij}|X_{k}^{j}|^{2}+\sum_{j=1}^{d}b_{ij}|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}^{j}|^{2}\right).$

Therefore

$\displaystyle\sum_{i=1}^{d}\frac{|X_{k}^{i}|^{2}}{p_{i}}$
$\displaystyle\leq\sum_{i=1}^{d}\frac{|F_{k}^{i}|^{2}}{p_{i}(1-\theta\Delta
a_{ii})}+\sum_{i=1}^{d}\frac{\theta\Delta}{1-\theta\Delta a_{ii}}\sum_{j\neq
i}\frac{a_{ij}p_{j}}{p_{i}}\frac{|X_{k}^{j}|^{2}}{p_{j}}$
$\displaystyle\quad+\sum_{j=1}^{d}\left(\sum_{i=1}^{d}\frac{\theta\Delta}{1-\theta\Delta
a_{ii}}\frac{b_{ij}p_{j}}{p_{i}}\right)\frac{|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}^{j}|^{2}}{p_{j}}$
$\displaystyle\leq\sum_{i=1}^{d}\frac{|F_{k}^{i}|^{2}}{p_{i}(1-\theta\Delta
a_{ii})}+\left(\sum_{i=1}^{d}\frac{\theta\Delta}{1-\theta\Delta
a_{ii}}\max_{j\neq
i}\frac{a_{ij}p_{j}}{p_{i}}\right)\sum_{j=1}^{d}\frac{|X_{k}^{j}|^{2}}{p_{j}}$
$\displaystyle\quad+\sum_{j=1}^{d}\left(\sum_{i=1}^{d}\frac{\theta\Delta}{1-\theta\Delta
a_{ii}}\frac{b_{ij}p_{j}}{p_{i}}\right)\frac{|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}^{j}|^{2}}{p_{j}}.$

Then

$\displaystyle\sum_{i=1}^{d}\frac{|X_{k}^{i}|^{2}}{p_{i}}$
$\displaystyle\leq\frac{1}{1-D}\left(\sum_{i=1}^{d}\frac{|F_{k}^{i}|^{2}}{p_{i}(1-\theta\Delta
a_{ii})}+\sum_{j=1}^{d}\left(\sum_{i=1}^{d}\frac{\theta\Delta}{1-\theta\Delta
a_{ii}}\frac{b_{ij}p_{j}}{p_{i}}\right)\frac{|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}^{j}|^{2}}{p_{j}}\right)$

where $D=\sum_{i=1}^{d}\frac{\theta\Delta}{1-\theta\Delta a_{ii}}\max_{j\neq
i}\frac{a_{ij}p_{j}}{p_{i}}$.

By (3.9) and inductive hypothesis (3.7), it follows that

$\displaystyle\sum_{i=1}^{d}\frac{E|X_{k}^{i}|^{2}}{p_{i}}$
$\displaystyle\leq\frac{1}{1-D}\left(\frac{dC^{\prime}\varepsilon^{\prime}e^{-\beta
k\Delta}}{1-\theta\Delta\max_{i}|a_{ii}|}+C^{\prime}e^{-\beta
k\Delta}e^{\beta\tau}\sum_{j=1}^{d}\sum_{i=1}^{d}\frac{\theta\Delta}{1-\theta\Delta
a_{ii}}\frac{b_{ij}p_{j}}{p_{i}}\right)$ $\displaystyle=C^{\prime}e^{-\beta
k\Delta}\frac{1}{1-D}\left(\frac{d\varepsilon^{\prime}}{1-\theta\Delta\max_{i}|a_{ii}|}+e^{\beta\tau}\sum_{j=1}^{d}\sum_{i=1}^{d}\frac{\theta\Delta}{1-\theta\Delta
a_{ii}}\frac{b_{ij}p_{j}}{p_{i}}\right).$

Denote
$G(\Delta):=\frac{1}{1-D}\left(\frac{d\varepsilon^{\prime}}{1-\theta\Delta\max_{i}|a_{ii}|}+e^{\beta\tau}\sum_{j=1}^{d}\sum_{i=1}^{d}\frac{\theta\Delta}{1-\theta\Delta
a_{ii}}\frac{b_{ij}p_{j}}{p_{i}}\right)$.

Since $G(0)=d\varepsilon^{\prime}<1$, and $G$ is continuous, then there exists
$\Delta^{*}$ small enough such that $G(\Delta)\leq 1$ for all
$\Delta\leq\Delta^{*}.$

Then we have

$\displaystyle\sum_{i=1}^{d}\frac{E|X_{k}^{i}|^{2}}{p_{i}}$ $\displaystyle\leq
C^{\prime}e^{-\beta k\Delta}.$

Thus

$E|X_{k}^{i}|^{2}\leq C^{\prime}p_{i}e^{-\beta k\Delta},i=1,2,...,d.$

That is, $\theta$-EM method is exponentially stable in mean square sense.

Almost surely exponential stability of $\theta$-EM method $X_{k}$ is a direct
consequence of mean square exponential stability by using Chebyshev inequality
and Borel-Cantelli Lemma (see e.g. [3]).

We complete the proof. $\square$

###### Theorem 3.3

Suppose all conditions in Theorem 3.1 hold, and there exists a constant $K>0$
such that

$|f(x,y)|\leq K(|x|+|y|).$ (3.10)

Then for any fixed $0\leq\theta\leq\frac{1}{2},$ $\theta$-EM method is mean
square and almost surely exponentially stable for sufficient small step size
$\Delta>0$.

Proof. Firstly, it still holds that

$\displaystyle|F^{i}_{k+1}|^{2}$
$\displaystyle\leq|F^{i}_{k}|^{2}+[2X^{i}_{k}f_{i}(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]})+\sum_{l=1}^{m}g_{il}(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]})^{2}$
(3.11)
$\displaystyle\quad+(1-2\theta)|f_{i}(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]})|^{2}\Delta]\Delta+M^{i}_{k}.$

Notice that linear growth condition (3.10) on $f$ implies

$\displaystyle|F^{i}_{k}|^{2}$ $\displaystyle=|X^{i}_{k}|^{2}-2\theta\Delta
X^{i}_{k}f_{i}(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]})+\theta^{2}\Delta^{2}|f_{i}(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]})|^{2}$
(3.12) $\displaystyle\leq|X^{i}_{k}|^{2}+2\theta\Delta
K|X^{i}_{k}|(|X_{k}|+|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}|)+2\theta^{2}\Delta^{2}K^{2}(|X_{k}|^{2}+|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}|^{2})$
$\displaystyle\leq(1+4\theta\Delta
K+2\theta^{2}\Delta^{2}K^{2})|X^{i}_{k}|^{2}+(\theta\Delta
K+2\theta^{2}\Delta^{2}K^{2})(\sum_{j\neq
i}^{d}|X^{j}_{k}|^{2}+|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}|^{2}),$

then we have

$|X^{i}_{k}|^{2}\geq\frac{|F^{i}_{k}|^{2}-(\theta\Delta
K+2\theta^{2}\Delta^{2}K^{2})(\sum_{j\neq
i}^{d}|X^{j}_{k}|^{2}+|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}|^{2})}{1+4\theta\Delta
K+2\theta^{2}\Delta^{2}K^{2}}.$

