# Stochastic differential variational inequalities with applications††thanks:
This work was supported by the National Natural Science Foundation of China
(11671282, 12171339).

Yao-Jia Zhang, Tao Chen, Nan-jing Huang and Xue-song Li
Department of Mathematics, Sichuan University, Chengdu, Sichuan, P.R. China
Corresponding author. E-mail address<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract: In this paper, we introduce and study a stochastic differential
variational inequality (SDVI) which consists of a stochastic differential
equation and a stochastic variational inequality. We obtain the existence and
uniqueness of the solutions for SDVI by using the iteration method and
Gronwall’s inequality. Moreover, we show the convergence of Euler scheme for
solving SDVI under some mild conditions. Finally, we apply the obtained
results to solve the electrical circuits with diodes and the collapse of the
bridge problems in stochastic environment.

Keywords: Stochastic differential variational inequality; Stochastic
differential equation; Stochastic variational inequality; Euler scheme;
Convergence.

2020 AMS Subject Classification: 34K50, 37A50, 49J40, 90C33, 91A15.

##  1 Introduction

Pang and Stewart first systematically investigated the differential
variational inequality (DVI) in [30]. They obtained solution existence results
and studied the numerical analysis of DVI by using a Euler time-stepping
method. They further showed the sensitivity theorems and Newton approximation
methods for DVI in [31] under some mild conditions. Since then, various
theoretical results, algorithms and applications concerned with DVIs have been
studied by many authors under different conditions (see, for example, [20, 13,
41, 14, 38, 39, 26, 40, 26, 30, 45, 40, 7] and the references therein).

As pointed out by Pang and Stewart [30], DVI provides a powerful mathematical
paradigm for modeling many real problems. By taking into account some
stochastic environmental effects in DVIs, this paper investigates the
following SDVI:

$\left\\{\begin{aligned}
&dx(t)=f(t,x(t),u(t))dt+g(t,x(t),u(t))dB_{t},\,t\in[0,T],\,\Gamma(x(0),x(T))=0,\\\
&\left\langle
F(t,\omega,x(t,\omega),u(t,\omega)),v-u(t,\omega)\right\rangle\geq 0,\,\forall
v\in K,\,a.e.\,t\in[0,T],\,a.s.\,\omega\in\Omega,\end{aligned}\right.$ (1.1)

where $B_{t}$ is an $l$-dimensional standard Brownian motion, $K$ is a closed
convex subset in $\mathbb{R}^{m}$,
$f:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$,
$g:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n\times
l}$, $F:[0,T]\times\Omega\times\mathbb{R}^{n}\times
K\rightarrow\mathbb{R}^{m}$, and
$\Gamma:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ are
measurable functions. As an extension of DVI under random environmental
effects, SDVI (1.1) provides a powerful mathematical tool for describing many
real problems arising in engineering sciences, economics, biology such as
stochastic electrical circuits, stochastic earthquake engineering, stochastic
spatial price equilibrium problems, stochastic differential Nash games and
stochastic batch fermentation problems (see, for example, [5, 4, 13, 14, 30,
40, 45, 44, 12, 35, 25]).

It is worth noting that there have been numerous studies concerning the
special cases of SDVI (1.1) under different conditions; for example,
stochastic differential algebraic equations [29, 28, 1], stochastic
differential complementarity problems [37], stochastic differential inclusions
[42, 18, 9] and stochastic variational inequalities [5, 4]. Nevertheless, to
the best of authors’ knowledge, there is few work addressing the existence and
uniqueness of the solutions for SDVIs. Thus, it would be significant and
interesting to obtain some conditions for ensuring the existence and
uniqueness of the solutions for SDVI (1.1). The first purpose of this paper is
to show the existence and uniqueness of the solutions for SDVI (1.1) under
some mild conditions.

On the other hand, fruitful results on numerical method of stochastic
differential equations (SDEs) have been obtained by many authors including
[36, 33, 22, 10, 32] in the earlier years and [21, 11, 24, 15] more recently.
However, there is few work concerned with the numerical approximation method
for solving SDVI (1.1). Therefore, it would be significant and attractive to
investigate numerical approximation method for solving SDVI (1.1). The second
goal of this paper is to propose and investigate a kind of Euler scheme for
solving SDVI (1.1) with $\Gamma(x(0),x(T))=x(0)-x_{0}$. Motivated and spirited
by Euler method for solving DVIs [30, 13, 14, 20, 26, 39, 38] and SDEs [33,
43, 32], we divide time interval $[0,T]$ into $N$ subintervals:

$\displaystyle 0=t_{0}<t_{1}<t_{2}<\ldots<t_{N}=T,$

where $t_{i+1}-t_{i}=h=\frac{T}{N}$ $(i=0,1,\cdots,N-1)$, and construct Euler
scheme for solving SDVI (1.1) as follows:

$\left\\{\begin{aligned}
&x_{h}(t_{i+1})=x_{h}(t_{i})+f(t_{i},x_{h}(t_{i}),u_{h}(t_{i}))(t_{i+1}-t_{i})\\\
&\qquad\quad\mbox{}+g(t_{i},x_{h}(t_{i}),u_{h}(t_{i}))(B_{t_{i+1}}-B_{t_{i}}),\\\
&x_{h}(t)=x_{h}(t_{i})+f(t_{i},x_{h}(t_{i}),u_{h}(t_{i}))(t-t_{i-1})\\\
&\qquad\quad\mbox{}+g(t_{i},x_{h}(t_{i}),u_{h}(t_{i}))(B_{t}-B_{t_{i}}),\,\forall
t\in[t_{i},t_{i+1}),\,x_{h}(0)=x_{0},\\\ &\left\langle
F(t_{i},\omega,x_{h}(t_{i}),u_{h}(t_{i})),v-u_{h}(t_{i})\right\rangle\geq
0,\;\forall v\in K,\;a.s.\;\omega\in\Omega,\\\
&u_{h}(t)=u_{h}(t_{i}),\quad\forall t\in[t_{i},t_{i+1}).\end{aligned}\right.$
(1.2)

In fact, we can compute $u_{h}(0)$ from the stochastic variational inequality
(SVI) in (1.2) for a given $x_{0}$, and then obtain $x_{h}(t_{1})$ from SDE in
(1.2) by using $x_{0}$ and $u_{h}(0)$. In the same way, we can compute
$u_{h}(t_{1})$ and $x_{h}(t_{2})$ by using $x_{h}(t_{1})$. Again, repeat the
above process.

The investigation of SDVIs is still rarely explored and much is desired to be
done. The current work is an attempt in this new direction. The contributions
of this paper are twofold. One is to obtain the existence and uniqueness of
the solutions for SDVI (1.1) by employing the iteration method and Gronwall’s
inequality. The other is to prove the convergence of Euler scheme (1.2) for
solving SDVI (1.1) under some mild conditions.

The outline of this paper is structured as follows. The next section recalls
some necessary notions, symbols and lemmas. After that in Section 3, we prove
the unique existence of the solutions for SDVI (1.1). In Section 4, we show
the convergence of Euler scheme constructed by (1.2). Finally, we give two
applications to the electrical circuits with diodes and the collapse of the
bridge problems in stochastic environment in Section 5.

##  2 Preliminaries

This section recalls some necessary notions, symbols and lemmas.

######  Definition 2.1.

[23, 2, 17]

  * •

Denote the norm and inner product of $\mathbb{R}^{n}$ (or $\mathbb{R}^{m}$) by
$\|\cdot\|$ and $\left\langle\cdot,\cdot\right\rangle$.

  * •

$B_{t}$ is an $l$-dimensional standard Brownian motion with probability
distribution $N(0,t)$.

  * •

$\mathcal{F}_{t}=\sigma\left\\{B_{t}:0\leq s\leq t\right\\}$ is a filtration
generated by $B_{t}$.

  * •

$L^{2}(\Omega,\mathbb{R}^{n})=L^{2}(\Omega,\mathcal{F},\mathbb{P},\mathbb{R}^{n})$,
the set which contains all the $\mathbb{R}^{n}$-valued measurable square-
integrable random variables, is a Hilbert space fitted out the norm
$\|\cdot\|_{L^{2}}=(E|\cdot|^{2})^{\frac{1}{2}}$.

  * •

$H_{[a,b]}=L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{n})$ is the space of all
stochastic processes $f(t,\omega)$, $0\leq a\leq b\leq T$ valued in
$\mathbb{R}^{n}$ such that $f(t,\omega)$ is adapted to the filtration
$\left\\{\mathcal{F}_{t}\right\\}$ and
$\int_{a}^{b}\mathbb{E}\|f(t,\omega)\|^{2}dt<\infty$. Moreover,
$L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{n})$ is a Hilbert space with the
inner product

$\left\langle u,v\right\rangle_{H_{[a,b]}}=\int_{a}^{b}\mathbb{E}[\langle
u(t,\omega),v(t,\omega)\rangle]dt,\quad\forall u,v\in
L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{n})$

and norm

$\|u\|_{H_{[a,b]}}=\left[\int_{a}^{b}\mathbb{E}\|u(t,\omega)\|^{2}dt\right]^{\frac{1}{2}},\quad\forall
u\in L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{n}).$

Let $K\subset\mathbb{R}^{m}$ be nonempty. For any given given interval
$[a,b]\subset[0,T]$, set

$U[a,b]=\left\\{u\in
L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{m}):u(s,\omega)\in
K,a.e.\;s\in[a,b],\;a.s.\;\omega\in\Omega\right\\}$

and

$U=\left\\{u\in L^{2}(\Omega,\mathbb{R}^{n}):u(\omega)\in
K,a.s.\;\omega\in\Omega\right\\}.$

######  Lemma 2.1.

Let $K\subset\mathbb{R}^{m}$ be nonempty, closed and convex. Then, the
following statements hold:

  * (i)

for any given interval $[a,b]\subset[0,T]$, $U[a,b]$ is nonempty, closed and
convex in $L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{m})$;

  * (ii)

$U$ is nonempty, closed and convex in $L^{2}(\Omega,\mathbb{R}^{m})$.

###### Proof.

For any $u(s,\omega)\in U[a,b]$, we have $u(s,\omega)\in
L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{m})$ and so it is
$\mathcal{F}_{s}$-measurable for all $s\in[a,b]$. It follows from the fact
$K\not=\emptyset$ that $U[a,b]\not=\emptyset$. Now we show that $U[a,b]$ is
convex in $L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{m})$. Indeed, for any
$u_{1}(s,\omega),u_{2}(s,\omega)\in U[a,b]$ and $0<\lambda<1$, one has
$u_{1}(s,\omega),u_{2}(s,\omega)\in
L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{m})$ and
$u_{1}(s,\omega),u_{2}(s,\omega)\in K$ for a.e. $t\in[a,b]$, a.s.
$\omega\in\Omega$. Since $K$ is nonempty closed convex, it is easy to check
that

$\lambda u_{1}(s,\omega)+(1-\lambda)u_{2}(s,\omega)\in
K,\;a.e.\;t\in[a,b],\;a.s.\;\omega\in\Omega$

and so $U[0,t]$ is convex.

Next we prove that $U[a,b]$ is closed in
$L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{m})$. In fact, let
$\left\\{u_{n}(s,\omega)\right\\}\subset U[a,b]$ be a sequence such that
$\|u_{n}(s,\omega)-u^{*}(s,\omega)\|_{H[a,b]}\to 0$. Then it is easy to see
that $u^{*}(t,\omega)\in L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{m})$.
Moreover, for any $s\in[a,b]$,

$\|u_{n}(s,\omega)-u^{*}(s,\omega)\|^{2}_{H[a,b]}=\int_{a}^{b}\mathbb{E}\|u_{n}(s,\omega)-u^{*}(s,\omega)\|^{2}ds\to
0,$

which implies that

$\|u_{n}(s,\omega)-u^{*}(s,\omega)\|\to
0,\;a.e.\;s\in[a,b],\;a.s.\;\omega\in\Omega.$

Now, the closedness of $K$ shows that $u^{*}(s,\omega)\in U[a,b]$ and so
$U[a,b]$ is closed. Thus the statement $(i)$ holds. Similarily, we can prove
$(ii)$. ∎

######  Lemma 2.2.

[6] Let $H$ be a Hilbert space and $K\subset H$ be nonempty, closed and
convex. Then, for each $m\in H$, there exists a unique $u\in K$, named as the
projection of $m$ onto $K$ and denoted by $u=P_{K}(m)$, such that

$\displaystyle\|m-u\|_{H}=\min_{v\in K}\|m-v\|_{H}=dist(m,K).$

Moreover, $u=P_{K}(m)$ if and only if

$u\in K,\quad\left\langle m-u,v-u\right\rangle_{H}\leq 0,\quad\forall v\in K.$

######  Lemma 2.3.

[27] Let $H$ be a Hilbert space, $K\subset H$ be nonempty, closed and convex,
and $T:H\rightarrow H$ be a mapping. Then $x\in K$ is a solution of the
following variational inequality (VI):

$\left\langle T(x),y-x\right\rangle_{H}\geq 0,\quad\forall y\in K$

if and only if $x=P_{K}(I-\alpha T)(x)$, where $\alpha$ is a positive constant
and $I$ is an identity operator.

