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\jyear{2021}%

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\begin{document}

	
	\title[ ]{An operational matrix method based on scaling function of Daubechies  wavelet to solve  stochastic differential equations }

	
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	\abstract{In our joint work, by introducing Quasi Block-Pulse (QBP) functions, we provide integration and stochastic integration operational matrices based on scaling function of Daubechies wavelet to solve linear and nonlinear stochastic differential equations. Due to interesting properties of the Daubechies wavelet such as orthogonality, compactly support and vanishing moments, these operational matrices have good properties. The convergence of the proposed method is presented and in order to verify the accuracy of the proposed method some numerical examples are provided.}

	
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	% \abstract{\textbf{Purpose:} The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. The abstract must not include subheadings (unless expressly permitted in the journal's Instructions to Authors), equations or citations. As a guide the abstract should not exceed 200 words. Most journals do not set a hard limit however authors are advised to check the author instructions for the journal they are submitting to.

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	\keywords{Stochastic differential equation, Scaling function, Daubechies wavelet, Quasi Block-Pulse functions, Integration operational matrix, Stochastic integration operational matrix.}






 
\maketitle
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\section{Introduction}
Many scientists and researchers in various fields such as 
finance and biology \cite{Braumann2019}, control \cite{Fuhrman20021397}, social science  \cite{Khodabin20121023}, boundary value problems \cite{Ismail2019,Khan201913,Xu1992123},  statistic analysis \cite{Liu20121136}, image processing \cite{Porwik200479}, and signal analysis \cite{Saritha200868}    
need to have enough knowledge about  stochastic calculus  in order to model and solve their problems.
Many real problems  in applied mathematics, physics and  engineering lead to stochastic models. So, being able to solve stochastic equations is a necessity. 

In order to model  stochastic  processes, it is unavoidable to deal with  stochastic differential or integral equations. It is not always possible to obtain the exact solution of these equations easily. So, in these cases we rely on the suitable numerical methods such as explicit and implicit  approximation schemes
\cite{Baldi2017,Mao2015370,Mao201314,Mikosch2000,Platen2010,Soheili201689}, Runge-Kutta method \cite{Robler2003,Robler20091713,Robler2010922,Solin2014,Tocino2002219}, Monte Carlo method \cite{Ermakov2019155}, weak Simpson method \cite{Adhikari2015}, and wavelets \cite{Le2014,Shiralashetti2021}. In recent years, some researchers used the operational matrices of wavelets to solve stochastic equations \cite{Heydari202081,Mirzaee201880,Mohammadi2015254,Mohammadi201613,Rostami201639,Shiralashetti2021395,Shiralashetti20192507}. %%%%%%%%%%%%%%%%%%%%
The Daubechies wavelet  has vanishing moments of order $N\in \mathbb{N}$ which becomes smoother when moment's degree increases. Ingrid Daubechies made it in $1988$ and used it in order to approximate functions.

In this paper, we are going to provide  operational matrix and stochastic operational matrix for scaling function of the Daubechies wavelet and use them to solve stochastic differential equations in the  form of
\begin{equation*}\label{equ18}
dX_t = a(t,X_t) dt + b(t,X_t) dW_t, \quad X_0 = x_0, \quad  t \in [0,T],
\end{equation*}
or equivalently  stochastic It\^o-Voltera integral equations in the following form 
\cite{Baldi2017,Gardiner2004,Kloeden1995,Kloeden2003,Platen2010}:
\begin{equation}\label{equ183}
X_t = X_0 + \int_{0}^{t} a(s,X_s) ds + \int_{0}^{t} b(s,X_s) dW_s, \quad 0 \le t \le T,
\end{equation}
such that  $W_t$ is the \textit{standard Brownian motion} and the \textit{random variable} $X=X(\omega)$ is a real-valued function defined on \textit{outcome space}  $\Omega$ \cite{Baldi2017,Durrett2005,Gardiner2004,Kloeden1995,Kloeden2003,Platen2010}.
The  first and the second integrals in  Eq. \eqref{equ183} are Riemann and  It\^o stochastic integrals, respectively. Numerical solution  for  Eq. \eqref{equ183} is a stochastic process $X^{(n)}=(X^{(n)}_t,t\in [0,T])$ which demonstrates an approximation for the exact solution $X=(X_t,t\in [0,T])$. In order to have such a numerical solution, we need a   discretization of the time interval 
\[ 0=t_0 <t_1<\dots<t_{n-1} <t_n=T,\]
where grid points are not equidistant necessarily.
This paper is planned  as follows: Section \ref{The preliminaries and basic definitions} provides some required information about stochastic calculus and wavelet theory that we need in the following. Then in Section  \ref{QB} quasi Block-Pulse functions are introduced and some of their important properties are declared. Next, in Section \ref{OPD}, integration and stochastic integration operational matrices of the scaling function for the Daubechies wavelet are provided. Section \ref{nm} explains the numerical method by using the introduced matrices in the previous section to solve differential or integral stochastic equations. In Section \ref{error}  the convergence of the numerical method is argued. Finally, several examples are presented in Section \ref{examples}. In conclusion, a  discussion is provided in Section \ref{Conclusion}.