If $\theta\in[0,\frac{1}{2}],$ then by Assumption 2.3 and (3.10), we have

$\displaystyle|F^{i}_{k+1}|^{2}$
$\displaystyle\leq|F^{i}_{k}|^{2}+[\sum_{j=1}^{d}a_{ij}|X_{k}^{j}|^{2}+\sum_{j=1}^{d}b_{ij}|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}^{j}|^{2}$
(3.13)
$\displaystyle\quad+2(1-2\theta)K^{2}(|X_{k}|^{2}+|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}|^{2})\Delta]\Delta+M^{i}_{k}$
$\displaystyle=|F^{i}_{k}|^{2}+(2(1-2\theta)K^{2}\Delta+a_{ii})|X^{i}_{k}|^{2}\Delta+\sum_{j\neq
i}(2(1-2\theta)K^{2}\Delta+a_{ij})|X^{j}_{k}|^{2}\Delta$
$\displaystyle\quad+\sum_{j=1}^{d}(2(1-2\theta)K^{2}\Delta+b_{ij})|X^{j}_{k-[\frac{\tau(k\Delta)}{\Delta}]}|^{2}\Delta+M^{i}_{k}$
$\displaystyle\leq|F^{i}_{k}|^{2}+(2(1-2\theta)K^{2}\Delta+a_{ii})\Delta$
$\displaystyle\quad\times\frac{|F^{i}_{k}|^{2}-(\theta\Delta
K+2\theta^{2}\Delta^{2}K^{2})(\sum_{j\neq
i}^{d}|X^{j}_{k}|^{2}+|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}|^{2})}{1+4\theta\Delta
K+2\theta^{2}\Delta^{2}K^{2}}$ $\displaystyle\quad+\sum_{j\neq
i}(2(1-2\theta)K^{2}\Delta+a_{ij})|X^{j}_{k}|^{2}\Delta$
$\displaystyle\quad+\sum_{j=1}^{d}(2(1-2\theta)K^{2}\Delta+b_{ij})|X^{j}_{k-[\frac{\tau(k\Delta)}{\Delta}]}|^{2}\Delta+M^{i}_{k}.$

We have used the fact that $2(1-2\theta)K^{2}\Delta+a_{ii}<0$ in the last
inequality.

Thus

$\displaystyle|F^{i}_{k+1}|^{2}$
$\displaystyle\leq|F^{i}_{k}|^{2}\left(1+\frac{(2(1-2\theta)K^{2}\Delta+a_{ii})\Delta}{1+4\theta\Delta
K+2\theta^{2}\Delta^{2}K^{2}}\right)$ $\displaystyle\quad+\sum_{j\neq
i}\left(2(1-2\theta)K^{2}\Delta-(\theta K+2\theta^{2}\Delta
K^{2})\frac{(2(1-2\theta)K^{2}\Delta+a_{ii})\Delta}{1+4\theta\Delta
K+2\theta^{2}\Delta^{2}K^{2}}+a_{ij}\right)|X^{i}_{k}|^{2}\Delta$
$\displaystyle\quad+\sum_{j=1}^{d}\left(2(1-2\theta)K^{2}\Delta-(\theta
K+2\theta^{2}\Delta
K^{2})\frac{(2(1-2\theta)K^{2}\Delta+a_{ii})\Delta}{1+4\theta\Delta
K+2\theta^{2}\Delta^{2}K^{2}}+b_{ij}\right)$
$\displaystyle\quad\times|X^{i}_{k-[\frac{\tau(k\Delta)}{\Delta}]}|^{2}\Delta+M^{i}_{k}.$

It is obvious that

$C_{1,\Delta}:=\frac{2(1-2\theta)K^{2}\Delta+a_{ii}}{1+4\theta\Delta
K+2\theta^{2}\Delta^{2}K^{2}}\to a_{ii},$
$C_{2,\Delta}:=2(1-2\theta)K^{2}\Delta-(\theta K+2\theta^{2}\Delta
K^{2})\frac{(2(1-2\theta)K^{2}\Delta+a_{ii})\Delta}{1+4\theta\Delta
K+2\theta^{2}\Delta^{2}K^{2}}+a_{ij}\to a_{ij}$

and

$C_{3,\Delta}:=2(1-2\theta)K^{2}\Delta-(\theta K+2\theta^{2}\Delta
K^{2})\frac{(2(1-2\theta)K^{2}\Delta+a_{ii})\Delta}{1+4\theta\Delta
K+2\theta^{2}\Delta^{2}K^{2}}+b_{ij}\to b_{ij}$

as $\Delta\to 0.$

Then for any $C<\min_{i}|a_{ii}|$, we can choose $\tilde{\varepsilon}>0$ and
$\Delta>0$ small enough such that

$|F^{i}_{k+1}|^{2}\leq|F^{i}_{k}|^{2}(1-C\Delta)+\sum_{j\neq
i}(a_{ij}+\tilde{\varepsilon})|X^{j}_{k}|^{2}\Delta+\sum_{j=1}^{d}(b_{ij}+\tilde{\varepsilon})|X^{j}_{k-[\frac{\tau(k\Delta)}{\Delta}]}|^{2}+M^{i}_{k}.$

Note that (3.1) implies that there exist $\beta>0$ small enough and
$\varepsilon^{\prime}>0$ such that
$\varepsilon\frac{\max_{i}|a_{ii}|}{\min_{i}|a_{ii}|}<\varepsilon^{\prime}<\frac{1}{d}$
and

$\varepsilon a_{ii}p_{i}+\sum_{j\neq
i}(a_{ij}+\tilde{\varepsilon})p_{j}+\sum_{j=1}^{d}(b_{ij}+\tilde{\varepsilon})p_{j}e^{\beta\tau}\leq-\varepsilon^{\prime}\beta
p_{i}$ (3.14)

for any $i\in\underline{d}$.

That is, (3.8) holds for $a_{ij}$ and $b_{ij}$ replaced by
$a_{ij}+\tilde{\varepsilon}$ and $b_{ij}+\tilde{\varepsilon}$, respectively.

Then repeat the following part of the proof of Theorem 3.1 from line to line,
we complete the proof. $\square$

##  4 Mean square exponential stability of MTEM method

In this Section, we will prove that MTEM method replicates the mean
exponential stability of exact solution under given conditions.

To obtain the mean exponential stability of MTEM method, we need the following
two Lemmas.

###### Lemma 4.1

Suppose the Assumption 2.1 and Assumption 2.3 hold. Then for any fixed
$\Delta>0$,there exist constant matrices $a_{ii}\in\mathbb{R};a_{ij}\geq
0,i\neq j;b_{ij}\geq 0.$ $i\in\underline{d}$, such that

$2x_{i}\cdot
f_{\Delta,i}(x,y)+\sum_{l=1}^{m}g^{2}_{\Delta,il}(x,y)\leq\sum_{j=1}^{d}a_{ij}x_{j}^{2}+\sum_{j=1}^{d}b_{ij}y_{j}^{2},\quad
i\in\underline{d}$ (4.1)

for any $x,y\in\mathbb{R}^{d}$.

Proof.

If $|x|\vee|y|\leq h(\Delta)$, then (4.1) holds naturally by (2.3). Now assume
$|x|\vee|y|>h(\Delta)$. Then by $\eqref{2.4}$, we have

$\displaystyle\quad
2x_{i}f_{\Delta,i}(x,y)+\sum_{l=1}^{m}g_{\Delta,il}^{2}(x,y)$
$\displaystyle=2x_{i}\frac{|x|\vee|y|}{h(\Delta)}\cdot
f_{i}(\frac{h(\Delta)}{|x|\vee|y|}(x,y))+\sum_{l=1}^{m}(\frac{|x|\vee|y|}{h(\Delta)}\cdot
g_{il}(\frac{h(\Delta)}{|x|\vee|y|}(x,y)))^{2}$
$\displaystyle=(\frac{|x|\vee|y|}{h(\Delta)})^{2}\cdot[2x_{i}\frac{h(\Delta)}{|x|\vee|y|}\cdot
f_{i}(\frac{h(\Delta)}{|x|\vee|y|}(x,y))+\sum_{l=1}^{m}g_{il}^{2}(\frac{h(\Delta)}{|x|\vee|y|}(x,y))]$
$\displaystyle\leq(\frac{|x|\vee|y|}{h(\Delta)})^{2}\cdot[\sum_{j=1}^{d}a_{ij}(x_{j}\frac{h(\Delta)}{|x|\vee|y|})^{2}+\sum_{j=1}^{d}b_{ij}(\frac{h(\Delta)}{|x|\vee|y|}x_{j})^{2}]$
$\displaystyle=\sum_{j=1}^{d}a_{ij}x_{j}^{2}+\sum_{j=1}^{d}b_{ij}y_{j}^{2}.$

Then (4.1) holds.