######  Lemma 2.4.

[8, 43] If there exist a non-negative sequence
$\left\\{z_{n}\right\\}_{n=1}^{N}$ and a non-negative and non-decreasing
sequence $\left\\{y_{n}\right\\}_{n=1}^{N}$ such that

$\displaystyle
z_{n}\leq\frac{Q_{A}}{N^{\beta}}\sum_{i=1}^{n-1}\frac{z_{i}}{(n-i)^{1-\beta}}+y_{n},\quad
1\leq n\leq N,0<\beta<1,Q_{A}>0,$

then the sequence $\left\\{z_{n}\right\\}_{n=1}^{N}$ can be bounded from above
by

$\displaystyle z_{n}\leq y_{n}(1+\mathbb{G}_{\beta}[Q_{A}\Gamma(\beta)],1\leq
n\leq N,$

where $\Gamma(x)=2\int_{0}^{+\infty}t^{2x-1}e^{-t^{2}}dt$ is the
$\Gamma$-function and
$\mathbb{G}_{\beta}(x)=\sum\limits_{k=0}^{\infty}\frac{x^{k}}{\Gamma(\beta
k+1)}$ with $x\in\mathbb{R}$.

######  Lemma 2.5.

[23] Let $\left\\{y_{n}\right\\}_{n=1}^{\infty}$ be a sequence of functions in
$L^{1}[a,b]$ satisfying equation

$\displaystyle y_{n+1}(t)\leq\alpha+\beta\int_{a}^{t}y_{n}(s)ds,\quad\forall
t\in[a,b],$

where $\alpha$ and $\beta$ are constants. If $y_{1}(t)\equiv c$ is a constant,
then the following inequality holds for any $n\geq 1$:

$\displaystyle y_{n+1}(t)\leq\alpha
e^{\beta(t-a)}+c\frac{\beta^{n}(t-a)^{n}}{n!}.$

######  Lemma 2.6.

For any fixed $x\in L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{n})$, if $u\in
U[a,b]$, then the following SVI

$\displaystyle\langle
F(t,\omega,x(t,\omega),u(t,\omega)),v-u(t,\omega)\rangle\geq 0,,\quad\forall
v\in K,a.e.\;t\in[a,b],\;a.s.\;\omega\in\Omega$ (2.1)

is equivalent to that, for all $v^{\prime}\in U[a,b]$ with $t\in(0,T]$,

$\displaystyle\left\langle\tilde{F}(x,u),v^{\prime}-u\right\rangle_{H_{[a,b]}}\geq
0,$ (2.2)

where $\tilde{F}:L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{n})\times U[a,b]\to
L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{m})$ is defined by

$\tilde{F}(x,u)(s,\omega):=F(s,\omega,x(s,\omega),u(s,\omega))$

for all $(x,u)\in L^{2}_{ad}([a,b]\times\Omega,\mathbb{R}^{n})\times U[a,b]$,
for all $s\in[a,b]$, and all $\omega\in\Omega$.

###### Proof.

We first show that (2.1) leads to (2.2). If (2.2) false, then there exists a
$v_{0}\in U[a,b]$ such that

$\displaystyle\mathbb{E}\int_{a}^{b}\langle
F(s,\omega,x(s,\omega),u(s,\omega)),v_{0}(s,\omega)-u(s,\omega)\rangle
ds<0,\quad\forall(s,\omega)\in[a,b]\times\Omega,$

which implies that

$\displaystyle\Lambda^{0}:=\left\\{(s,\omega)\in[a,b]\times\Omega:\langle
F(s,\omega,x(s,\omega),u(s,\omega)),v_{0}(s,\omega)-u(s,\omega)\rangle<0\right\\}\neq\emptyset$

with $\int_{\Omega}\int_{a}^{b}I_{\Lambda^{0}}dsdP(\omega)>0$. From the
definition of $U[a,b]$, we know $v_{0}(s,\omega)\in K$ for a.e. $t\in[a,b]$
and a.s. $\omega\in\Omega$. Thus, there exists $v^{\prime}\in K$ such that

$\displaystyle\langle
F(s,\omega,x(s,\omega),u(s,\omega)),v^{\prime}-u(s,\omega)\rangle<0,\forall(s,\omega)\in\Lambda^{0},$

which contradicts with (2.1).

Next we prove that (2.2) leads to (2.1). On the contrary, suppose that (2.1)
fails. Then there exists a $u_{0}\in K$ and $\epsilon>0$ such that

$\displaystyle\Lambda_{\epsilon}:=\left\\{(s,\omega)\in[a,b]\times\Omega:\langle
F(s,\omega,x(s,\omega),u(s,\omega)),v_{0}(s,\omega)-u(s,\omega)\rangle<\epsilon\right\\}$

with $\int_{\Omega}\int_{a}^{b}I_{\Lambda_{\epsilon}}dsdP(\omega)>0$. Since
$\langle
F(s,\omega,x(s,\omega),u(s,\omega)),v_{0}(s,\omega)-u(s,\omega)\rangle$ is
$\mathcal{F}_{s}$-adapted, so is $I_{\Lambda_{\epsilon}}(s,\omega)$. Let

$\tilde{u}(s,\omega)=u_{0}I_{\Lambda_{\epsilon}}(s,\omega)+u(s,\omega)I_{\Lambda_{\epsilon}^{c}}(s,\omega),\quad\forall(t,\omega)\in[a,b]\times\Omega.$

Clearly, $\tilde{u}\in U[a,b]$. By taking $v=\tilde{u}$ in (2.2) , one has

$\displaystyle\quad\;\mathbb{E}\int_{a}^{b}\langle
F(s,\omega,x(s,\omega),u(s,\omega)),\tilde{u}(s,\omega)-u(s,\omega)\rangle ds$
$\displaystyle=\int_{\Omega}\int_{a}^{b}I_{\Lambda_{\epsilon}}(s,\omega)\langle
F(s,\omega,x(s,\omega),u(s,\omega)),u_{0}-u(s,\omega)\rangle dsdP(\omega)$
$\displaystyle\leq-\epsilon\int_{\Omega}\int_{a}^{b}I_{\Lambda_{\epsilon}}dsdP(\omega)<0,$

which is a contradiction. ∎

##  3 Existence and uniqueness of the solutions for SDVI

In this section, we prove the existence and uniqueness of the solutions for
SDVI. To this end, we need the following assumption and lemmas.

######  Assumption 3.1.

Suppose that there exist a few positive constants $C$, $L_{f}$, $K_{1}$,
$K_{2}$, $L_{g}$, $L_{F}$ with $L_{F}>C$ such that

  1. ($i$)

$\|f(t,x,u)\|\leq K_{1}(1+\|x\|+\|u\|)$;

  2. ($ii$)

$\|g(t,x,u)\|_{\mathbb{R}^{n\times m}}\leq K_{2}(1+\|x\|+\|u\|)$;

  3. ($iii$)

$\|f(t_{1},x_{1},u_{1})-f(t_{2},x_{2},u_{2})\|\leq
L_{f}(|t_{1}-t_{2}|+\|x_{1}-x_{2}\|+\|u_{1}-u_{2}\|)$;

  4. ($iv$)

$\|g(t_{1},x_{1},u_{1})-g(t_{2},x_{2},u_{2})\|_{\mathbb{R}^{n\times m}}\leq
L_{g}(|t_{1}-t_{2}|+\|x_{1}-x_{2}\|+\|u_{1}-u_{2}\|)$;

  5. ($v$)

$\|\tilde{F}(x^{\prime}_{1},u^{\prime}_{1})-\tilde{F}(x^{\prime}_{2},u^{\prime}_{2})\|_{H[0,t]}\leq
L_{F}(\|x^{\prime}_{1}-x^{\prime}_{2}\|_{H[0,t]}+\|u^{\prime}_{1}-u^{\prime}_{2}\|_{H[0,t]})$;

  6. ($vi$)

$\left\langle\tilde{F}(x^{\prime},u^{\prime}_{1})-\tilde{F}(x^{\prime},u^{\prime}_{2}),u^{\prime}_{1}-u^{\prime}_{2}\right\rangle_{H[0,t]}\geq
C\|u^{\prime}_{1}-u^{\prime}_{2}\|_{H[0,t]}^{2}$,

for all $t,t_{1},t_{2}\in[0,T]$, all $x,x_{1},x_{2}\in\mathbb{R}^{n}$, all
$u,u_{1},u_{2}\in\mathbb{R}^{m}$, all
$x^{\prime},x^{\prime}_{1},x^{\prime}_{2}\in
L^{2}_{ad}([0,t]\times\Omega,\mathbb{R}^{n})$, all
$u^{\prime}_{1},u^{\prime}_{2}\in U[0,t]$.

######  Lemma 3.1.

For any fixed $x\in L^{2}_{ad}([0,b]\times\Omega,\mathbb{R}^{n})$ $(0<b\leq
T)$, if $\tilde{F}$ satisfies the conditions $(v)$ and $(vi)$ in Assumption
3.1, then there exists a unique $u\in U[0,b]$ such that

$\langle F(t,\omega,x(t,\omega),u(t,\omega)),v-u(t,\omega)\rangle\geq
0,\quad\forall v\in K,a.e.\;t\in[0,b],\;a.s.\;\omega\in\Omega.$

###### Proof.

From Lemma 2.6, we know that the problem is equivalent to finding a unique
$u\in U[0,b]$ such that

$\left\langle\tilde{F}(x,u),v^{\prime}-u\right\rangle_{H_{[0,b]}}\geq
0,\quad\forall v^{\prime}\in U[0,b].$

Consider the following iterative sequence:

$u_{n+1}=P_{U[0,b]}[u_{n}-\rho\tilde{F}(x,u_{n})],\quad\rho>0.$

By the conditions $(v)$ and $(vi)$ in Assumption 3.1, one has

$\displaystyle\|u_{n+1}-u_{n}\|^{2}_{H_{[0,b]}}$
$\displaystyle=\|P_{U[0,b]}[u_{n}-\rho\tilde{F}(x,u_{n})]-P_{U[0,b]}[u_{n-1}-\rho\tilde{F}(x,u_{n-1})]\|^{2}_{H_{[0,b]}}$
$\displaystyle\leq\|u_{n}-u_{n-1}-\rho\tilde{F}(x,u_{n})+\rho\tilde{F}(x,u_{n-1})\|^{2}_{H_{[0,b]}}$
$\displaystyle=\|u_{n}-u_{n-1}\|^{2}_{H_{[0,b]}}+\rho^{2}\|\tilde{F}(x,u_{n-1})-\tilde{F}(x,u_{n})\|^{2}_{H_{[0,b]}}$
$\displaystyle\quad\>-2\rho\langle\tilde{F}(x,u_{n})-\tilde{F}(x,u_{n-1}),u_{n}-u_{n-1}\rangle_{H_{[0,b]}}$
$\displaystyle\leq(1-2\rho
C+\rho^{2}L^{2}_{F})\|u_{n+1}-u_{n}\|^{2}_{H_{[0,b]}},$

where $0<\rho<\frac{2C}{L^{2}_{F}}$, which implies that there exists a unique
$u^{*}\in L^{2}_{ad}([0,b]\times\Omega,\mathbb{R}^{m})$ such that
$u^{*}=P_{U[0,b]}[u^{*}-\rho\tilde{F}(x,u^{*})]$. It follows from Lemma 2.3
that

$\left\langle\tilde{F}(x,u^{*}),v^{\prime}-u^{*}\right\rangle_{H_{[0,b]}}\geq
0,\quad\forall v^{\prime}\in U[0,b].$

This ends the proof. ∎

######  Lemma 3.2.

For each $j=1,2$ and fixed $x_{j}\in
L^{2}_{ad}([0,T]\times\Omega,\mathbb{R}^{n})$, under the conditions $(v)$ and
$(vi)$ in Assumption 3.1, there exists a unique $u_{j}\in[0,T]$ such that