\section{Preliminaries }\label{The preliminaries and basic definitions} 
\subsection{Stochastic calculus}\label{Sc}

\begin{definition}
	\cite{Kloeden1995} A stochastic process $W = (W_t, t\in [0,+\infty))$ is called \textit{standard Brownian motion } or a \textit{Wiener process} if  following conditions
	are satisfied:
	\begin{enumerate}[(1)]
		\item $W_0 = 0.$
		\item $W_t$ has stationary and independent increments.
		\item For all $t>0$, $W_t$ has $N(0,t)$ which $N$ means the normal distribution. 
		\item $W_t$ has continuous sample paths.
	\end{enumerate}
\end{definition}


\begin{definition}
	\cite{Kloeden1995} The \textit{It\^o stochastic integral for a simple process C} on $[0,T]$ is defined in the following form:
	\begin{equation*}\label{Itoint}
	\int_{0}^{T} C_s dW_s := \sum_{i=1}^{n}	 C_{t_{i-1}}(W_{t_i}-W_{t_{i-1}}). 
	\end{equation*}	
\end{definition}

The  It\^o stochastic integral has the \textit{isometry property}:
\begin{equation}\label{Iso}
\mathbb{E}  \bigg( \int_{0}^{t} C_s dW_s\bigg)^2 = \int_{0}^{t} \mathbb{E}	{ C_s}^2 ds, \quad t \in [0,T].
\end{equation}

Assume $C$ is adapted to Brownian motion on $[0,T]$, in the other words, $C_t$ is dependent on $W_s, s\le t$. Also assume  
$\int_{0}^{T} \mathbb{E} {C_s}^2 ds$ is finite, there exists a sequence $\Big(C^{(n)}\Big)$ of simple processes such that

\[\int_{0}^{T} \mathbb{E} [C_s - {C_s}^{(n)}]^2 ds \to 0.\]



\begin{theorem}\label{th1}
	Let   $\mathbb{E}({X_0}^2)<\infty$ and $X_0$ is independent of the standard Brownian motion. If for all $t \in [0,T]$ and  $x,y \in \mathbb{R}$, the functions $a(t,x)$ and $b(t,x)$ are continuous and \textit{Lipschitz condition}  with respect to the second variable is hold for them, then Eq. \eqref{equ18} has a unique strong solution on $[0,T]$.  
	\begin{proof}
		The proof is available in  \cite{Oksendahl2003}.
	\end{proof}
\end{theorem}



\subsection{Wavelet theory}\label{Wt}

\begin{definition}
	\cite{Kasnazani2020,Pan2003,Pendar2015} A multiresolution analysis of $L^2(\mathbb{R})$ is a sequence of subspaces $\big\{V_j\big\}_{j\in \mathbb{Z}}$ of
	$L^2(\mathbb{R})$ that the following properties are hold:
	\begin{enumerate}
		\item $\dots \subset V_{-1} \subset V_0 \subset V_1 \subset \dots \subset L^2 (\mathbb{R})$.
		\item $f(t)\in V_0 \Leftrightarrow f(2^j t) \in V_{j}$.
		\item $f(2^j t) \in V_j \Leftrightarrow f(2^j t+1) \in V_j$.
		\item $\bigcap_j V_j = \{0\}, (\overline{\bigcup_j V_j}) = L^2(\mathbb{R})$. 
		\item There exists a function $\phi(t) \in V_0$ 
		with nonzero integral which  $\{\phi(t-n)\}$ is an orthonormal base for $V_0$. 
		The function $\phi$ is called  scaling function.
	\end{enumerate}
\end{definition} 


\begin{definition}
	\cite{Walnut2002} The function $f(t)$ has vanishing moments of order   $N$ 	if
	\[
	\int_{\mathbb{R}}^{ }  t^ m f(t) dt = 0, \quad 0\le m \le N-1,
	\]
	but
	\[
	\int_{\mathbb{R}}^{ }  t^ {N} f(t) dt \neq 0.
	\]
\end{definition}