We complete the proof.

###### Lemma 4.2

Suppose Assumption 2.1 holds. Then for any fixed $\Delta>0,$ the modified
truncated functions $f_{\Delta}$ and $g_{\Delta}$ are linear growing with
coefficient $L_{h(\Delta)}$. That is

$|f_{\Delta}(x,y)|\vee|g_{\Delta}(x,y)|\leq L_{h(\Delta)}(|x|+|y|)$ (4.2)

holds for all $x,y\in\mathbb{R}^{d}.$

Proof. Since $f_{\Delta}$ and $g_{\Delta}$ are defined in the same way, we
only need prove linear growth condition of $f$. If $|x|\vee|y|\leq h(\Delta)$,
by definition of $f_{\Delta}$ and $g_{\Delta}$, it follows that

$|f_{\Delta}(x,y)|=|f(x,y)|=|f(x,y)-f(0,0)|\leq L_{h(\Delta)}(|x-0|+|y-0|).$

If $|x|\vee|y|>h(\Delta)$, denote $a=\frac{h(\Delta)}{|x|\vee|y|}$. Since
$|ax|\vee|ay|\leq h(\Delta)$, then

$|f_{\Delta}(x,y)|=\frac{1}{a}|f(ax,ay)|=\frac{1}{a}|f(ax,ay)-f(0,0)|\leq\frac{1}{a}L_{h(\Delta)}(|ax-0|+|ay-0|).$

We complete the proof.

###### Remark 4.3

In [3], the author proved that both $f_{\Delta}$ and $g_{\Delta}$ are globally
Lipschitz continuous for any fixed $\Delta>0$. However, we can only obtain
$|f_{\Delta}(x,y)|\leq 5L_{h(\Delta)}(|x|+|y|)$ while by Lemma 4.2, we have
$|f_{\Delta}(x,y)|\leq L_{h(\Delta)}(|x|+|y|)$.

Now we are ready to present the main result in this section.

###### Theorem 4.4

Suppose Assumptions 2.1 and 2.3 hold. Then the MTEM method (2.6) is mean quare
and almost surely exponentially stable.

Proof. By (2.6), for any $k\geqslant 0$, we have

$\displaystyle
X_{k+1}^{i}=X_{k}^{i}+f_{\Delta,i}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta+\sum_{l=1}^{m}g_{\Delta,il}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta
B^{l}_{k}.$ (4.3)

So,

$\displaystyle|X_{k+1}^{i}|^{2}$
$\displaystyle=|X_{k}^{i}|^{2}+(2X_{k}^{i}f_{\Delta,i}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)+\sum_{l=1}^{m}g_{\Delta,il}^{2}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right))\Delta$
(4.4)
$\displaystyle\quad+f^{2}_{\Delta,i}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta^{2}+M^{i}_{k}$

where

$\displaystyle M^{i}_{k}$ $\displaystyle=2\langle
X_{k}^{i},\sum_{l=1}^{m}g_{\Delta,il}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta
B^{l}_{k}\rangle$ $\displaystyle\quad+2\langle
f_{\Delta,i}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right),\sum_{l=1}^{m}g_{\Delta,il}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta
B^{l}_{k}\rangle\Delta$
$\displaystyle\quad+\left(\left|\sum_{l=1}^{m}g_{\Delta,il}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta
B^{l}_{k}\right|^{2}-\sum_{l=1}^{m}g^{2}_{\Delta,il}\left(X_{k},X_{k-[\frac{\tau(k\Delta)}{\Delta}]}\right)\Delta\right).$

It is obvious that $M^{i}_{k}$ is a $\mathscr{F}_{k\Delta}$ martingale and
$EM^{i}_{k}=0$. Then, by Lemmas 4.1 and 4.2, we have

$\displaystyle\quad e^{\alpha(k+1)\Delta}E|X_{k+1}^{i}|^{2}-e^{\alpha
k\Delta}E|X_{k}^{i}|^{2}$ $\displaystyle\leq
e^{\alpha(k+1)\Delta}(1-e^{-\alpha\Delta})E|X_{k}^{i}|^{2}+(\sum_{j=1}^{d}a_{ij}E|X_{k}^{j}|^{2}+\sum_{j=1}^{d}b_{ij}E|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}^{j}|^{2})e^{\alpha(k+1)\Delta}\Delta$
$\displaystyle\quad+2L_{h(\Delta)}^{2}\Delta^{2}(E|X_{k}|^{2}+E|X_{k-[\frac{\tau(k\Delta)}{\Delta}]}|^{2})e^{\alpha(k+1)\Delta}$
$\displaystyle\leq
e^{\alpha(k+1)\Delta}(1-e^{-\alpha\Delta})E|X_{k}^{i}|^{2}+\sum_{j=1}^{d}(a_{ij}+2L_{h(\Delta)}^{2}\Delta)E|X_{k}^{j}|^{2}e^{\alpha(k+1)\Delta}\Delta$
$\displaystyle\quad+(\sum_{j=1}^{d}b_{ij}+2L^{2}_{h(\Delta)}\Delta)E|X^{j}_{k-[\frac{\tau(k\Delta)}{\Delta}]}|^{2}e^{\alpha(k+1)\Delta}\Delta.$

Since (2.9) holds, similar to the proof of Theorem 2.4, there exist $\beta>0$
and $\varepsilon>0$ small enough such that

$\sum_{j=1}^{d}\left(a_{ij}+\varepsilon+e^{\beta\tau}(b_{ij}+\varepsilon)\right)p_{j}\leq-\beta
p_{i}$ (4.5)

for any $i\in\underline{d}$.

Obviously, for the same $\bar{K}$, we have

$E|X_{k}^{i}|^{2}\leq\bar{K}p_{i}e^{-\beta k\Delta}E||\xi||^{2},\quad
k=-\bar{m},-\bar{m}+1,\ldots,0.$ (4.6)

Now assume that

$E|X_{k}^{i}|^{2}\leq\bar{K}p_{i}e^{-\beta k\Delta}E||\xi||^{2},\quad
k\leqslant n-1,i=1,...,d$ (4.7)

for some $\beta>0$.

Now for $k=n$, it follows that

$\displaystyle e^{\alpha n\Delta}E|X_{n}^{i}|^{2}$ $\displaystyle\leq
E|X_{0}^{i}|^{2}+\sum_{l=0}^{n-1}e^{\alpha(l+1)\Delta}(1-e^{-\alpha\Delta})E|X_{l}^{i}|^{2}$
$\displaystyle\quad+\sum_{l=0}^{n-1}\sum_{j=1}^{d}(a_{ij}+2L_{h(\Delta)}^{2}\Delta)\Delta
E|X_{l}^{j}|^{2}e^{\alpha(l+1)\Delta}$
$\displaystyle\quad+\sum_{l=0}^{n-1}\sum_{j=1}^{d}(b_{ij}+2L_{h(\Delta)}^{2}\Delta)\Delta
E|X_{l-[\frac{\tau(l\Delta)}{\Delta}]}^{j}|^{2}e^{\alpha(l+1)\Delta}$
$\displaystyle=E|X_{0}^{i}|^{2}+\sum_{l=0}^{n-1}e^{\alpha(l+1)\Delta}(1-e^{-\alpha\Delta}+(a_{ii}+2L_{h(\Delta)}^{2}\Delta)\Delta)E|X_{l}^{i}|^{2}$
$\displaystyle\quad+\sum_{l=0}^{n-1}\sum_{i\neq
j}(a_{ij}+2L_{h(\Delta)}^{2}\Delta)\Delta
e^{\alpha(l+1)\Delta}E|X_{l}^{j}|^{2}$
$\displaystyle\quad+\sum_{l=0}^{n-1}\sum_{j=1}^{d}(b_{ij}+2L_{h(\Delta)}^{2}\Delta)\Delta
e^{\alpha(l+1)\Delta}E|X_{l-[\frac{\tau(l\Delta)}{\Delta}]}^{j}|^{2}.$