$\langle
F(t,\omega,x_{j}(t,\omega),u_{j}(t,\omega)),v-u_{j}(t,\omega)\rangle\geq
0,\quad\forall v\in K,a.e.\;t\in[0,T],\;a.s.\;\omega\in\Omega.$

Moreover, there exists a constant $M^{\prime}>0$ such that

$\displaystyle\mathbb{E}\int_{0}^{t}\|u_{1}(s,\omega)-u_{2}(s,\omega)\|^{2}ds\leq
M^{\prime}\mathbb{E}\int_{0}^{t}\|x_{1}(s,\omega)-x_{2}(s,\omega)\|^{2}ds,\forall
t\in[0,T].$

###### Proof.

For each $j=1,2$, it follows from Lemma 3.1 that there exists a unique
$u_{j}\in U[0,T]$ such that

$\langle
F(s,\omega,x_{j}(s,\omega),u_{j}(s,\omega)),v-u_{j}(s,\omega)\rangle\geq
0,\quad\forall v\in K,a.e.\;s\in[0,T],\;a.s.\;\omega\in\Omega.$

By Lemmas 2.3 and 2.6, we obtain
$u_{j}=P_{U[0,t]}[u_{j}-\rho\tilde{F}(x_{j},u_{j})]$ for all $t\in[0,T]$,
$j=1,2$. Now the conditions $(v)$ and $(vi)$ in Assumption 3.1 implies that

$\displaystyle\|u_{1}-u_{2}+\rho\tilde{F}(x_{1},u_{2})-\rho\tilde{F}(x_{1},u_{1})\|^{2}_{H[0,t]}$
(3.1) $\displaystyle=$
$\displaystyle\|u_{1}-u_{2}\|^{2}_{H[0,t]}+\rho^{2}\|\tilde{F}(x_{1},u_{2})-\rho\tilde{F}(x_{1},u_{1})\|^{2}_{H[0,t]}$
$\displaystyle-2\rho\langle\tilde{F}(x_{1},u_{1})-\tilde{F}(x_{1},u_{2}),u_{1}-u_{2}\rangle_{H[0,t]}$
$\displaystyle\leq$ $\displaystyle(1-2\rho
C+\rho^{2}L^{2}_{F})\mathbb{E}\|u_{1}-u_{2}\|^{2}_{H[0,t]},$

where $0<\rho<\frac{2C}{L^{2}_{F}}$. Applying (3.1) and the nonexpansiveness
of $P_{U[0,t]}$, one has

$\displaystyle\|u_{1}-u_{2}\|_{H[0,t]}$ $\displaystyle=$
$\displaystyle\|P_{U[0,t]}[u_{1}-\rho\tilde{F}(x_{1},u_{1})]-P_{U[0,t]}[u_{1}-\rho\tilde{F}(x_{1},u_{1})]\|_{H[0,t]}$
$\displaystyle\leq$
$\displaystyle\|u_{1}-u_{2}-\rho\tilde{F}(x_{1},u_{1})+\rho\tilde{F}(x_{2},u_{2})\|_{H[0,t]}$
$\displaystyle=$
$\displaystyle\|u_{1}-u_{2}+\rho\tilde{F}(x_{2},u_{2})-\rho\tilde{F}(x_{1},u_{2})+rho\tilde{F}(x_{1},u_{2})-\rho\tilde{F}(x_{2},u_{2})\|_{H[0,t]}$
$\displaystyle\leq$
$\displaystyle\|u_{1}-u_{2}+\rho\tilde{F}(x_{2},u_{2})-\rho\tilde{F}(x_{1},u_{2})\|_{H[0,t]}+\rho\|\tilde{F}(x_{1},u_{2})-\tilde{F}(x_{2},u_{2})\|_{H[0,t]}$
$\displaystyle\leq$ $\displaystyle\sqrt{1-2\rho
C+\rho^{2}L^{2}_{F}}\|u_{1}-u_{2}\|_{H[0,t]}+\rho
L_{F}\|x_{1}-x_{2}\|_{H[0,t]},$

which implies that

$\displaystyle\mathbb{E}\int_{0}^{t}\|u_{1}(s,\omega)-u_{2}(s,\omega)\|^{2}ds\leq
M^{\prime}\mathbb{E}\int_{0}^{t}\|x_{1}(s,\omega)-x_{2}(s,\omega)\|^{2}ds,\forall
t\in[0,T],$

where $M^{\prime}=\frac{\rho^{2}L^{2}_{F}}{(1-\sqrt{1-2\rho
C+\rho^{2}L^{2}_{F}})^{2}}$. ∎

In the sequel, we omit $\omega$ in $x(t,\omega)$ and $u(t,\omega)$ sometimes
without ambiguity.

Let $SOL(U[0,T],F(t,\omega,x(t),u(t)))$ be the set of solutions for following
SVI: find $u(t)\in U[0,T]$ such that

$\displaystyle\left\langle F(t,\omega,x(t),u(t)),v-u(t)\right\rangle\geq
0,\quad\forall v\in K,a.e.\;t\in[0,T],\;a.s.\;\omega\in\Omega.$ (3.2)

A pair $(x(t),u(t))$ is called a Carathéodory solution of problem SDVI (1.1)
with $\Gamma(x(0),x(T))=x(0)-x_{0}$ if and only if $x(t)$ belongs to
$L^{2}_{ad}([0,T]\times\Omega,\mathbb{R}^{n})$ such that, for any $t\in[0,T]$,

$dx(t)=f(t,x(t),u(t))dt+g(t,x(t),u(t))dB_{t},\quad x(0)=x_{0}$

and $u(t)\in SOL(U[0,T],F(t,\omega,x(t),u(t)))$.

Now we give the main result of this section as follows.

######  Theorem 3.1.

Under Assumption 3.1, SDVI (1.1) admits a unique Carathéodory solution
$(x(t),u(t))$.

###### Proof.

Taking $x^{(1)}_{t}=x_{0}$ for all $t\in[0,T]$, it follows from Lemmas 2.3,
2.6 and 3.1 that there exists a unique $u^{(1)}_{t}\in U[0,T]$ such that
$u^{(1)}_{t}=P_{U[0,T]}[u^{(1)}_{t}-\rho\tilde{F}(x^{(1)}_{t},u^{(1)}_{t})]$.
Let
$x^{(2)}_{t}=x_{0}+\int_{0}^{t}f(s,x^{(1)}_{s},u^{(1)}_{s})ds+\int_{0}^{t}g(s,x^{(1)}_{s},u^{(1)}_{s})dB_{s}$.
Again, by Lemmas 2.3, 2.6 and 3.1, we know there exists a unique
$u^{(2)}_{t}\in U[0,T]$ such that
$u^{(2)}_{t}=P_{U[0,T]}[u^{(2)}_{t}-\rho\tilde{F}(x^{(2)}_{t},u^{(2)}_{t})]$.
Repeating this process, we can construct the sequence
$\left\\{(x^{(n)}_{t},u^{(n)}_{t})\right\\}$ satisfying

$\left\\{\begin{aligned}
&u^{(n)}_{t}=P_{U[0,T]}\left(u^{(n)}_{t}-\rho\tilde{F}(x^{(n)}_{t},u^{(n)}_{t})\right),\\\
&x^{(n+1)}_{t}=x_{0}+\int_{0}^{t}f(s,x^{(n)}_{s},u^{(n)}_{s})ds+\int_{0}^{t}g(s,x^{(n)}_{s},u^{(n)}_{s})dB_{s},\end{aligned}\right.$
(3.3)

where $0<\rho<\frac{2C}{L^{2}_{F}}$, $n=1,2,\cdots$.

First we show that $(x^{(n+1)}_{t},u^{(n+1)}_{t})\in
L^{2}_{ad}([0,T]\times\Omega,\mathbb{R}^{n})\times{U[0,T]}$ by the induction.
Obviously, $x^{(1)}_{t}=x_{0}$ for all $t\in[0,T]$ and
$u^{(1)}_{t}=P_{U[0,T]}[u^{(1)}_{t}-\rho\tilde{F}(x^{(1)}_{t},u^{(1)}_{t})]$
lead to $(x^{(1)}_{t},u^{(1)}_{t})\in
L^{2}_{ad}([0,T]\times\Omega,\mathbb{R}^{n})\times{U[0,T]}$. Assume by the
induction that $(x^{(n)}_{t},u^{(n)}_{t})\in
L^{2}_{ad}([0,T]\times\Omega,\mathbb{R}^{n})\times{U[0,T]}$. Then it follows
from conditions $(i)$ and $(ii)$ in Assumption 3.1 that

$\displaystyle\quad\;\mathbb{E}\left[\int_{0}^{T}g(t,x^{(n)}_{t},u^{(n)}_{t})dB_{t}\right]^{2}=\mathbb{E}\left[\int_{0}^{T}\|g(t,x^{(n)}_{t},u^{(n)}_{t})\|_{\mathbb{R}^{n\times
m}}^{2}dt\right]$ (3.4) $\displaystyle\leq
3K_{2}^{2}T+3K_{2}^{2}\mathbb{E}\left[\int_{0}^{T}\left(\|x^{(n)}_{t}\|^{2}+\|u^{(n)}_{t}\|^{2}\right)dt\right]<+\infty$

and

$\displaystyle\quad\;\mathbb{E}\left[\int_{0}^{T}f(t,x^{(n)}_{t},u^{(n)}_{t})dt\right]^{2}$
(3.5) $\displaystyle\leq
3K_{1}^{2}T\mathbb{E}\left[\int_{0}^{T}\left(1+\|x^{(n)}_{t}\|^{2}+\|u^{(n)}_{t}\|^{2}\right)dt\right]<+\infty.$

Moreover, we have

$\displaystyle\quad\;\mathbb{E}\|x^{(n+1)}_{t}\|^{2}$ $\displaystyle\leq
3\mathbb{E}\left[x_{0}^{2}+\left[\int_{0}^{T}f(t,x^{(n)}_{t},u^{(n)}_{t})dt\right]^{2}+\left[\int_{0}^{T}g(t,x^{(n)}_{t},u^{(n)}_{t})dB_{t}\right]^{2}\right].$

Combining (3.4) to (3), we know that
$\mathbb{E}\left[\int_{0}^{T}|x^{(n+1)}_{t}|^{2}dt\right]<\infty$ and so
$x^{(n+1)}_{t}\in L^{2}_{ad}([0,T]\times\Omega,\mathbb{R}^{n})$. By (3.3) and
the definition of $P_{U[0,T]}$, one has $u^{(n+1)}_{t}\in U[0,T]$.