Ingrid Daubechies made a basis from smooth wavelet  with vanishing moments in its support \cite{Daubechies1988909}. Also, if Daubechies wavelet has   vanishing moments of order $N\in \mathbb{N}$, the support of scaling function is $[0,2N-1]$. So, dilation relation for the scaling function $\phi(t)$ can be have the following form
\begin{equation}	\label{dil1}
\phi(t) = \sqrt{2}\sum_{k=0}^{D-1} h_k \phi (2t-k),
\end{equation} 
where $D=2N$ and $\{h_k\}_{k\in\mathbb{Z}}$ are named filter coefficients. Their values can be obtained by using    some useful relations which are related to properties of the Daubechies wavelet. Of course, Daubechies calculated and set them in tables for various values of $N$. They are available in some books \cite{Daubechies1992,Pan2003}.

From definition of the multiresolution analysis, it is clear that 
\[
\{\phi_{j,k}(t):=2^{j/2}\phi(2^j t - k)\vert j,k\in\mathbb{Z}\},
\]
is an orthonormal base for $V_j$. So we can write an orthogonal projection of function $f(t)$ belongs $L^2(\mathbb{R})$ in subspace $V_j$ by using scaling function \cite{Pendar2015,Sweldens19941240}:
\begin{equation}\label{appp}
\mathcal{P}_j f(t) = \sum_{k=-\infty}^{+\infty} c_{j,k} \phi_{j,k} (t),
\end{equation}

where
\[
c_{j,k} = \int_{-\infty}^{+\infty} f(t) \phi_{j,k} (t) dt.
\]

Assume $f\in V_j$ and $t \in [a,b]$ such that $a,b\in \mathbb{Z}$, due to compactly support of the scaling function of the Daubechies wavelet, \eqref{appp} can be written in the following form
\begin{equation*}
\mathcal{P}_j f(t) = \sum_{k=2^j a +1-L}^{2^j b -1} c_{j,k} \phi_{j,k} (t).
\end{equation*}

From the forth property of multiresolution analysis, it can be found
\[
\lim_{j\to \infty} \Vert f - \mathcal{P}_j f \Vert =0.
\]


\begin{theorem}\label{thD}
	If we use the Daubechies wavelet with vanishing moments of order $N$ to approximate the smooth function $f\in \mathcal{C}^N$ in space $V_j$, then
	\[	
	\exists C>0, \quad \Vert f - \mathcal{P}_j  f\Vert_{L^2} \le C 2^{-jN}.
	\]
	\begin{proof}
		The proof can be found in \cite{Cohen2003}.	
	\end{proof}
\end{theorem}


\subsubsection{Scaling function in integer and dyadic points}\label{SfIDP}
Dyadic  points  are defined in the following form \cite{Walnut2002}
\[
t = \frac{k}{2^j}, \quad j,k \in \mathbb{Z}.
\]	
For  $j\ge 0 $ we have the dyadic points in $[0,D-1]$  as follows
\[
t = \frac{n}{2^j}, \quad n= 0,1,2,\cdots,2^j(D-1).
\]
As we can see, when $j=0$ the dyadic points are the same integer points. 

Since the scaling function of the Daubechies wavelet does not have an explicit form, first we can obtain the values of the scaling function in  integer points and then in dyadic points.   Substituting integer points of support interval in relation \eqref{dil1} 
results in a homogenous system
\[
(\mathrm{A-I})\Phi = 0,
\]
such that
\begin{equation*}
\mathrm{A}_{D\times D}=\sqrt{2}
\begin{pmatrix}
h_0 & 0 & 0 & \cdots & 0 & 0 & 0 \\
h_2 & h_1 & h_0 & \cdots & 0 & 0 & 0 \\
h_4 & h_3 & h_2 & \cdots & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & h_{D-3} & h_{D-4} & h_{D-5} \\
0 & 0 & 0 & \cdots & h_{D-1} & h_{D-2} & h_{D-3} \\
0 & 0 & 0 & \cdots & 0 & 0 & h_{D-1}
\end{pmatrix},
\end{equation*}
and
\begin{equation*}
\Phi_{D\times 1}=
\begin{pmatrix}
\phi(0) \\
\phi(1) \\
\phi(2) \\
\vdots \\
\phi(D-3) \\
\phi(D-2) \\
\phi(D-1) \\
\end{pmatrix},
\end{equation*}
and $I$ is the identity matrix by the same order of matrix $A$.