If we choose $\alpha>\max\limits_{i\in\underline{d}}{|a_{ii}|}\vee\beta$, then
it is obvious that
$1-e^{-\alpha\Delta}+(a_{ii}+2L_{h(\Delta)}^{2}\Delta)\Delta>0$ holds for any
$\Delta>0.$ Therefore, by the induction hypothesis (4.7), we have

$\displaystyle e^{\alpha n\Delta}E|X_{n}^{i}|^{2}$ $\displaystyle\leq
E|X_{0}^{i}|^{2}+\sum_{l=0}^{n-1}e^{\alpha(l+1)\Delta}(1-e^{-\alpha\Delta}+(a_{ii}+2L_{h(\Delta)}^{2}\Delta)\Delta)K_{1}p_{i}e^{-\beta
l\Delta}$ $\displaystyle\quad+\sum_{l=0}^{n-1}\sum_{i\neq
j}(a_{ij}+2L_{h(\Delta)}^{2}\Delta)\Delta
e^{\alpha(l+1)\Delta}K_{1}p_{j}e^{-\beta l\Delta}$
$\displaystyle\quad+\sum_{l=0}^{n-1}\sum_{j=1}^{d}(b_{ij}+2L_{h(\Delta)}^{2}\Delta)\Delta
e^{\alpha(l+1)\Delta}K_{1}p_{j}e^{-\beta(l-[\frac{\tau(l\Delta)}{\Delta}])\Delta}$
$\displaystyle\leq
E|X_{0}^{i}|^{2}+\sum_{l=0}^{n-1}(e^{\alpha\Delta}-1)K_{1}p_{i}\cdot
e^{(\alpha-\beta)l\Delta}$
$\displaystyle\quad+\sum_{l=0}^{n-1}\sum_{j=1}^{d}((a_{ij}+2L_{h(\Delta)}^{2}\Delta)+(b_{ij}+2L_{h(\Delta)}^{2}\Delta)e^{\beta\tau})p_{j}\Delta
K_{1}e^{\alpha\Delta}e^{(\alpha-\beta)l\Delta}.$

where $K_{1}=\bar{K}E||\xi||^{2}.$

Then if we choose the stepsize $\Delta>0$ sufficiently small such that
$2L_{h(\Delta)}^{2}\Delta\leq\varepsilon$, (4.5) yields

$\displaystyle e^{\alpha n\Delta}E|X_{n}^{i}|^{2}$ $\displaystyle\leq
E|X_{0}^{i}|^{2}+K_{1}p_{i}\cdot(e^{\alpha\Delta}-1)\cdot\frac{e^{(\alpha-\beta)n\Delta}-1}{e^{(\alpha-\beta)\Delta}-1}-K_{1}\beta
p_{i}\Delta
e^{\alpha\Delta}\frac{e^{(\alpha-\beta)n\Delta}-1}{e^{(\alpha-\beta)\Delta}-1}$
$\displaystyle=E|X_{0}^{i}|^{2}+K_{1}p_{i}\cdot(e^{\alpha\Delta}-1-\beta\Delta
e^{\alpha\Delta})\cdot\frac{e^{(\alpha-\beta)n\Delta}-1}{e^{(\alpha-\beta)\Delta}-1}.$

On the other hand, it is obvious that

$\displaystyle e^{\alpha\Delta}-1-\beta\Delta
e^{\alpha\Delta}=(1-\beta\Delta)e^{\alpha\Delta}-1\leq e^{-\beta\Delta}\cdot
e^{\alpha\Delta}-1=e^{(\alpha-\beta)\Delta}-1.$ (4.8)

Therefore,

$\displaystyle e^{\alpha n\Delta}E|X^{i}_{n}|^{2}$ $\displaystyle\leq
E|X_{0}^{i}|^{2}+K_{1}p_{i}\cdot(e^{(\alpha-\beta)n\Delta}-1)$ (4.9)
$\displaystyle=E|X_{0}^{i}|^{2}-K_{1}p_{i}+K_{1}p_{i}\cdot
e^{(\alpha-\beta)n\Delta}$ $\displaystyle\leq K_{1}p_{i}\cdot
e^{(\alpha-\beta)n\Delta}.$

Thus,

$\displaystyle E|X_{n}^{i}|^{2}\leq K_{1}p_{i}\cdot e^{-\beta n\Delta}.$
(4.10)

Similar to almost sure exponential stability of $\theta$-EM method, a standard
procedure of using Chebyshev inequality and Borel-Cantelli’s Lemma implies
that

$\frac{\log|X_{n}^{i}|}{n\Delta}\leq-\frac{\beta}{2}.$

We complete the proof. $\square$

##  5 Examples

Now let us present two examples to interpret our theory.

Example 1 Consider a 2-D stochastic differential delay equation given by

$\displaystyle dx_{1}(t)$
$\displaystyle=(-\frac{1}{5}x_{1}(t)-\frac{2}{9}x_{1}^{3}(t)+\frac{4}{5}x_{2}(t)+\frac{1}{10^{4}}x_{1}(t-\tau(t))+\frac{1}{10^{4}}x_{2}(t-\tau(t)))dt$
(5.1) $\displaystyle\quad+\frac{2}{3}x^{2}_{1}(t)dB_{1}(t)$ $\displaystyle
dx_{2}(t)$
$\displaystyle=(\frac{\sqrt{38}}{625}x_{1}(t)-x_{2}(t)+\frac{1}{10^{4}}x_{1}(t-\tau(t))+\frac{1}{10^{4}}x_{2}(t-\tau(t)))dt+\sqrt{\frac{2}{5}}x_{2}(t)dB_{2}(t)$

for $t\geq 0$, where the initial value $\xi(s)\in
C([-\tau,0],\mathbb{R}^{d}),\tau(t)=0.1(1-|\sin(t)|)\leq\tau=0.1$, and
$B(\cdot)=\left(B_{1}(\cdot),B_{2}(\cdot)\right)^{T}$ is a 2-D Brownian
motion.

Obviously, (5.1) is of the form (1.1) with
$f(x,y):=\left(f_{1}(x,y),f_{2}(x,y)\right)^{T}\in\mathbb{R}^{2}$ defined by

$\displaystyle
f_{1}(x,y):=-\frac{1}{5}x_{1}-\frac{2}{9}x_{1}^{3}+\frac{4}{5}x_{2}+\frac{1}{10^{4}}y_{1}+\frac{1}{10^{4}}y_{2},$
$\displaystyle
f_{2}(x,y):=\frac{\sqrt{38}}{625}x_{1}-x_{2}+\frac{1}{10^{4}}y_{1}+\frac{1}{10^{4}}y_{2}$

for
$x:=\left(x_{1},x_{2}\right)^{T}\in\mathbb{R}^{2},y:=\left(y_{1},y_{2}\right)^{T}\in\mathbb{R}^{2}$,
and

$g(x,y):=\left(\begin{array}[]{cc}\frac{2}{3}x^{2}_{1}&0\\\
0&\sqrt{\frac{2}{5}}x_{2}\end{array}\right)$

for
$x:=\left(x_{1},x_{2}\right)^{T}\in\mathbb{R}^{2},y:=\left(y_{1},y_{2}\right)^{T}\in\mathbb{R}^{2}$.
Clearly

$\displaystyle 2x_{1}f_{1}(x,y)+g_{11}^{2}(x,y)+g_{12}^{2}(x,y)$
$\displaystyle=-\frac{2}{5}x_{1}^{2}-\frac{4}{9}x_{1}x_{1}^{3}+\frac{8}{5}x_{1}x_{2}+\frac{2}{10^{4}}x_{1}y_{1}+\frac{2}{10^{4}}x_{1}y_{2}+\frac{4}{9}x_{1}^{4}$
$\displaystyle\leq-\frac{399}{5000}x_{1}^{2}+2x_{2}^{2}+\frac{1}{10^{4}}y_{1}^{2}+\frac{1}{10^{4}}y_{2}^{2}$

for any $x,y\in\mathbb{R}^{2}$ and

$\displaystyle 2x_{2}f_{2}(x,y)+g_{21}^{2}(x,y)+g_{22}^{2}(x,y)$
$\displaystyle=\frac{2\sqrt{38}}{625}x_{1}x_{2}-\frac{8}{5}x_{2}^{2}+\frac{2}{10^{4}}x_{2}y_{1}+\frac{2}{10^{4}}x_{2}y_{2}$
$\displaystyle\leq\frac{1}{5^{6}}x_{1}^{2}-\frac{399}{5000}x_{2}^{2}+\frac{1}{10^{4}}y_{1}^{2}+\frac{1}{10^{4}}y_{2}^{2}$

for any $y\in\mathbb{R}^{2}$.