Next we prove that $\left\\{(x^{n}_{t},u^{n}_{t})\right\\}$ is a Cauchy
sequence in $L^{2}_{ad}([0,T]\times\Omega,\mathbb{R}^{n})\times U[0,T]$. It
follows from Cauchy-Schwartz’s inequality and the conditions $(iii)$ and
$(iv)$ in Assumption 3.1 that

$\displaystyle\quad\;\mathbb{E}\|x^{(n+1)}_{t}-x^{(n)}_{t}\|^{2}$ (3.7)
$\displaystyle\leq
2\mathbb{E}\Bigg{[}\left(\int_{0}^{t}\left(f(s,x^{(n)}_{s},u^{(n)}_{s})-f(s,x^{(n-1)}_{s},u^{(n-1)}_{s})\right)ds\right)^{2}$
$\displaystyle\quad\;+\left(\int_{0}^{t}\left(g(s,x^{(n)}_{s},u^{(n)}_{s})-g(s,x^{(n-1)}_{s},u^{(n-1)}_{s})\right)dB_{s}\right)^{2}\Bigg{]}$
$\displaystyle\leq
4L_{g}^{2}\left(\int_{0}^{t}\mathbb{E}\|x^{(n)}_{s}-x^{(n-1)}_{s}\|^{2}ds+\int_{0}^{t}\mathbb{E}\|u^{(n)}_{s}-u^{(n-1)}_{s}\|^{2}ds\right)$
$\displaystyle\quad\;+4t_{0}L_{f}^{2}\left(\int_{0}^{t}\mathbb{E}\|x^{(n)}_{s}-x^{(n-1)}_{s}\|^{2}ds+\int_{0}^{t}\mathbb{E}\|u^{(n)}_{s}-u^{(n-1)}_{s}\|^{2}ds\right)$
$\displaystyle=4(L_{g}^{2}+TL_{f}^{2})\left(\int_{0}^{t}\left(\mathbb{E}\|x^{(n)}_{s}-x^{(n-1)}_{s}\|^{2}+\mathbb{E}\|u^{(n)}_{s}-u^{(n-1)}_{s}\|^{2}\right)ds\right).$

On the other hand, from Lemma 3.2, we know there exists a constant
$M^{\prime}>0$, for all $t\in[0,T]$, such that

$\displaystyle\mathbb{E}\int_{0}^{t}\|u^{(n)}(s,\omega)-u^{(n-1)}(s,\omega)\|^{2}ds\leq
M^{\prime}\mathbb{E}\int_{0}^{t}\|x^{(n)}(s,\omega)-x^{(n-1)}(s,\omega)\|^{2}ds.$
(3.8)

Combining (3.7) and (3.8), for all $t\in[0,T]$, one has

$\displaystyle\mathbb{E}\|x^{(n+1)}_{t}-x^{(n)}_{t}\|^{2}\leq\beta\mathbb{E}\int_{0}^{t}\|x^{(n)}(s,\omega)-x^{(n-1)}(s,\omega)\|^{2}ds,$
(3.9)

where $\beta=4(L_{g}^{2}+TL_{f}^{2})(1+M^{\prime})$. Moreover, the conditions
$(i)$ and $(iii)$ in Assumption 3.1 lead to

$\displaystyle\mathbb{E}\|x^{(2)}_{t}-x^{(2)}_{t}\|^{2}\leq
6(K^{2}_{1}+K^{2}_{2})\mathbb{E}\int_{0}^{t}(1+\|x_{0}\|^{2}+\|u^{(1)}_{t}\|^{2})ds.$
(3.10)

It follows from (3.9), (3.10) and Lemma 2.5 that

$\displaystyle\mathbb{E}\|x^{(n+1)}_{t}-x^{(n)}_{t}\|^{2}\leq\eta\frac{\beta^{n-1}t^{n}}{n!},$

which implies that

$\|x^{(n+1)}-x^{(n)}\|^{2}_{H[0,T]}\leq\eta\frac{\beta^{n-1}T^{n+1}}{n!}\rightarrow
0,\quad(n\rightarrow\infty),$

where
$\eta=6(K^{2}_{1}+K^{2}_{2})(1+\|x_{0}\|^{2}_{H[0,T]}+\|u^{(1)}_{t}\|^{2}_{H[0,T]})$.
Now Lemma 3.2 implies that $\left\\{(x^{n}_{t},u^{n}_{t})\right\\}$ is a
Cauchy sequence in $L^{2}_{ad}([0,T]\times\Omega,\mathbb{R}^{n})\times
U[0,T]$.

Therefore, there exists a couple $(x^{*}(t),u^{*}(t))\in
L^{2}_{ad}([0,T]\times\Omega,\mathbb{R}^{n})\times U[0,T]$ such that
$(x^{(n)}(t),u^{(n)}(t))\to(x^{*}(t),u^{*}(t))$. Furthermore, it follows from
the continuity of $P_{U[0,T]}$ and dominated convergence theorem that

$\displaystyle\left\\{\begin{aligned}
u^{*}_{t}&=\lim\limits_{n\rightarrow\infty}P_{U[0,T]}\left(u^{(n)}_{t}-\rho\tilde{F}(x^{(n)}_{t},u^{(n)}_{t})\right)\\\
&=P_{U[0,T]}\bigg{(}u^{*}_{t}-\rho\tilde{F}(x^{*}_{t},u^{*}_{t})\bigg{)},\\\
x^{*}_{t}&=x_{0}+\lim\limits_{n\rightarrow\infty}\int_{0}^{t}f(s,x^{(n)}_{s},u^{(n)}_{s})ds+\lim\limits_{n\rightarrow\infty}\int_{0}^{t}g(s,x^{(n)}_{s},u^{(n)}_{s})dB_{s}\\\
&=x_{0}+\int_{0}^{t}f(s,x^{*}_{s},u^{*}_{s})ds+\int_{0}^{t}g(s,x^{*}_{s},u^{*}_{s})dB_{s},\end{aligned}\right.$

which implies that the limit $(x^{*}_{t},u^{*}_{t})$ is a Carathéodory
solution for SDVI (1.1) with $\Gamma(x(0),x(T))=x(0)-x_{0}$ in
$L^{2}_{ad}([0,T]\times\Omega,\mathbb{R}^{n})\times U[0,T]$.

Finally, we show the uniqueness of the Carathéodory solutions for SDVI (1.1)
with $\Gamma(x(0),x(T))=x(0)-x_{0}$. Suppose there exist $(x_{i},u_{i})\in
L^{2}_{ad}([0,T]\times\Omega,\mathbb{R}^{n})\times U[0,T]$ $(i=1,2)$
satisfying

$\displaystyle\left\\{\begin{aligned}
u_{i}(t)&=P_{U[0,T]}\left(u_{i}(t)-\rho\tilde{F}(x_{i}(t),u_{i}(t))\right),\\\
x_{i}(t)&=x_{0}+\int_{0}^{t}f(s,x_{i}(s),u_{i}(s))ds+\int_{0}^{t}g(s,x_{i}(s),u_{i}(s))dB_{s}.\end{aligned}\right.$
(3.11)

Then, similar to the proof of (3.9), it follows from (3.11) and the conditions
$(i)$ to $(iv)$ in Assumption 3.1 that

$\displaystyle\mathbb{E}\|x_{1}-x_{2}\|^{2}\leq\beta\mathbb{E}\int_{0}^{t}\|x_{1}(s,\omega)-x_{2}(s,\omega)\|^{2}ds,\forall
t\in[0,T],$

where $\beta=4(L_{g}^{2}+TL_{f}^{2})(1+M^{\prime})$. By Gronwall’s inequality,
we have $\mathbb{E}\|x_{1}-x_{2}\|^{2}=0$. Now Lemma 3.2 leads to

$\|x_{1}-x_{2}\|_{H[0,T]}=0,\quad\|u_{1}-u_{2}\|_{H[0,T]}=0.$

This show the uniqueness of the Carathéodory solutions. ∎

##  4 Convergence of Euler scheme for solving SDVI

In this section, we prove the convergence of Euler scheme constructed by
(1.2). To this end, we need the following hypotheses and lemmas.

######  Assumption 4.1.

Suppose that there exist positive constants $C$, $L_{f}$, $L_{g}$, and $L_{F}$
with $L_{F}>C$ such that

  1. ($i$)

$\|f(t_{1},x_{1},u_{1})-f(t_{2},x_{2},u_{2})\|\leq
L_{f}(|t_{1}-t_{2}|+\|x_{1}-x_{2}\|+\|u_{1}-u_{2}\|)$;

  2. ($ii$)

$\|g(t_{1},x_{1},u_{1})-g(t_{2},x_{2},u_{2})\|_{\mathbb{R}^{n\times m}}\leq
L_{g}(|t_{1}-t_{2}|+\|x_{1}-x_{2}\|+\|u_{1}-u_{2}\|)$;

  3. ($iii$)

$\|F(t_{1},\omega,x_{1},u_{1})-F(t_{2},\omega,x_{2},u_{2})\|\leq
L_{F}(|t_{1}-t_{2}|+\|x_{1}-x_{2}\|+\|u_{1}-u_{2}\|)$, a.s. $\omega\in\Omega$;

  4. ($iv$)

$\left\langle
F(t,\omega,x,u_{1})-F(t,\omega,x,u_{2}),u_{1}-u_{2}\right\rangle\geq
C\|u_{1}-u_{2}\|^{2}$, a.s. $\omega\in\Omega$,

for all $t,t_{1},t_{2}\in[0,T]$, all $x,x_{1},x_{2}\in\mathbb{R}^{n}$, all
$u_{1},u_{2}\in\mathbb{R}^{m}$.

######  Remark 1.

Clearly, $(i)$ and $(ii)$ imply that $f(t,x,u)$ and $g(t,x,u)$ have linear
growth in $x$ and $u$, i.e., there exist two constants $K_{1}>0$ and $K_{2}>0$
conforming to $\|f(t,x,u)\|\leq K_{1}(1+\|x\|+\|u\|)$ and
$\|g(t,x,u)\|_{\mathbb{R}^{n\times m}}\leq K_{2}(1+\|x\|+\|u\|)$ for all
$(t,\omega)\in[0,T]\times\Omega$. It is worth noting that Assumption 3.1 is a
speacil case of Assumption 4.1.

######  Lemma 4.1.

Let $F$ satisfy the conditions $(iii)$ and $(iv)$ in Assumption 4.1. Then
$u_{h}(t)$ solves the following stochastic variational inequality

$\displaystyle\mathbb{E}\int_{t_{i}}^{t}\left\langle
F(t_{i},\omega,x_{h}(t_{i},\omega),u_{h}(s,\omega)),v(s,\omega)-u_{h}(s,\omega)\right\rangle
ds\geq 0$ (4.1)

for all $v\in U[t_{i},t]$ and all $t\in[t_{i},t_{i+1})$ with $i=0,1,\cdots,N$.

###### Proof.

First, for fixed $t\in[0,T]$, define a mapping
$\bar{F}:L^{2}(\Omega,\mathbb{R}^{n})\times U\to L^{2}(\Omega,\mathbb{R}^{m})$
by

$\bar{F}(x,u)(\omega):=F(t,\omega,x(\omega),u(\omega)),\;\forall(x,u)\in
L^{2}(\Omega,\mathbb{R}^{n})\times U,\;\forall\omega\in\Omega.$

Then, for each $i=0,1,\cdots,N$, by the conditions $(iii)$ and $(iv)$ in
Assumption 4.1, Lemma 3.1 implies that there is a unique $u_{h}(t_{i})$
satisfying

$\left\langle\bar{F}(x_{h}(t_{i}),u_{h}(t_{i})),v-u_{h}(t_{i})\right\rangle_{L^{2}(\Omega)}\geq
0,\;\forall v\in U.$

Let $u_{h}(t)=u_{h}(t_{i})$ for all $t\in[t_{i},t_{i+1})$ in (1.2) with
$i=0,1,\cdots,N$. Thus, by the definition of $U[a,b]$, we know that
$u_{h}(s)\in U[t_{i},t]$ for all $t\in[t_{i},t_{i+1})$. Clearly, (1.2) implies
that

$\displaystyle\left\langle
F(t_{i},\omega,x_{h}(t_{i},\omega),u_{h}(t,\omega)),v-u_{h}(t,\omega)\right\rangle\geq
0,\;\forall v\in K,\forall t\in[t_{i},t_{i+1}),\;a.s.\;\omega\in\Omega.$

For any $t\in[t_{i},t_{i+1})$, by taking
$F(t,\omega,x(t,\omega),u(t,\omega))=F(t_{i},\omega,x_{h}(t_{i},\omega),u_{h}(t,\omega)))$
in Lemma 2.6, we know that,

$\displaystyle\mathbb{E}\int_{t_{i}}^{t}\left\langle
F(t_{i},\omega,x_{h}(t_{i},\omega),u_{h}(s,\omega)),v(s,\omega)-u_{h}(s,\omega)\right\rangle
ds\geq 0$

for all $v\in U[t_{i},t]$ with $t\in[t_{i},t_{i+1})$. This ends the proof. ∎

######  Lemma 4.2.

Let $K\subset\mathbb{R}^{n}$ be closed convex. Under Assumption 4.1, there
exists a constant $M\geq 0$ such that, for $i=1,\cdots,N$,

$\displaystyle\|u_{h}(t_{i})\|_{L^{2}(\Omega)}\leq
M(1+\|x_{h}(t_{i})\|_{L^{2}(\Omega)}).$

###### Proof.

By Lemmas 2.1, 2.3 and 4.1, there exists a unique $u_{h}(t_{n})\in U$
satisfying

$\displaystyle
u_{h}(t_{i})=P_{U}[u_{h}(t_{i})-\rho\bar{F}(x_{h}(t_{i}),u_{h}(t_{i}))],$
(4.2)

where $0<\rho<\frac{2C}{L^{2}_{F}}$. It follows from (4.2) and the conditions
(iii) and (iv) in Assumption 4.1 that