It is obvious that both $f$ and $g$ are locally Lipschitz continuous, and
(2.8) holds for $K=2$. Thus, there exists a unique global solution to equation
(5.1).

On the other hand, if we take

$\displaystyle
a_{11}=-\frac{399}{5000},a_{12}=2,a_{21}=\frac{1}{5^{6}},a_{22}=-\frac{399}{5000},b_{11}=\frac{1}{10^{4}},b_{12}=\frac{1}{10^{4}},b_{21}=\frac{1}{10^{4}},b_{22}=\frac{1}{10^{4}},$

and choose
$\varepsilon=\frac{499}{1000}<\frac{1}{2}=\frac{\min_{i}a_{ii}}{d\max_{i}a_{ii}},p_{1}=500,p_{2}=1$,
we obtain that

$\displaystyle\varepsilon a_{11}p_{1}+a_{12}p_{2}+b_{11}p_{1}+b_{12}p_{2}<0$
$\displaystyle a_{21}p_{1}+\varepsilon a_{22}p_{2}+b_{21}p_{1}+b_{22}p_{2}<0.$

That is, (2.7) holds with $a_{ij},b_{ij}$ satisfies (3.1) with given $p_{i}$
and $\varepsilon$, which clearly implies (2.9) for the same $a_{ij},b_{ij}$
and $p_{i}$. Therefore, the trivial solution to (5.1) is exponentially stable
in mean square sense by Theorem 2.4. Moreover, it is easy to verify that (2.2)
holds for $L=1$. Thus $\theta$-EM method is well defined in this case if
$\Delta<1$. Then mean square and almost surely exponentially stable for the
$\theta$-EM method for any fixed $\theta\in(\frac{1}{2},1]$ by Theorem 3.1.
Note that all assumptions in Theorem 4.4 hold. Then Theorem 4.4 implies that
the corresponding MTEM method of (5.1) is also exponentially stable in mean
square.

Meanwhile, we can find

$\displaystyle 2\langle x,f(x,y)\rangle+||g(x,y)||^{2}$ (5.2) $\displaystyle=$
$\displaystyle
2x_{1}f_{1}(x,y)+2x_{2}f_{2}(x,y)+g_{11}^{2}(x,y)+g_{12}^{2}(x,y)+g_{21}^{2}(x,y)+g_{22}^{2}(x,y)$
$\displaystyle=$
$\displaystyle-\frac{2}{5}x_{1}^{2}+(\frac{2\sqrt{38}}{625}+\frac{8}{5})x_{1}x_{2}-\frac{8}{5}x_{2}^{2}+\frac{2}{10^{4}}x_{1}y_{1}+\frac{2}{10^{4}}x_{1}y_{2}+\frac{2}{10^{4}}x_{2}y_{1}+\frac{2}{10^{4}}x_{2}y_{2}.$

We claim that (5.2) could not be written as Khasminskii-type condition (1.2).
Indeed, it follows that

$\displaystyle 2\langle x,f(x,y)\rangle+||g(x,y)||^{2}$ $\displaystyle=$
$\displaystyle-\frac{2}{5}x_{1}^{2}+(\frac{2\sqrt{38}}{625}+\frac{8}{5})x_{1}x_{2}-\frac{8}{5}x_{2}^{2}+\frac{2}{10^{4}}x_{1}y_{1}+\frac{2}{10^{4}}x_{1}y_{2}+\frac{2}{10^{4}}x_{2}y_{1}+\frac{2}{10^{4}}x_{2}y_{2}$
$\displaystyle\leq$
$\displaystyle-\frac{2}{5}x_{1}^{2}+(\frac{\sqrt{38}}{625}+\frac{4}{5})(n_{1}x_{1}^{2}+\frac{1}{n_{1}}x_{2}^{2})-\frac{8}{5}x_{2}^{2}+\frac{1}{10^{4}}(n_{2}x_{1}^{2}+\frac{1}{n_{2}}y_{1}^{2})+\frac{1}{10^{4}}(n_{3}x_{1}^{2}+\frac{1}{n_{3}}y_{2}^{2})$
$\displaystyle+\frac{1}{10^{4}}(n_{4}x_{2}^{2}+\frac{1}{n_{4}}y_{1}^{2})+\frac{1}{10^{4}}(n_{5}x_{2}^{2}+\frac{1}{n_{5}}y_{2}^{2})$
$\displaystyle=$
$\displaystyle-\frac{1}{10^{4}}(4000-(16\sqrt{38}+8000)n_{1}-n_{2}-n_{3})x_{1}^{2}-\frac{1}{10^{4}}(16000-\frac{8000+16\sqrt{38}}{n_{1}}-n_{4}-n_{5})x_{2}^{2}$
$\displaystyle+\frac{1}{10^{4}}(\frac{1}{n_{2}}+\frac{1}{n_{4}})y_{1}^{2}+\frac{1}{10^{4}}(\frac{1}{n_{3}}+\frac{1}{n_{5}})y_{2}^{2}.$

for any $n_{1},n_{2},n_{3},n_{4},n_{5}>0$.

However, if (1.2) holds for some $C_{1}>C_{2}>0$, then we must have

$\frac{1}{10^{4}}\min\left\\{4000-(16\sqrt{38}+8000)n_{1}-n_{2}-n_{3},16000-\frac{8000+16\sqrt{38}}{n_{1}}-n_{4}-n_{5}\right\\}\geq
C_{1}$ (5.3)

for some $n_{1},n_{2},n_{3},n_{4},n_{5}>0.$

Since there is no $n_{1}>0$ such that $4000-(16\sqrt{38}+8000)n_{1}>0$ and
$16000-\frac{8000+16\sqrt{38}}{n_{1}}>0,$ (5.2) can never be written in the
form of Khasminskii-type conditions (1.2). However, we can get mean square
exponential stability of both the exact solution $x(t)$ and the $\theta$-EM
method $X_{k}$ for any fixed $\theta\in(\frac{1}{2},1]$ by Theorem 2.4 and
Theorem 3.1. And $X_{k}$ is also almost surely exponentially stable.

Example 2 Consider the following 2-D stochastic differential delay equation

$\displaystyle dx_{1}(t)$
$\displaystyle=(-\frac{2}{5}x_{1}(t)+\frac{4}{5}x_{2}(t)+\frac{1}{10^{4}}x_{1}(t-\tau(t))+\frac{1}{10^{4}}x_{2}(t-\tau(t)))dt$
(5.4) $\displaystyle\quad+\frac{\sqrt{10}}{5}x_{1}(t)dB_{1}(t)$ $\displaystyle
dx_{2}(t)$
$\displaystyle=(\frac{\sqrt{38}}{625}x_{1}(t)-x_{2}(t)+\frac{1}{10^{4}}x_{1}(t-\tau(t))+\frac{1}{10^{4}}x_{2}(t-\tau(t)))dt$
$\displaystyle\quad+\frac{\sqrt{10}}{5}x_{2}(t)dB_{2}(t)$

for $t\geq 0$, with initial value $\xi(s)\in C([-\tau,0],\mathbb{R}^{d})$,
where $\tau=0.1,\theta\in[-\tau,0],$ $\tau(t)=0.1|\sin(t)|\leq\tau$.

Obviously, (5.4) is of the form (1.1) with
$f(x,y):=\left(f_{1}(x,y),f_{2}(x,y)\right)^{T}\in\mathbb{R}^{2}$ defined by

$\displaystyle
f_{1}(x,y):=-\frac{2}{5}x_{1}+\frac{4}{5}x_{2}+\frac{1}{10^{4}}y_{1}+\frac{1}{10^{4}}y_{2},$
$\displaystyle
f_{2}(x,y):=\frac{\sqrt{38}}{625}x_{1}-x_{2}+\frac{1}{10^{4}}y_{1}+\frac{1}{10^{4}}y_{2}$

for
$x:=\left(x_{1},x_{2}\right)^{T}\in\mathbb{R}^{2},y:=\left(y_{1},y_{2}\right)^{T}\in\mathbb{R}^{2}$,
and

$g(x,y):=\left(\begin{array}[]{cc}\frac{\sqrt{10}}{5}x_{1}&0\\\
0&\frac{\sqrt{10}}{5}x_{2}\end{array}\right)$

for
$x:=\left(x_{1},x_{2}\right)^{T}\in\mathbb{R}^{2},y:=\left(y_{1},y_{2}\right)^{T}\in\mathbb{R}^{2}$.
Similar to Example 1, it is easy to verify that (2.7) holds for

$\displaystyle
a_{11}=-\frac{399}{5000},a_{12}=2,a_{21}=\frac{1}{5^{6}},a_{22}=-\frac{399}{5000},b_{11}=\frac{1}{10^{4}},b_{12}=\frac{1}{10^{4}},b_{21}=\frac{1}{10^{4}},b_{22}=\frac{1}{10^{4}},$

and (3.1) holds for $\varepsilon=\frac{499}{1000},p_{1}=500,p_{2}=1$. It is
easy to see $f$ satisfies (3.10). Therefore, it follows that the $\theta$-EM
method (it is well defined since $f$ is global Lipschitz continuous in this
case) is mean square and almost surely exponentially stable for any fixed
$0\leq\theta\leq\frac{1}{2}$ by Theorem 3.3.

Meanwhile, we can verify that (1.2) does not hold in the same way as in
Example 1. However, we obtain that mean square exponential stability of both
the exact solution $x(t)$ and the $\theta$-EM method $X_{k}$ for any fixed
$0\leq\theta\leq\frac{1}{2}$ by Theorem 2.4 and Theorem 3.3. And $X_{k}$ is
also almost surely exponentially stable.

## References

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# On transversal numbers of intersecting straight line systems and
intersecting segment systems

Adrián Vázquez Ávila
Subdirección de Ingeniería y Posgrado
Universidad Aeronáutica en Querétaro
<EMAIL_ADDRESS>
###### Abstract

An intersecting $r$-uniform straight line system is an intersecting linear
system whose lines consist of $r$ points on straight line segment of
$\mathbb{R}^{2}$ and any two lines share a point.

Recently, the author [A. Vázquez-Ávila, _On intersecting straight line
systems_ , J. Discret. Math. Sci. Cryptogr. Accepted] proved that any
intersecting $r$-uniform straight line system $(P,\mathcal{L})$ has
transversal number at most $\nu_{2}-1$, with $r\geq\nu_{2}$, where $\nu_{2}$
is the maximum cardinality of a subset of lines $R\subseteq\mathcal{L}$ such
that every triplet of different elements of $R$ do not have a common point.

In this paper, we improve such upper bound if the intersecting $r$-uniform
straight line system satisfies $r=\nu_{2}$. This result has immediate
consequences for some questions given by Oliveros et al. [D. Oliveros, C.
O’Neill and S. Zerbib, _The geometry and combinatorics of discrete line
segment hypergraphs_ , Discrete Math. 343 (2020), no. 6, 111825].

Keywords. Linear systems, straight line systems, segment systems, transversal
number, 2-packing number.

Math. Subj. Class.: 05C65, 05C69.

##  1 Introduction

A _linear system_ is a pair $(P,\mathcal{L})$ where $\mathcal{L}$ is a family
of subsets on a ground finite set $P$, such that $|l\cap l^{\prime}|\leq 1$,
for every pair of distinct subsets $l,l^{\prime}\in\mathcal{L}$. The linear
system $(P,\mathcal{L})$ is _intersecting_ if $|l\cap l^{\prime}|=1$, for
every pair of distinct subsets $l,l^{\prime}\in\mathcal{L}$. The elements of
$P$ and $\mathcal{L}$ are called _points_ and _lines_ , respectively. A line
with exactly $r$ points is called a _$r$ -line_; when all the lines of
$\mathcal{L}$ are $r$-lines the linear system is called _$r$ -uniform_. In
this context, a _simple graph_ is an 2-uniform linear system.

Let $(P^{\prime},\mathcal{L}^{\prime})$ and $(P,\mathcal{L})$ be two linear
systems. The linear systems $(P^{\prime},\mathcal{L}^{\prime})$ and
$(P,\mathcal{L})$ are _isomorphic_ ,
$(P^{\prime},\mathcal{L}^{\prime})\simeq(P,\mathcal{L})$, if after of deleting
points of degree 1 or 0 from both, the linear systems
$(P^{\prime},\mathcal{L}^{\prime})$ and $(P,\mathcal{L})$ are isomorphic as
hypergraphs, see [4].

A subset $T\subseteq P$ is a _transversal_ (also called _vertex cover_ or
_hitting set_ in many papers, as example [6, 7, 9, 10, 11, 13, 14, 15, 16, 17,
19]) of $(P,\mathcal{L})$ if every line $l\in\mathcal{L}$ satisfies $T\cap
l\neq\emptyset$. The _transversal number_ of $(P,\mathcal{L})$,
$\tau=\tau(P,\mathcal{L})$, is the smallest possible cardinality of a
transversal of $(P,\mathcal{L})$. On the other hand, a subset of lines $R$ of
$\mathcal{L}$ is called a _2-packing_ of $(P,\mathcal{L})$ if every triplet of
different elements of $R$ do not have a common point. The maximum cardinality
of a 2-packing of $(P,\mathcal{L})$, $\nu_{2}=\nu_{2}(P,\mathcal{L})$, is
called _2-packing number_ of $(P,\mathcal{L})$. There are some papers which
have been studied this parameter, see for example [1, 2, 3, 4, 20, 21, 22,
23].

An interesting problem is found an upper bound of $\tau$ in terms of a
function of $\nu_{2}$ for any linear system. Araujo-Pardo et al. in [4] proved
a relationship between the transversal and the 2-packing numbers for linear
systems

$\lceil\nu_{2}/2\rceil\leq\tau\leq\frac{\nu_{2}(\nu_{2}-1)}{2}.$ (1)

The transversal number of any linear system is upper bounded by a quadratic
function of their 2-packing number. For some linear systems, the transversal
number is bounded above by a linear function of their 2-packing number, see
[1, 2, 3, 4, 23]. However, Eustis and Verstraëte in [8] proved, using
probabilistic methods, the existence of $k$-uniform linear systems
$(P,\mathcal{L})$ for infinitely many $k$’s and $n=|P|$ large enough, which
transversal number is $\tau=n-o(n)$. This $k$-uniform linear systems has
2-packing number upper bounded by $\frac{2n}{k}$.

Through this paper, every linear systems $(P,\mathcal{L})$ are considered with
$|\mathcal{L}|>\nu_{2}$ due to the fact $|\mathcal{L}|=\nu_{2}$ if and only if
$\Delta\leq 2$. Alfaro and Vázquez-Ávila in [1] proved that any linear system
with $\Delta=2$ satisfies $\tau\leq\nu_{2}-1$.