$\displaystyle\quad\;\mathbb{E}\|u_{h}(t_{i})-u_{h}(t_{0})+\rho
F(t_{0},\omega,x_{h}(t_{0}),u_{h}(t_{0}))-\rho
F(t_{0},\omega,x_{h}(t_{0}),u_{h}(t_{i}))\|^{2}$ (4.3)
$\displaystyle\leq\mathbb{E}\|u_{h}(t_{i})-u_{h}(t_{0})\|^{2}+\rho^{2}\mathbb{E}\|\bar{F}(x_{h}(t_{0}),u_{h}(t_{0}))-\rho\bar{F}(x_{h}(t_{0}),u_{h}(t_{i}))\|^{2}$
$\displaystyle\mbox{}\quad-2\rho\mathbb{E}\Big{[}\langle
u_{h}(t_{i})-u_{h}(t_{0}),F(t_{0},\omega,x_{h}(t_{0}),u_{h}(t_{i}))-F(t_{0},\omega,x_{h}(t_{0}),u_{h}(t_{0}))\rangle\Big{]}$
$\displaystyle\leq(1-2\rho
C+\rho^{2}L^{2}_{F})\mathbb{E}\|u_{h}(t_{i})-u_{h}(t_{0})\|^{2}.$

Applying (4.3) and the condition $(iii)$ in Assumption 4.1, we obtain

$\displaystyle\mbox{}\quad\;\|u_{h}(t_{i})-u_{h}(t_{0})\|_{L^{2}(\Omega)}$
$\displaystyle=\left[\mathbb{E}\|P_{U}[u_{h}(t_{i})-\rho\bar{F}(x(t_{i}),u(t_{i}))]-P_{U}[u(t_{0})-\rho\bar{F}(x_{h}(t_{0}),u_{h}(t_{0}))]\|^{2}\right]^{\frac{1}{2}}$
$\displaystyle\leq\left[\mathbb{E}\|u_{h}(t_{i})-u_{h}(t_{0})+\rho
F(t_{0},\omega,x_{h}(t_{0}),u_{h}(t_{0}))-\rho
F(t_{i},\omega,x(t_{i}),u(t_{i}))\|^{2}\right]^{\frac{1}{2}}$
$\displaystyle=\Big{[}\mathbb{E}\|u_{h}(t_{i})-u_{h}(t_{0})+\rho
F(t_{0},\omega,x_{h}(t_{0}),u_{h}(t_{0}))-\rho
F(t_{0},\omega,x_{h}(t_{0}),u_{h}(t_{i}))$ $\displaystyle\mbox{}\quad+\rho
F(t_{0},\omega,x_{h}(t_{0}),u_{h}(t_{i}))-\rho
F(t_{i},\omega,x_{h}(t_{n}),u_{h}(t_{i}))\|^{2}\Big{]}^{\frac{1}{2}}$
$\displaystyle\leq\left[\mathbb{E}\|u_{h}(t_{i})-u_{h}(t_{0})+\rho
F(t_{0},\omega,x_{h}(t_{0}),u_{h}(t_{0}))-\rho
F(t_{0},\omega,x(t_{0}),u(t_{i}))\|^{2}\right]^{\frac{1}{2}}$
$\displaystyle\mbox{}\quad+\left[\mathbb{E}\|\rho
F(t_{0},\omega,x(t_{0}),u(t_{i}))-\rho
F(t_{i},\omega,x(t_{i}),u(t_{i}))\|^{2}\right]^{\frac{1}{2}}$
$\displaystyle\leq\sqrt{(1-2\rho
C+\rho^{2}L^{2}_{F})}\|u_{h}(t_{i})-u_{h}(t_{0})\|_{L^{2}(\Omega)}$
$\displaystyle\mbox{}\quad+\sqrt{2}\rho L_{F}ih+\sqrt{2}\rho
L_{F}\|x_{h}(t_{i})-x_{h}(t_{0})\|_{L^{2}(\Omega)},$

which implies that

$\displaystyle\|u_{h}(t_{i})-u_{h}(t_{0})\|_{L^{2}(\Omega)}\leq
M(ih+\|x_{h}(t_{i})-x_{h}(t_{0})\|_{L^{2}(\Omega)}),$ (4.4)

where $M=\frac{\sqrt{2}\rho L_{F}}{1-\sqrt{(1-2\rho C+\rho^{2}L^{2}_{F})}}$
and $u_{h}(t_{0})=P_{U}[u_{h}(t_{0})-\rho\bar{F}(x_{h}(t_{0}),u_{h}(t_{0}))]$.
The inequality (4.4) shows that there exists a constant $M>0$ such that

$\displaystyle\|u_{h}(t_{i})\|_{L^{2}(\Omega)}\leq
M(1+\|x_{h}(t_{i})\|_{L^{2}(\Omega)}).$ (4.5)

This ends the proof. ∎

At the end of this section, we show our main results as follows.

######  Theorem 4.1.

Let $K\subset\mathbb{R}^{n}$ be closed convex. Under Assumptions 4.1, one has

$\|x_{h}-x\|_{H_{[0,T]}}\rightarrow 0,\quad\|u_{h}-u\|_{H_{[0,T]}}\rightarrow
0\quad\mbox{as}\quad h\rightarrow 0,$

where $(x_{h}(t),u_{h}(t))$ is given by (1.2) and $(x(t),u(t))$ is a unique
solution of SDVI (1.1). Moreover, there exists a constant $C>0$ such that

$\displaystyle\|x_{h}-x\|^{2}_{H_{[0,T]}}\leq
C[h^{2}+h],\quad\|u-u_{h}\|_{H_{[0,T]}}^{2}\leq C[h^{2}+h].$

###### Proof.

We first show the measurability of $x_{h}(t)$ and $u_{h}(t)$. In fact, define
$\hat{s}=\hat{s}(s)$ by setting

$\hat{s}(s)=\left\\{\begin{array}[]{ll}t_{i},&\quad s\in[t_{i},t_{i+1});\\\
0,&\quad\mbox{else}.\end{array}\right.$ (4.6)

Then, it follows from (1.2) and (4.6) that

$\left\\{\begin{aligned}
&x_{h}(t)=x_{0}+\int_{0}^{t}f(\hat{s},x_{h}(\hat{s}),u_{h}(\hat{s}))ds+\int_{0}^{t}g(\hat{s},x_{h}(\hat{s}),u_{h}(\hat{s}))dB_{s},\\\
&\left\langle
F(\hat{s},\omega,x_{h}(\hat{s}),u_{h}(\hat{s})),v-u_{h}(\hat{s})\right\rangle\geq
0,\quad\forall v\in K,\;a.s.\;\omega\in\Omega.\end{aligned}\right.$ (4.7)

Similar to the proof of Theorem 3.1, we know that $x_{h}(t)$ is
$\mathcal{F}_{t}$-measurable and
$\mathbb{E}\int_{0}^{T}(x_{h}(t))^{2}dt<\infty$, which implies that
$x_{h}(t)\in L^{2}_{ad}([0,T]\times\Omega,\mathbb{R}^{n})$. Define a mapping
$\tilde{F}_{i}:L^{2}_{ad}([t_{i},t_{i+1}]\times\Omega,\mathbb{R}^{n})\times
U[t_{i},t_{i+1}]\to L^{2}_{ad}([t_{i},t_{i+1}]\times\Omega,\mathbb{R}^{m})$ by

$\tilde{F}_{i}(x,u)(s,\omega):=F(s,\omega,x(s,\omega),u(s,\omega))$

for all $(x,u)\in L^{2}_{ad}([t_{i},t_{i+1}]\times\Omega,\mathbb{R}^{n})\times
U[t_{i},t_{i+1}]$, for all $s\in[t_{i},t_{i+1})$, and for all
$\omega\in\Omega$. Then, by (1.2) Lemmas 2.3 and 4.1, we have

$\displaystyle
u_{h}(\hat{s})=P_{U[t_{i},t_{i+1}]}[u_{h}(\hat{s})-\rho\tilde{F}_{n}(x_{h}(\hat{s}),u_{h}(\hat{s}))],$
(4.8)

where $0<\rho<\frac{2C}{L^{2}_{F}}$ is a constant. Thus, (4.8) and (1.2) imply
that $u_{h}(t)$ is $\mathcal{F}_{t_{i}}$-measurable for all
$t\in[t_{i},t_{i+1})$ and $\mathbb{E}\int_{0}^{T}\|u_{h}(t)\|^{2}dt<\infty$.

Next we consider $\mathbb{E}\|x_{h}(t)-x(t)\|^{2}$. To this end, by employing
(1.2), one has

$\displaystyle
x_{h}(t_{i})=x_{0}+\sum_{k=1}^{i-1}f(t_{k-1},x_{h}(t_{k-1}),u_{h}(t_{k-1}))(t_{k}-t_{k-1})$
$\displaystyle\qquad\qquad\mbox{}+\sum_{k=1}^{i-1}g(t_{k-1},x_{h}(t_{k-1}),u_{h}(t_{k-1}))(B_{t_{k}}-B_{t_{k-1}}),$

which implies that

$\displaystyle\mathbb{E}\|x_{h}(t_{i})\|^{2}$ $\displaystyle\leq
3\mathbb{E}\|x_{0}\|^{2}+3\mathbb{E}\left[\sum_{k=1}^{i-1}f(t_{k-1},x_{h}(t_{k-1}),u_{h}(t_{k-1}))(t_{k}-t_{k-1})\right]^{2}$
(4.9)
$\displaystyle\mbox{}\quad+3\mathbb{E}\left[\sum_{k=1}^{i-1}g(t_{k-1},x_{h}(t_{k-1}),u_{h}(t_{k-1}))(B_{t_{k}}-B_{t_{k-1}})\right]^{2}$
$\displaystyle\leq 3\mathbb{E}(x_{0})^{2}+3H_{1}+3H_{2},$

where

$H_{1}=h^{2}\mathbb{E}\left[\sum\limits_{k=1}^{i-1}f(t_{k-1},x_{h}(t_{k-1}),u_{h}(t_{k-1}))\right]^{2},$
$H_{2}=h\sum\limits_{k=1}^{i-1}\mathbb{E}\|g(t_{k-1},x_{h}(t_{k-1}),u_{h}(t_{k-1}))\|_{\mathbb{R}^{n\times
m}}^{2}.$

By Remark 1, there exist two constants $K_{1},K_{2}>0$ such that, for all
$(t,\omega)\in[0,T]\times\Omega$,

$\|f(t,x,u)\|\leq K_{1}(1+\|x\|+\|u\|),\quad\|g(t,x,u)\|_{\mathbb{R}^{n\times
m}}\leq K_{2}(1+\|x\|+\|u\|).$

Thus, we have

$\displaystyle H_{1}\leq$ $\displaystyle
h^{2}K_{1}^{2}\mathbb{E}\left[\sum_{k=1}^{i-1}(1+\|x_{h}(t_{k-1})\|+\|u_{h}(t_{k-1})\|)\right]^{2}$
(4.10) $\displaystyle\leq$ $\displaystyle
h^{2}K_{1}^{2}N\sum_{k=1}^{i-1}\mathbb{E}\left[1+\|x_{h}(t_{k-1})\|+\|u_{h}(t_{k-1})\|\right]^{2}$
$\displaystyle\leq$ $\displaystyle
3ThK_{1}^{2}\sum_{k=1}^{i-1}\mathbb{E}[1+\|x_{h}(t_{k-1})\|^{2}+\|u_{h}(t_{k-1})\|^{2}]$
$\displaystyle\leq$ $\displaystyle
3ThK_{1}^{2}+3ThK_{1}^{2}\sum_{k=1}^{i-1}\mathbb{E}\|x_{h}(t_{k-1})\|^{2}+3ThK_{1}^{2}\sum_{k=1}^{i-1}\mathbb{E}\|u_{h}(t_{k-1})\|^{2}$

and

$\displaystyle H_{2}$ $\displaystyle\leq
3hK_{2}^{2}\sum_{k=1}^{i-1}\mathbb{E}[1+\|x_{h}(t_{k-1})\|^{2}+\|u_{h}(t_{k-1})\|^{2}]$
(4.11) $\displaystyle\leq
3hK_{2}^{2}+3hK_{2}^{2}\sum_{k=1}^{i-1}\mathbb{E}\|x_{h}(t_{k-1})\|^{2}+3hK_{2}^{2}\sum_{k=1}^{i-1}\mathbb{E}\|u_{h}(t_{k-1})\|^{2}.$