A _straight line representation_ on $\mathbb{R}^{2}$ of a linear system
$(P,\mathcal{L})$ maps each point $x\in P$ to a point $p(x)$ of
$\mathbb{R}^{2}$, and each line $L\in\mathcal{L}$ to a straight line segment
$l(L)$ of $\mathbb{R}^{2}$ in such a way that for each point $x\in P$ and line
$L\in\mathcal{L}$ satisfies $p(x)\in l(L)$ if and only if $x\in L$, and for
each pair of distinct lines $L,L^{\prime}\in\mathcal{L}$ satisfies $l(L)\cap
l(L^{\prime})=\\{p(x):x\in L\cap L^{\prime}\\}$. A _straight line system_
$(P,\mathcal{L})$ is a linear system, such that it has a straight line
representation on $\mathbb{R}^{2}$, see [4].

Araujo-Pardo et al. [4] proved that any linear system with
$\nu_{2}\in\\{2,3\\}$ satisfies $\tau\leq\nu_{2}-1$. Also, they proved any
linear system with $\nu_{2}=4$ satisfies $\tau\leq 4$; the inequality holds
for a certain linear subsystems of the finite projective plane of order 3,
$\Pi_{3}$; and they proved that, this linear subsystem have not a straight
line representation on $\mathbb{R}^{2}$.

######  Theorem 1.1.

[4] Let $(P,\mathcal{L})$ be an straight line system. If
$\nu_{2}\in\\{2,3,4\\}$, then $\tau\leq\nu_{2}-1$.

As a simple consequence of Theorem 1.1 is the following:

######  Corollary 1.2.

Let $(P,\mathcal{L})$ be an intersecting $r$-uniform straight line system with
$r\geq\nu_{2}$. If $\nu_{2}\in\\{2,3,4\\}$, then $\tau\leq\nu_{2}-1$.

Recently, the author [23] proved the following:

######  Theorem 1.3.

[23] Any intersecting $r$-uniform straight line system satisfies
$\tau\leq\nu_{2}-1$, where $r\geq\nu_{2}$.

In this paper, we improve the upper bound given in Theorem 1.3 when
$r=\nu_{2}$.

######  Theorem 1.4.

Any intersecting $\nu_{2}$-uniform straight line system with $r\geq\nu_{2}$
satisfies $\tau\leq\frac{2}{3}(\nu_{2}+1)$.

And we present some others interesting results. Also, we present some
consequences for certain intersecting subsystems of uniform straight line
systems called _segment systems_.

##  2 Main results

In this section we present the main results of the paper. Before to this, we
need some definitions and results.

According to [12], any straight line system is Zykov-planar, see also [25].
Zykov proposed to represent the lines of a set system by a subset of the faces
of a planar map on $\mathbb{R}^{2}$, i.e., a set system $(X,\mathcal{F})$ is
Zykov-planar if there exists a planar graph $G$ (not necessarily a simple
graph) such that $V(G)=X$ and $G$ can be drawn in the plane with faces of $G$
two-colored (say red and blue) so that there exists a bijection between the
red faces of $G$ and the subsets of $\mathcal{F}$ such that a point $x$ is
incident with a red face if and only if it is incident with the corresponding
subset. Walsh in [24] shown the Zykov’s definition is equivalent to the
following: A set system $(X,\mathcal{F})$ is Zykov-planar if and only if the
Levi graph $B(X,\mathcal{F})$ is planar. It is well known for any planar graph
$G$ the size of $G$, $|E(G)|$, is upper bounded by
$\frac{\kappa(|V(G)|-2)}{\kappa-2}$ (see for example [5]), where $\kappa$ is
the _girth_ of $G$ (the length of a shortest cycle contained in the graph
$G$).

The _Levi graph_ of a linear system $(P,\mathcal{L})$, $B(P,\mathcal{L})$, is
a bipartite graph with vertex set $V=P\cup\mathcal{L}$, where two vertices
$p\in P$, and $L\in\mathcal{L}$ are adjacent if and only if $p\in L$.

######  Theorem 2.1.

If $(P,\mathcal{L})$ is an intersecting $\nu_{2}$-uniform straight line system
with $\nu_{2}\geq 3$, then $|\mathcal{L}|\leq\frac{3\nu_{2}+1}{2}$.

Proof. Let $B=B(P,\mathcal{L})$ be the Levi graph of $(P,\mathcal{L})$. Since
the linear system is a straight line system, then the Levi graph $B$ is a
planar graph. It is not hard to see the size of the girth of $B$ is $6$ (since
$\nu_{2}\geq 3$). We have $|P|\leq\nu_{2}^{2}-\nu_{2}+1$. Hence

$\frac{\kappa(|V(B)|-2)}{\kappa-2}=\frac{3}{2}(|P|+|\mathcal{L}|-2)=\frac{3}{2}(\nu_{2}^{2}-\nu_{2}+|\mathcal{L}|-1).$

On the other hand, since every line has $\nu_{2}$ points, then
$|E(B)|=\nu_{2}|\mathcal{L}|$. Since the Levi graph is planar, then
$\nu_{2}|\mathcal{L}|\leq\frac{3}{2}(|P|+|\mathcal{L}|-2)$, which implies that

$|\mathcal{L}|\leq\frac{3(\nu_{2}^{2}-\nu_{2}-1)}{2\nu_{2}-3}\leq\frac{3\nu_{2}+1}{2}.$

Let $(P_{\nu_{2}},\mathcal{L}_{\nu_{2}})$ be a $\nu_{2}$-uniform linear
system. The $r$-uniform linear system $(P_{r},\mathcal{L}_{r})$, with
$r+1\geq\nu_{2}$, is _$\nu_{2}$ -isomorphic_ to
$(P_{\nu_{2}},\mathcal{L}_{\nu_{2}})$,
$(P_{r},\mathcal{L}_{r})\simeq_{\nu_{2}}(P_{\nu_{2}},\mathcal{L}_{\nu_{2}})$,
if after of either adding or deleting some points of degree 1 from
$(P_{r},\mathcal{L}_{r})$ the resultant linear systems is isomorphic to
$(P_{\nu_{2}},\mathcal{L}_{\nu_{2}})$. Notice that, if
$(P_{r},\mathcal{L}_{r})\simeq_{\nu_{2}}(P_{\nu_{2}},\mathcal{L}_{\nu_{2}})$,
then (trivially) $\nu_{2}(P_{r},\mathcal{L}_{r})=\nu_{2}$.

######  Corollary 2.2.

Let $(P_{\nu_{2}},\mathcal{L}_{\nu_{2}})$ be an intersecting $\nu_{2}$-uniform
straight line system with $\nu_{2}\geq 3$. If
$(P_{r},\mathcal{L}_{r})\simeq_{\nu_{2}}(P_{\nu_{2}},\mathcal{L}_{\nu_{2}})$,
then $|\mathcal{L}_{r}|=|\mathcal{L}_{\nu_{2}}|\leq\frac{3\nu_{2}+1}{2}$.

######  Theorem 2.3.

If $(P,\mathcal{L})$ is an intersecting $\nu_{2}$-uniform straight line system
with $\nu_{2}\geq 3$, then $\tau\leq\frac{2}{3}(\nu_{2}+1)$.

Proof. Suppose that $\nu_{2}\geq 2$ is an even integer. By Theorem 2.1

$|\mathcal{L}|\leq\frac{3}{2}\nu_{2}=\nu_{2}+\frac{\nu_{2}}{2}.$

Suppose that $\frac{\nu_{2}}{2}=3l+i$, for some $l\in\mathbb{Z}$ and
$i\in\\{0,1,2\\}$. Analyzing all cases for $i$, it can be proved that
$\tau\leq\frac{2\nu_{2}+1}{3}$ (see case $i=2$).