Combining inequalities (4.9) to (4.11), we obtain

$\displaystyle\mathbb{E}\|x_{h}(t_{i})\|^{2}\leq
3\mathbb{E}\|x_{0}\|^{2}+Ch\left[1+\sum_{k=1}^{i-1}\mathbb{E}\|x_{h}(t_{k-1})\|^{2}+\sum_{k=1}^{i-1}\mathbb{E}\|u_{h}(t_{k-1})\|^{2}\right],$
(4.12)

where $C=9TK_{1}^{2}+9K_{2}^{2}$. By Lemma 4.2, there exists a constant $M>0$
such that $\mathbb{E}\|u_{h}(t_{i})\|^{2}\leq
M(1+\mathbb{E}\|x_{h}(t_{i})\|^{2})$ and so (4.12) implies that

$\displaystyle\mathbb{E}\|x_{h}(t_{i})\|^{2}\leq
3\mathbb{E}\|x_{0}\|^{2}+CT+C^{\prime}h\sum_{k=1}^{i-1}\mathbb{E}\|x_{h}(t_{k-1})\|^{2},$
(4.13)

where $C^{\prime}=CM+C$. Clearly,
$h\leq\frac{T}{N^{\frac{1}{2}}i^{\frac{1}{2}}}$. Now applying (4.13) and Lemma
2.4, one has

$\displaystyle\mathbb{E}\|x_{h}(t_{i})\|^{2}\leq
3\mathbb{E}\|x_{0}\|^{2}+CT+\frac{C^{\prime}T}{N^{\frac{1}{2}}}\sum_{k=1}^{i-1}\frac{\mathbb{E}\|x_{h}(t_{k-1})\|^{2}}{(i-k)^{\frac{1}{2}}}\leq
A,$ (4.14)

where
$A=(3\mathbb{E}(x_{0})^{2}+CT)[1+\mathbb{G}_{\frac{1}{2}}(C^{\prime}T\Gamma(\frac{1}{2}))]$.
Moreover, for any fixed $t\in[t_{i-1},t_{i})$, it follows from (4.14) and
Assumption 4.1 that

$\displaystyle\mathbb{E}\|x_{h}(t)-x_{h}(t_{i-1})\|^{2}$ (4.15)
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\int_{t_{i-1}}^{t}f(t_{i-1},x_{h}(t_{i-1}),u_{h}(t_{i-1}))ds+\int_{t_{i-1}}^{t}g(t_{i-1},x_{h}(t_{i-1}),u_{h}(t_{i-1}))dB_{s}\right]^{2}$
$\displaystyle\leq$ $\displaystyle
2\mathbb{E}\left[\int_{t_{i-1}}^{t}f(t_{i-1},x_{h}(t_{i-1}),u_{h}(t_{i-1}))ds\right]^{2}$
$\displaystyle+2\mathbb{E}\left[\int_{t_{i-1}}^{t}g(t_{i-1},x_{h}(t_{i-1}),u_{h}(t_{i-1}))dB_{s}\right]^{2}$
$\displaystyle=$ $\displaystyle
2\mathbb{E}[\|f(t_{i-1},x_{h}(t_{i-1}),u_{h}(t_{i-1})\|^{2}(t-t_{i-1})^{2}]$
$\displaystyle+2\mathbb{E}[\|g(t_{i-1},x_{h}(t_{i-1}),u_{h}(t_{i-1})\|_{\mathbb{R}^{n\times
m}}^{2}(t-t_{i-1})]$ $\displaystyle\leq$
$\displaystyle\mathbb{E}[6K_{1}^{2}(1+\|x_{h}(t_{i-1})\|^{2}+\|u_{h}(t_{i-1})\|^{2})(t-t_{i-1})^{2}]$
$\displaystyle+\mathbb{E}[6K_{2}^{2}(1+\|x_{h}(t_{i-1})\|^{2}+\|u_{h}(t_{i-1})\|^{2})(t-t_{i-1})]$
$\displaystyle\leq$ $\displaystyle B_{1}h^{2}+B_{2}h,$

where $B_{1}=6K_{1}^{2}(1+A+M+AM)$ and $B_{2}=6K_{2}^{2}(1+A+M+AM)$. Thus, by
(4.7), we have

$\displaystyle\mathbb{E}\|x_{h}(t)-x(t)\|^{2}\leq I_{1}+I_{2},$ (4.16)

where

$\left\\{\begin{aligned}
&I_{1}=2\mathbb{E}\left[\int_{0}^{t}(f(s,x(s),u(s))-f(\hat{s},x_{h}(\hat{s}),u_{h}(\hat{s})))ds\right]^{2},\\\
&I_{2}=2\mathbb{E}\left[\int_{0}^{t}(g(s,x(s),u(s))-g(\hat{s},x_{h}(\hat{s}),u_{h}(\hat{s})))dB_{s}\right]^{2}.\end{aligned}\right.$

From (4.15), Cauchy-Schwartz’s inequality and Assumption 4.1, one has

$\displaystyle I_{1}$
$\displaystyle=2\mathbb{E}\left[\int_{0}^{t}(f(s,x(s),u(s))-f(\hat{s},x_{h}(\hat{s}),u_{h}(\hat{s})))ds\right]^{2}$
(4.17) $\displaystyle\leq
2T\mathbb{E}\left[\int_{0}^{t}\|f(s,x(s),u(s))-f(\hat{s},x_{h}(\hat{s}),u_{h}(\hat{s}))\|^{2}ds\right]$
$\displaystyle\leq
2TL_{f}^{2}\mathbb{E}\left[\int_{0}^{t}|s-\hat{s}|+\|x(s)-x_{h}(\hat{s})\|+\|u(s)-u_{h}(\hat{s})\|\right]^{2}ds$
$\displaystyle\leq
6TL_{f}^{2}\mathbb{E}\left[\int_{0}^{t}|s-\hat{s}|^{2}+\|x(s)-x_{h}(\hat{s})\|^{2}+\|u(s)-u_{h}(\hat{s})\|^{2}\right]ds$
$\displaystyle\leq
6T^{2}L_{f}^{2}h^{2}+12TL_{f}^{2}\mathbb{E}\left[\int_{0}^{t}\|x(s)-x_{h}(s)\|^{2}ds\right]$
$\displaystyle\mbox{}\quad+12TL_{f}^{2}\mathbb{E}\left[\int_{0}^{t}\|x_{h}(s)-x_{h}(\hat{s})\|^{2}ds\right]+6TL_{f}^{2}\mathbb{E}\left[\int_{0}^{t}\|u(s)-u_{h}(\hat{s})\|^{2}ds\right]$
$\displaystyle\leq
6T^{2}L_{f}^{2}h^{2}+12T^{2}L_{f}^{2}B_{1}h^{2}+12T^{2}L_{f}^{2}B_{2}h$
$\displaystyle\mbox{\quad}+6TL_{f}^{2}\mathbb{E}\left[\int_{0}^{t}\|u(s)-u_{h}(\hat{s})\|^{2}ds\right]++12TL_{f}^{2}\mathbb{E}\left[\int_{0}^{t}\|x_{h}(s)-x_{h}(\hat{s})\|^{2}ds\right]$

and

$\displaystyle I_{2}$
$\displaystyle=2\mathbb{E}\left[\int_{0}^{t}(g(s,x(s),u(s))-g(\hat{s},x_{h}(\hat{s}),u_{h}(\hat{s})))dB_{s}\right]^{2}$
(4.18)
$\displaystyle=2\mathbb{E}\left[\int_{0}^{t}\|g(s,x(s),u(s))-g(\hat{s},x_{h}(\hat{s}),u_{h}(\hat{s}))\|^{2}_{\mathbb{R}^{n\times
m}}ds\right]$ $\displaystyle\leq
2L_{g}^{2}\mathbb{E}\int_{0}^{t}\left[|s-\hat{s}|+\|x(s)-x_{h}(\hat{s})\|+\|u(s)-u_{h}(\hat{s})\|\right]^{2}ds$
$\displaystyle\leq
6L_{g}^{2}\mathbb{E}\left[\int_{0}^{t}|s-\hat{s}|^{2}+\|x(s)-x_{h}(\hat{s})\|^{2}+\|u(s)-u_{h}(\hat{s})\|^{2}\right]ds$
$\displaystyle\leq
6TL_{g}^{2}h^{2}+12L_{g}^{2}\mathbb{E}\left[\int_{0}^{t}\|x(s)-x_{h}(s)\|^{2}ds\right]$
$\displaystyle\mbox{\quad}+12L_{g}^{2}\mathbb{E}\left[\int_{0}^{t}\|x_{h}(s)-x_{h}(\hat{s})\|^{2}ds\right]+6L_{g}^{2}\mathbb{E}\left[\int_{0}^{t}\|u(s)-u_{h}(\hat{s})\|^{2}ds\right]$
$\displaystyle\leq
6TL_{g}^{2}h^{2}+12TL_{g}^{2}B_{1}^{2}h^{2}+12TL_{g}^{2}B_{2}h+12L_{g}^{2}\mathbb{E}\left[\int_{0}^{t}\|x(s)-x_{h}(s)\|^{2}ds\right]$
$\displaystyle\mbox{\quad}+6L_{g}^{2}\mathbb{E}\left[\int_{0}^{t}\|u(s)-u_{h}(\hat{s})\|^{2}ds\right].$

Combining (4.16) to (4.18), we obtain

$\displaystyle\mathbb{E}\|x_{h}(t)-x(t)\|^{2}$ $\displaystyle\leq
C_{1}h^{2}+C_{2}h+C_{3}\mathbb{E}\left[\int_{0}^{t}\|x(s)-x_{h}(s)\|^{2}ds\right]$
(4.19)
$\displaystyle\quad\;+C_{4}\mathbb{E}\left[\int_{0}^{t}\|u(s)-u_{h}(\hat{s})\|^{2}ds\right],$

where
$C_{1}=6T^{2}L_{f}^{2}+6TL_{g}^{2}+12T^{2}L_{f}^{2}B_{1}+12TL_{g}^{2}B_{1}^{2}$,
$C_{2}=12T^{2}L_{f}^{2}B_{2}+12TL_{g}^{2}B_{2}$,
$C_{3}=12TL_{f}^{2}+12L_{g}^{2}$, and $C_{4}=6TL_{f}^{2}+6L_{g}^{2}$.

Finally, we estimate
$\mathbb{E}\left[\int_{0}^{t}\|u(s)-u_{h}(\hat{s})\|^{2}ds\right]$. To do
this, it follows from (4.8) and Assumption 4.1 that

$\displaystyle\mbox{}\quad\;\mathbb{E}\int_{t_{i-1}}^{t}\|u(s)-u(\hat{s})+\rho
F(s,\omega,x(s),u_{h}(\hat{s}))-\rho F(s,\omega,x(s),u(s))\|^{2}ds$ (4.20)
$\displaystyle\leq\mathbb{E}\int_{t_{i-1}}^{t}\|u(s)-u(\hat{s})\|^{2}ds$
$\displaystyle\mbox{}\quad+\rho^{2}\mathbb{E}\int_{t_{i-1}}^{t}\|F(s,\omega,x(s),u_{h}(\hat{s}))-F(s,\omega,x(s),u(s))\|^{2}ds$
$\displaystyle\mbox{}\quad-2\rho\mathbb{E}\left[\int_{t_{i-1}}^{t}\langle
u(s)-u_{h}(\hat{s}),F(s,\omega,x(s),u(s))-F(s,\omega,x(s),u_{h}(\hat{s}))\rangle
ds\right]$ $\displaystyle\leq(1-2\rho
C+\rho^{2}L^{2}_{F})\mathbb{E}\int_{t_{i-1}}^{t}\|u(t)-u_{h}(\hat{s})\|^{2}ds.$