On the other hand, suppose that $\nu_{2}\geq 3$ is an odd integer. By Theorem
2.1

$|\mathcal{L}|\leq\nu_{2}+\frac{\nu_{2}+1}{2}.$

Suppose that $\frac{\nu_{2}+1}{2}=3l+i$, for some $l\in\mathbb{Z}$ and
$i\in\\{0,1,2\\}$. Analyzing all cases for $i$, it can be proved that
$\tau\leq\frac{2}{3}(\nu_{2}+1)$ (see case $i=1$).

######  Corollary 2.4.

Let $(P,\mathcal{L})$ be an intersecting $\nu_{2}$-uniform straight line
system with $|\mathcal{L}|>\nu_{2}$ and $\nu_{2}\geq 3$ an integer. If
$\Delta\geq\lfloor\frac{\nu_{2}}{2}\rfloor+2$ then
$\tau=\lceil\frac{\nu_{2}}{2}\rceil$.

Proof. Suppose that $\nu_{2}\geq 4$ is an even integer. By Theorem 2.1

$|\mathcal{L}|\leq\nu_{2}+\frac{\nu_{2}}{2}=\nu_{2}-2+\frac{\nu_{2}}{2}+2.$

Hence

$\tau\leq\frac{\nu_{2}-2}{2}+\frac{\frac{\nu_{2}}{2}+2}{\Delta}\leq\frac{\nu_{2}}{2}=\lceil\frac{\nu_{2}}{2}\rceil.$

On the other hand, suppose that $\nu_{2}\geq 3$ is an odd integer. By Theorem
2.1

$|\mathcal{L}|\leq\nu_{2}+\frac{\nu_{2}+1}{2}=\nu_{2}-1+\frac{\nu_{2}-1}{2}+2.$

Hence

$\tau\leq\frac{\nu_{2}-1}{2}+\frac{\frac{\nu_{2}-1}{2}+2}{\Delta}\leq\frac{\nu_{2}+1}{2}=\lceil\frac{\nu_{2}}{2}\rceil.$

Therefore, by Equation (1), we have

$\lceil\frac{\nu_{2}}{2}\rceil\leq\tau\leq\lceil\frac{\nu_{2}}{2}\rceil,$

which implies $\tau=\lceil\frac{\nu_{2}}{2}\rceil$.

##  3 $r$-segment systems

Let $r\geq 2$ be an integer. A linear system $(P,\mathcal{L})$, where
$P\subseteq\mathbb{Z}^{2}$, is an _$r$ -segment system_ if every line in
$\mathcal{L}$ consists of $r$ consecutive integer points on some line in
$\mathbb{R}^{2}$, and every line in $\mathbb{R}^{2}$ contains at most one line
of $\mathcal{L}$, see [18]. It is easy to see that any $r$-segment system is
an $r$-uniform straight line system, see [18]. An immediate consequence of
this definition is the following:

######  Corollary 3.1.

Let $(P,\mathcal{L})$ be an intersecting $r$-segment system and
$l\in\mathcal{L}$ any line. If $(P^{\prime},\mathcal{L}^{\prime})$ is the
linear subsystem induced by $\mathcal{L}\setminus\\{l\\}$ taking away the
points of degree 0, then $(P^{\prime},\mathcal{L}^{\prime})$ is an
intersecting $r$-segment system.

Oliveros et al. [18] proved the following (which is a simple consequence of
Corollary 2.4):

######  Theorem 3.2.

[18] Any intersecting $r$-segment system with $r\geq 3$ have a point of degree
1. Consequently, $\tau\leq r-1$.

As a simple consequence of Theorem 3.2 and Corollary 3.1 is the following:

######  Corollary 3.3.

Let $(P,\mathcal{L})$ be an intersecting $r$-uniform linear system with
$\nu_{2}\geq 3$. If $r=\nu_{2}-1$, then $(P,\mathcal{L})$ can be isomorphic to
an intersecting $r$-segment system.

######  Corollary 3.4.

If $(P,\mathcal{L})$ is an intersecting $r$-segment system with $\nu_{2}\geq
3$, then $r\geq\nu_{2}$.

As a simple consequence of Theorem 2.1 and Theorem 2.3 are the following:

######  Corollary 3.5.

If $(P,\mathcal{L})$ is an intersecting $\nu_{2}$-segment system with
$r\geq\nu_{2}\geq 3$, then $|\mathcal{L}|\leq\frac{3\nu_{2}+1}{2}$.

######  Corollary 3.6.

If $(P,\mathcal{L})$ is an intersecting $\nu_{2}$-segment system with
$\nu_{2}\geq 3$, then $\tau\leq\frac{2}{3}(\nu_{2}+1)$.

######  Example 1.

The linear system depicted in Figure 1 (a) is an intersecting 4-uniform
straight line system with $\nu_{2}=4$ (red lines); moreover this linear system
is isomorphic to an intersecting 4-segment system (see Figure 1 (b)) which
satisfies the transversal number obtained from Corollary 3.6. According to
[18], this is the intersecting 4-segment system with the maximum number of
lines (6 lines). This number can be obtained from Corollary 3.5.

Let $(P,\mathcal{L})$ an intersecting $r$-segment system with
$r\geq\nu_{2}\geq 3$ be an integer. A _triangle_ $T$ of $(P,\mathcal{L})$, is
obtained by three points of $P$ non collinear and the three lines induced by
them such that any two points of $T$ are consecutive integer points, see [18].

Oliveros et al. [18] proved that any intersecting 5-segment system satisfies
$\tau\leq 3$ (see Theorem 3.7 in [18]). Also, they proved that any
intersecting 5-segment has at most 6 lines if it contains a triangle; and
moreover, those linear systems satisfy $\nu_{2}=4$. Hence, if
$(P_{5},\mathcal{L}_{5})$ is any intersecting 5-segment system containing a
triangle, then $(P_{5},\mathcal{L}_{5})\simeq_{4}(P_{4},\mathcal{L}_{4})$ (see
Theorem 3.7 in [18]). Therefore, $|\mathcal{L}_{5}|=|\mathcal{L}_{4}|\leq 6$,
by Corollary 2.2.

Hence, Corollary 3.5, Corollary 2.2 and Theorem 2.1 gives a partial answer to
the Question 3.9 given in [18]:

Question 3.9. What is the maximum number of lines that can occur in an
intersecting $r$-segment system containing a triangle?.

Also, Oliveros et al. in [18] conjectured the following:

######  Conjecture 3.1.

Any intersecting $r$-segment system satisfies $\tau\leq\lceil r/2\rceil$, for
$r\geq 5$ an integer.

Figure 1: Projective plane of order 2 (Fano’s plane) without a line (any) with
adding a single point for each one line and (b) is the corresponding
isomorphic intersecting 4-segment system (Fig. 2. (b) from [18]).

Hence, if Conjecture 3.1 is true, then by Equation (1), we have

$\lceil\nu_{2}/2\rceil\leq\tau\leq\lceil r/2\rceil,$

and the equality holds if and only if either $\nu_{2}=r$ or $\nu_{2}=r-1$ if
$r$ is an even integer.

Alfaro and Vázquez-Ávila in [1] proved that any intersecting linear system
with $\Delta=2$ satisfies $\tau=\lceil\nu_{2}/2\rceil$. Oliveros et al. [18]
constructed an $r$-segment system with $\tau=\lceil r/2\rceil$, for every
$r\geq 3$ an integer. Those $r$-segment systems are such that $\nu_{2}=r$ and
$\Delta=2$.

Finally, as a simple consequence of Corollary 2.4 is the following:

######  Corollary 3.7.

Let $(P,\mathcal{L})$ be an intersecting $\nu_{2}$-segment system with
$\nu_{2}\geq 3$ an integer. If $\Delta\geq\lfloor\frac{\nu_{2}}{2}\rfloor+2$
then $\tau=\lceil\frac{\nu_{2}}{2}\rceil$.

Corollary 3.7 shows examples of intersecting segment systems satisfying the
Conjecture 3.1.

Acknowledgment

Research was partially supported by SNI and CONACyT.

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