Applying (4.20) and Assumption 4.1, we get

$\displaystyle\mbox{}\quad\;\left[\mathbb{E}\int_{t_{i-1}}^{t}\|u(s)-u_{h}(\hat{s})\|^{2}ds\right]^{\frac{1}{2}}$
$\displaystyle=\Bigg{[}\mathbb{E}\int_{t_{i-1}}^{t}\|P_{U[t_{n-1},t]}(u-\rho\tilde{F}_{n}(x,u))$
$\displaystyle\mbox{}\quad-
P_{U[t_{i-1},t]}(u(\hat{s})-\rho\tilde{F}_{n}(x_{h}(\hat{s}),u_{h}(\hat{s})))\|^{2}ds\Bigg{]}^{\frac{1}{2}}$
$\displaystyle\leq\Bigg{[}\mathbb{E}\int_{t_{i-1}}^{t}\|u(s)-u_{h}(\hat{s})+\rho
F(\hat{s},\omega,x_{h}(\hat{s}),u_{h}(\hat{s}))-\rho
F(s,\omega,x(s),u(s))\|^{2}ds\Bigg{]}^{\frac{1}{2}}$
$\displaystyle=\Bigg{[}\mathbb{E}\int_{t_{i-1}}^{t}\|u(s)-u_{h}(\hat{s})+\rho
F(\hat{s},\omega,x_{h}(\hat{s}),u_{h}(\hat{s}))-\rho
F(s,\omega,x(s),u_{h}(\hat{s}))$ $\displaystyle\mbox{}\quad+\rho
F(s,\omega,x(s),u_{h}(\hat{s}))-\rho
F(s,\omega,x(s),u(s))\|^{2}ds\Bigg{]}^{\frac{1}{2}}$
$\displaystyle\leq\left[\mathbb{E}\int_{t_{i-1}}^{t}\|u(s)-u_{h}(\hat{s})+\rho
F(s,\omega,x(s),u_{h}(\hat{s}))-\rho
F(s,\omega,x(s),u(s))\|^{2}ds\right]^{\frac{1}{2}}$
$\displaystyle\mbox{}\quad+\left[\mathbb{E}\int_{t_{i-1}}^{t}\|\rho
F(\hat{s},\omega,x_{h}(\hat{s}),u_{h}(\hat{s}))-\rho
F(s,\omega,x(s),u_{h}(\hat{s}))\|^{2}ds\right]^{\frac{1}{2}}$
$\displaystyle\leq\sqrt{(1-2\rho
C+\rho^{2}L^{2}_{F})}\mathbb{E}\left[\int_{t_{i-1}}^{t}\|u(s)-u_{h}(\hat{s})\|^{2}ds\right]^{\frac{1}{2}}+\sqrt{2}\rho
L_{F}h^{\frac{3}{2}}$ $\displaystyle\mbox{}\quad+\sqrt{2}\rho
L_{F}\left[\mathbb{E}\int_{t_{i-1}}^{t}\|x(s)-x_{h}(\hat{s})\|^{2}ds\right]^{\frac{1}{2}}$

and so

$\displaystyle\mathbb{E}\left[\int_{t_{i-1}}^{t}\|u(s)-u_{h}(\hat{s})\|^{2}ds\right]^{\frac{1}{2}}\leq
A_{F}\mathbb{E}\left[\int_{t_{i-1}}^{t}\|x_{h}(\hat{s})-x(s)\|^{2}ds\right]^{\frac{1}{2}}+A_{F}h^{\frac{3}{2}},$

which implies

$\displaystyle\mathbb{E}\int_{t_{i-1}}^{t}\|u(s)-u_{h}(\hat{s})\|^{2}ds\leq
2A^{2}_{F}\mathbb{E}\int_{t_{i-1}}^{t}\|x_{h}(\hat{s})-x(s)\|^{2}ds+2A^{2}_{F}h^{3}.$

Then,

$\displaystyle\mathbb{E}\int_{0}^{t}\|u(s)-u_{h}(\hat{s})\|^{2}ds$
$\displaystyle=\mathbb{E}\sum_{k=0}^{i-1}\int_{t_{k}}^{t_{k+1}}\|u(s)-u_{h}(\hat{s})\|^{2}ds$
(4.21)
$\displaystyle\quad\;+\mathbb{E}\int_{t_{i-1}}^{t}\|u(s)-u_{h}(\hat{s})\|^{2}ds$
$\displaystyle\leq
2A^{2}_{F}\mathbb{E}\sum_{k=0}^{i-1}\int_{t_{k}}^{t_{k+1}}\|x_{h}(\hat{s})-x(s)\|^{2}ds$
$\displaystyle\quad\;+2A^{2}_{F}\mathbb{E}\int_{t_{i-1}}^{t}\|x_{h}(\hat{s})-x(s)\|^{2}ds+2iA^{2}_{F}h^{3}$
$\displaystyle\leq
2A^{2}_{F}\mathbb{E}\int_{0}^{t}\|x_{h}(\hat{s})-x(s)\|^{2}ds+2iA^{2}_{F}h^{3},$

where $A_{F}=\frac{\sqrt{2}\rho L_{F}}{1-\sqrt{(1-2\rho
C+\rho^{2}L^{2}_{F})}}$. It follows from (4.15) and (4.21) that

$\displaystyle\mbox{}\quad\mathbb{E}\left[\int_{0}^{t}\|u(s)-u_{h}(\hat{s})\|^{2}ds\right]$
(4.22) $\displaystyle\leq
2A^{2}_{F}\mathbb{E}\left[\int_{0}^{t}\|x_{h}(\hat{s})-x(s)\|^{2}ds\right]+2iA^{2}_{F}h^{3}$
$\displaystyle\leq
2A^{2}_{F}\mathbb{E}\left[\int_{0}^{t}\|x_{h}(\hat{s})-x_{h}(s)+x_{h}(s)-x(s)\|^{2}ds\right]+2iA^{2}_{F}h^{3}$
$\displaystyle\leq
4A^{2}_{F}TB_{1}h^{2}+4A^{2}_{F}TB_{2}h+4A^{2}_{F}\mathbb{E}\left[\int_{0}^{t}\|x_{h}(s)-x(s)\|^{2}ds\right]+2iA^{2}_{F}h^{3}$

Combining (4.19) and (4.22), one has

$\displaystyle\mathbb{E}\|x_{h}(t)-x(t)\|^{2}\leq
C^{\prime}_{1}h^{2}+C^{\prime}_{2}h+C^{\prime}_{3}\mathbb{E}\left[\int_{0}^{t}\|x(s)-x_{h}(s)\|^{2}ds\right],$
(4.23)

where $C^{\prime}_{1}=C_{1}+4C_{4}A^{2}_{F}TB_{1}+2A^{2}_{F}T$,
$C^{\prime}_{2}=C_{2}+4C_{4}A^{2}_{F}TB_{2}$, and
$C^{\prime}_{3}=C_{3}+4C_{4}A^{2}_{F}$. Applying Gronwall’s inequality in
(4.23), we get

$\displaystyle\mathbb{E}\|x_{h}(t)-x(t)\|^{2}\leq(C^{\prime}_{1}h^{2}+C^{\prime}_{2}h)+(C^{\prime}_{1}h^{2}+C^{\prime}_{2}h)(e^{C^{\prime}_{3}t}-1).$
(4.24)

Thus, it follows from (4.24) and (4.22) that

$\displaystyle\|x_{h}-x\|^{2}_{H_{[0,T]}}$
$\displaystyle\leq(C^{\prime}_{1}h^{2}+C^{\prime}_{2}h)T+(C^{\prime}_{1}h^{2}+C^{\prime}_{2}h)T(e^{C^{\prime}_{3}T}-1)$
$\displaystyle\leq(C^{\prime}_{1}T+C^{\prime}_{1}T(e^{C^{\prime}_{3}T}-1))h^{2}+(C^{\prime}_{2}T+C^{\prime}_{2}T(e^{C^{\prime}_{3}T}-1))h$

and

$\displaystyle\|u-u_{h}\|^{2}_{H_{[0,T]}}$ $\displaystyle=$
$\displaystyle\mathbb{E}\int_{0}^{T}\|u(s)-u_{h}(s)\|^{2}ds$ $\displaystyle=$
$\displaystyle\mathbb{E}\int_{0}^{T}\|u(s)-u_{h}(\hat{s})\|^{2}ds$
$\displaystyle\leq$
$\displaystyle(4A^{2}_{F}TB_{1}+2A^{2}_{F}T)h^{2}+4A^{2}_{F}TB_{2}h+4A^{2}_{F}\|x_{h}(s)-x(s)\|_{H_{[0,T]}}^{2}$
$\displaystyle\leq$
$\displaystyle(4A^{2}_{F}B_{1}+2A^{2}_{F}+4A^{2}_{F}C_{1}^{\prime})h^{2}T+(4A^{2}_{F}B_{2}+4A^{2}_{F}C_{2}^{\prime})Th$
$\displaystyle\;+4A^{2}_{F}(C^{\prime}_{1}h^{2}+C^{\prime}_{2}h)T(e^{C^{\prime}_{3}T}-1)$
$\displaystyle\leq$
$\displaystyle(4A^{2}_{F}TB_{1}+2A^{2}_{F}T+4A^{2}_{F}C^{\prime}_{1}T+4A^{2}_{F}C^{\prime}_{1}T(e^{C^{\prime}_{3}T}-1))h^{2}$
$\displaystyle\;+(4A^{2}_{F}TB_{2}+4A^{2}_{F}C^{\prime}_{2}T+4A^{2}_{F}C^{\prime}_{2}T(e^{C^{\prime}_{3}T}-1))h.$

Thus, we know that there exists a constant $C>0$ such that

$\displaystyle\|x_{h}-x\|^{2}_{H_{[0,T]}}\leq
C[h^{2}+h],\quad\|u-u_{h}\|_{H_{[0,T]}}^{2}\leq C[h^{2}+h].$

Clearly, $\|x_{h}-x\|_{H_{[0,T]}}\rightarrow 0$ and
$\|x_{h}-x\|_{H_{[0,T]}}\rightarrow 0$ as $h\rightarrow 0$. ∎

##  5 Two applications

In this section, we first propose Algorithm 5.1 for solving SDVI (1.1) and
then give two applications of Algorithm 5.1 to solve the electrical circuits
with diodes and the collapse of the bridge problems in stochastic environment.

From [19], by Monte Carlo method, we consider the following approximation of a
standard $l$-dimensional Brownian motion $B_{t}$:

$\displaystyle B_{t_{k+1}}-B_{t_{k}}=\sqrt{t_{k+1}-t_{k}}Z_{t_{k+1}},\quad
k=0,1,\cdots,N-1,$ (5.1)

where $\\{Z_{t_{k+1}}\\}_{k=0}^{N-1}$ are independent $l$-dimensional standard
normal random variables. It is well known that

$\displaystyle\mathbb{E}(B_{t_{k+1}}-B_{t_{k}})=\mathbb{E}(\sqrt{t_{k+1}-t_{k}}Z_{t_{k+1}})=0,$
$\displaystyle\mathbb{E}(B_{t_{k+1}}-B_{t_{k}})^{2}=\mathbb{E}(\sqrt{t_{k+1}-t_{k}}Z_{t_{k+1}})^{2}=t_{k+1}-t_{k}.$

Now we propose the following algorithm for solving SDVI (1.1).

######  Algorithm 5.1.

  * Step 0:

First divide the time $[0,T]$ $(h=\frac{T}{N})$ into $N$ intervals,

$\displaystyle t_{0}=0<t_{1}=h<\cdots<t_{N}=T.$

  * Step 1:

Let $x_{h}(t_{0})=x_{h}(0)$ and $u_{h}(t_{0})=u_{h}(0)\in K$.

  * Step 2:

For $k=0,1,2,\cdots,N-1$, let $Z(t_{k+1})$ be a standard normal random
variable obeying $N(0,t_{k+1})$ and

$\displaystyle
x_{h}(t_{k+1})=x_{h}(t_{k})+h\left[f(t_{k},x_{h}(t_{k}),u_{h}(t_{k}))\right]+\left[g(t_{k},x_{h}(t_{k}),u_{h}(t_{k}))\right]\sqrt{h}Z_{t_{k+1}}.$

Compute $u_{h}(t_{k+1})\in K$ satisfying the following stochastic variational
inequality problem:

$\displaystyle\left\langle
F(t_{k+1},\omega,x_{h}(t_{k+1}),u_{h}(t_{k+1})),v-u_{h}(t_{k+1})\right\rangle\geq
0,\quad\forall v\in K,\;a.s.\;\omega\in\Omega.$

  * Step 3:

By the recursion, for $k=0,1,2,\cdots,N-1$, we link the points of
$x_{h}(t_{k})$ when $t\in[t_{k},t_{k+1})$ by the following procedure

$\displaystyle x_{h}(t)=$ $\displaystyle
x_{h}(t_{k})+\left[f(t_{k},x_{h}(t_{k}),u_{h}(t_{k}))\right](t-t_{k})$
$\displaystyle+\left[g(t_{k},x_{h}(t_{k}),u_{h}(t_{k}))\right]\sqrt{t-t_{k}}Z_{t_{k+1}}$

and link the points of $u_{h}(t_{k})$ by the piecewise constant method.

Clearly, Theorem 4.1 ensures the convergence of Euler scheme constructed by
Algorithm 5.1.

###  5.1 The electrical circuits with diodes

In this subsection, we consider the electrical circuits with (ideal) diodes in
the stochastic environment. Following [34, 20, 14], it is well known that DVIs
can be used to model the electrical circuits with (ideal) diodes. Assume that
$x_{1}$ is the electric current across the inductor and $x_{2}$ is the
electric tension through the capacitor. Let $(v_{D_{i}},i_{D_{i}})$
(i=1,2,3,4) denote the voltage-current pairs associated to the $i$-th diode.
Setting $x(t)=(x_{1}(t),x_{2}(t))^{T}$ and
$u(t)=(u_{1}(t),u_{2}(t),u_{3}(t),u_{4}(t))^{T}=(i_{D_{1}},v_{D_{2}},v_{D_{3}},i_{D_{4}})^{T}$,
then the electrical circuits with (ideal) diodes can be described as the
following DVI (see [20, 14])

$\left\\{\begin{aligned} &dx(t)=[Ax(t)+Bu(t)+f(t)]dt,\quad x(0)=x_{0},\\\
&\left\langle Qx(t)+Mu(t),v-u(t)\right\rangle\geq 0,\quad\forall v\in
K,a.e.\;t\in[0,T],\end{aligned}\right.$ (5.2)

where

$\displaystyle A=\left(\begin{matrix}-\frac{2}{3}&0\\\
0&-\frac{1}{5}\end{matrix}\right),\quad
B=\left(\begin{matrix}0&\frac{1}{3}&-\frac{1}{3}&0\\\
1&0&0&1\end{matrix}\right),\quad f(t)=(2\sin(3t-\frac{\pi}{3}),0)^{T},$
$\displaystyle Q=\left(\begin{matrix}0&1\\\ 1&0\\\ -1&0\\\
0&1\end{matrix}\right),\quad M=\left(\begin{matrix}\epsilon&0&-1&0\\\
0&\epsilon&0&1\\\ 1&0&\epsilon&0\\\
0&-1&0&\epsilon\end{matrix}\right)(\epsilon>0),$ $\displaystyle
K=\left\\{y\in\mathbb{R}^{n}:-10\leq y_{1},y_{2}\leq 10,\;0\leq
y_{3},y_{4}\leq 20\right\\},\quad x_{0}=(-1,0)^{T},\quad T=1.5.$

Here the coefficient $\epsilon$ describes the level of the strong
monotonicity. By taking into account some stochastic environmental effects in
the electrical circuits with (ideal) diodes, DVI (5.2) should be changed into
the following SDVI:

$\left\\{\begin{aligned}
&dx(t)=[Ax(t)+Bu(t)+f(t)]dt+[Cx(t)+Du(t)+g(t)]dB_{t},\quad x(0)=x_{0},\\\
&\left\langle Qx(t)+Mu(t),v-u(t)\right\rangle\geq 0,\quad\forall v\in
K,\;a.e.\;t\in[0,T],\;a.s.\;\omega\in\Omega,\end{aligned}\right.$ (5.3)

where $C=\left(\begin{matrix}a&0\\\ 0&a\end{matrix}\right)$,
$D=\left(\begin{matrix}b&b&b&b\\\ b&b&b&b\end{matrix}\right)$,
$g(t)=(c\sin(t),0)^{T}$, and $B_{t}$ is a $1$-dimensional standard Brownian
motion. Here the constants $a,b,c\in\mathbb{R}$ are the diffusion coefficients
related to $x(t)$, $u(t)$, and time $t$, respectively.

It follows from Theorem 3.1 that (5.3) admits a unique Carathéodory solution
$(x(t),u(t))$ for fixed $a,b,c\in\mathbb{R}$ and $\epsilon>0$. Moreover, since
(5.3) satisfies Assumption 4.1, we can use Algorithm 5.1 to simulate paths for
(5.3). Especially, let $T=1.5$, $N=30(h=0.05)$, $x_{0.05}(0)=(-1,0)^{T}$ and
$u_{0.05}(0)=(0,0,0,-1)^{T}$ in Algorithm 5.1. By employing MATLAB R2018a, we
can obtain some numerical results as shown in Figures 5.1-5.6.

Figures 5.1-5.3 indicate that the electric current through the inductor, the
electric tension across the capacitor, the electric current through the first
diode, the voltage across the second diode, the voltage across the third
diode, and the electric current through the fourth diode change with time
respectively, for $\epsilon=0.001$ in different $a,b,c\in R$.

Figure 5.1: The electric current through the inductor and the electric tension
across the capacitor change with time in different $a,b,c\in R$.

Figure 5.2: The electric current across the first diode and the voltage
through the second diode change with time in different $a,b,c\in R$.

Figure 5.3: The voltage across the third diode and the electric current
through the fourth diode change with time in different $a,b,c\in R$.

Figures 5.1-5.3 show the following facts: (i) the curves of SDVI (5.3) and DVI
are coincident when $a=0$, $b=0$, $c=0$, which is drawn by the black line;
(ii) the diffusion coefficients have a great influence on the solution of SDVI
(5.3) as time going on. Moreover, Figure 5.1 tells us that the random effect
of $x$ has few effects in time $[0,1.2]$ and has great influence in time
$(1.2,1.5]$ on the voltage of the capacitor, and the random effects of $x$,
$u$ and time $t$ have few effects on the current of inductor and the voltage
of capacitor in time $[0,0.5]$; Figures 5.2 and 5.3 indicate that the random
effects of $u$ has the largest influence on $i_{D_{1}}$, $v_{D_{3}}$ and
$i_{D_{4}}$ and the random effects of $x$ has the largest effect on
$v_{D_{2}}$.

On the other hand, Figures 5.4-5.6 display that the electric current across
the inductor, the electric tension through the capacitor, the electric current
through the first diode, the electric tension across the second diode, the
electric tension across the third diode, and the electric current through the
fourth diode change with time respectively, for $a=b=c=1$ in different
$\epsilon\geq 0$. Moreover, Figures 5.4, 5.5(a), and 5.6(a) depict that
$\epsilon$ has little influence on the current and voltage across the
capacitor; Figures 5.5(b) and 5.6(b) show that $\epsilon$ has great influence
on the voltage. Clearly, matrix $M$ in SDVI (5.3) does not satisfy the strong
monotonicity condition in Assumption 4.1 when $\epsilon=0$. Nevertheless,
Figures 5.4 to 5.6 illustrate that the solution of SDVI (5.3) when
$\epsilon>0$ can approach the solution of SDVI (5.3) when $\epsilon=0$ by
taking $\epsilon$ small enough.

Figure 5.4: The electric current across the inductor and the electric tension
through the capacitor $C$ change with time in different $\epsilon$.

Figure 5.5: The current through the first diode and the voltage across the
second diode change with time in different $\epsilon$.

Figure 5.6: The voltage across the third diode and the current through the
fourth diode change with time in different $\epsilon$.

###  5.2 The collapse of the bridge

In this subsection, we consider an SDVI to model the collapse of the bridge
problem in stochastic environment. The following model was introduced in [13]
to describe the collapse of bridge:

$\displaystyle m\ddot{x}+q(x)=g(x),\quad t\in[0,T],$ (5.4)

where $m$ is the mass of the bridge floor, $g$ is the applied force, and

$q(x)=\left\\{\begin{aligned} &\alpha x,\quad\mbox{if}\quad x\geq 0,\\\ &\beta
x,\quad\mbox{if}\quad x<0\end{aligned}\right.$

is the restoring force for displacement in different directions. Here $\alpha$
and $\beta$ are Hooke’s constants for the tension and compression,
respectively. From [16, 4, 3], by taking into account the plasticity and white
noise in (5.4), one has

$\displaystyle m\ddot{x}+\tau\dot{x}+q(x)=g(x)+k\frac{dB_{t}}{dt},\quad
t\in[0,T],$ (5.5)

where $k\geq 0$ is the diffusion coefficient to describe the degree of the
white noise, $\tau\geq 0$ is the viscous damping coefficient to describe the
level of the plasticity of the bridge. If $\alpha\geq\beta$, then

$\displaystyle q(x)=\alpha x+u,$ (5.6)

where $u=\max\left\\{0,(\beta-\alpha)x\right\\}$. Thus,

$\displaystyle 0\leq u\bot u+(\alpha-\beta)x\geq 0.$ (5.7)

Let $y=(y_{1},y_{2})^{T}$ with $y_{1}=x$ and $y_{2}=\frac{dy_{1}}{dt}$. Take
$\theta\geq 0$, $\alpha=4$, $m=10$, $\beta=1$, and $T=1$. Then (5.5) can be
changed into the following SDVI:

$\left\\{\begin{aligned} &dy(t)=f(t,y(t),u(t))dt+g(t,y(t),u(t))dB_{t},\\\
&0\leq u\bot u+3y_{1}\geq 0,\\\ &y(0)=(0,\theta)^{T},\quad
t\in[0,1],\end{aligned}\right.$ (5.8)

where

$\begin{cases}f(t,y(t),u(t))=\left(y_{2}(t),-\frac{2}{5}y_{1}(t)-\tau
y_{2}(t)-\frac{1}{10}u(t)+sin(4t)\right)^{T},\\\
g(t,y(t),u(t))=(0,k)^{T},\quad\tau\geq 0,\quad k\geq 0,\quad\theta\geq
0.\end{cases}$

Here $\theta$ is initial value of $y_{2}$ which describes the initial velocity
of the oscillation of the bridge.

It follows from Theorem 3.1 that (5.8) admits a unique Carathéodory solution
$(x(t),u(t))$ for fixed $\tau\geq 0$, $k\geq 0$ and $\theta\geq 0$. Moreover,
since (5.8) satisfies Assumption 4.1, we can use Algorithm 5.1 to simulate
paths for (5.3). Particularly, take $T=1$, $N=50(h=0.02)$,
$x_{0.02}(0)=(0,\theta)^{T}$ and $u_{0.02}(0)=0$ in Algorithm 5.1. By using
MATLAB R2018a, we can obtain the numerical results as shown in Figures
5.7-5.12.

Figures 5.7 and 5.8 show the displacement and velocity of the bridge, and the
maximum between relative restoring force of the bridge and 0 change with time
respectively, for $\tau=1$ and $\theta=0$ in different $k\geq 0$. In fact,
Figures 5.7 and 5.8 tell us that (i) the curves of the SDVI (5.8) and a DVI
are coincident when $k=0$, which is drawn by the black line with hollow
circles; (ii) the effect of the white noise is very small when the diffusion
coefficient $k$ is small and the effect of the white noise can be large when
the diffusion coefficient $k$ is big enough, especially the blue line when
$k=10$; (iii) the little and appropriate white noise (the green line) can
reduce the oscillation of the bridge to some extent and large white noise can
impact greatly the oscillation of the bridge. However, it is unknown whether
large white noise reduce or increase the oscillation of the bridge in this
experiment.

Figure 5.7: The displacement and velocity of the bridge change with time in
different $k\geq 0$. Figure 5.8: The maximum between relative restoring force
of the bridge and 0 changes with time in different $k\geq 0$.

On the other hand, Figures 5.9 and 5.10 depict the displacement and velocity
of the bridge, and the maximum between relative restoring force of the bridge
and 0 change with time respectively, for $k=1$ and $\theta=0$ in different
$\tau$. Indeed, Figure 5.9 tells us that the $\tau$ bigger, the $y_{1}$ and
$y_{2}$ are smaller, which implies that the high plasticity of the bridge can
reduce the oscillation of the bridge, while Figure 5.10 displays that the
plasticity does not influence $u$.

Figure 5.9: The displacement and velocity of the bridge change with time in
different $\tau$. Figure 5.10: The maximum between relative restoring force of
the bridge and 0 changes with time in different $\tau$.

Moreover, Figures 5.11 and 5.12 show that the displacement and velocity of the
bridge, and the maximum between relative restoring force of the bridge and 0
change with time respectively, for $k=1$ and $\tau=1$ in different $\theta$.
In fact, Figure 5.11 indicates that the $\theta$ smaller, the $y_{1}$ and
$y_{2}$ are smaller, which implies that the small initial velocity can reduce
the oscillation of the bridge, while Figure 5.12 shows that the initial
velocity does not influence $u$.

Figure 5.11: The displacement and velocity of the bridge change with time in
different $\theta$. Figure 5.12: The maximum between relative restoring force
of the bridge and 0 changes with time in different $\theta$.

The numerical results obtained above illustrate that SDVI is useful and
promising for solving some real problems in the stochastic environment.

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