In order to have a unique solution, we need  a normalization condition which is
\cite{Pan2003}

\[
\sum_{k=0}^{D-1} \phi(x-k) = 1.
\]

Now, by solving the linear equations system, we can obtain values of scaling function in integer points. Then, we can determine values of scaling function in dyadic points via  a recursive process named cascade algorithm which is as follows
\[
\phi\Big(\frac{n}{2^j}\Big) = \sqrt{2} \sum_{k=0}^{D-1} h_k \phi\Big(\frac{n}{2^{j-1}}-k\Big), \quad n = 1,3,5,\cdots, 2^j(D-1)-1.
\]

\section{Quasi Block-Pulse Functions}\label{QB}
\begin{definition}
	For
	$m\in \mathbb{N}$ we define
	an  $m$ element partition of quasi Block-Pulse functions on 
	$[0,T)$:
	\begin{equation}\label{b1} 
	b_i(t) = \begin{array}{l}
	\left\{ \begin{array}{l}
	1, \quad t_i \le t < t_{i+1}, \\
	0, \quad \text{otherwise}, 
	\end{array} \right.
	\end{array}
	\end{equation}
	where $i=1,\dots,m$, and
	\begin{align*}
	0=t_1<t_2<\dots<t_m<t_{m+1}=T,
	\end{align*}
	these grid points are not equidistant necessarily.
\end{definition}

The functions in 
\eqref{b1}
have disjointness and orthogonality properties, in the other words, for 
$i,j=1,\dots,m$,
\begin{align*} 
b_i(t)b_j(t) = 
\begin{cases}
b_i(t),  &i = j,\\
0,  &i\neq j, 
\end{cases} 
\end{align*}
and
\begin{align*} 
\int_{0}^{T}	b_i(t)b_j(t)dt = \begin{cases}
\Delta_i t,  &i = j,\\
0,  &i\neq j, 
\end{cases}
\end{align*}
where  	
$\Delta_i t = t_{i+1}-t_i$.

A real and bounded function 
$f(t) \in L^2\big( [0,T)\big)$ 
can be expanded by quasi Block-Pulse functions
\begin{equation*}
f(t) = \sum_{i=1}^{\infty} f_i b_i(t),
\end{equation*}
where
\begin{equation*}
f_i = \frac{1}{\Delta_i t} \int_{0}^{T} f(t) b_i(t) dt.
\end{equation*}

For any function 
$f \in L^2 \big([0,T)\big)$,
we have
\begin{equation*}
\int_{0}^{T}f^2(t)dt = \sum_{i=1}^{\infty} {f_i}^2 \Vert b_i(t)\Vert ^2.
\end{equation*}


Let vector of the quasi Block-Pulse functions in the following form
\begin{equation*}
\textbf{B}_m(t)  = \begin{bmatrix}
b_1(t) ,
b_2(t),\dots,b_m(t)
\end{bmatrix}^T,
\end{equation*}
then
\begin{equation*}
\textbf{B}_m(t) \Big(\textbf{B}_m(t)\Big) ^ T = \begin{bmatrix}
b_1(t)  & & &\textbf{0}\\
& b_2(t) &&\\
&&\ddots&\\\textbf{0}&&&b_m(t)
\end{bmatrix},
\end{equation*}
and
\begin{equation*}
\Big(\textbf{B}_m(t)\Big)^T \textbf{B}_m(t)  =1, \quad \forall t\in[0,T).
\end{equation*}

Also for a given vector
$V = [v_1,v_2,\dots,v_m]^T$, 
we can write
\begin{equation*}
\textbf{B}_m(t) \Big(\textbf{B}_m(t)\Big) ^ T V = \begin{bmatrix}
v_1&&\textbf{0} \\ & \ddots \\ \textbf{0}& & v_m
\end{bmatrix}\begin{bmatrix}
b_1(t) \\ \vdots \\ b_m(t)
\end{bmatrix}=  \text{diag}(V) \textbf{B}_m(t).
\end{equation*}

In order to approximate functions, we  use truncated series of quasi Block-Pulse functions with
$m$ 
terms:
\begin{equation*}
f(t)   \approx \sum_{i=1}^{m} f_i b_i(t) = F^T \textbf{B}_m(t) = \textbf{B}_m(t)^T F,
\end{equation*}
where
$F = [f_1,f_2.\dots,f_m]^T$.

\subsection{Integration Operational Matrix for the quasi Block-Pulse Functions}
Integrating on
$[0,t]$ from functions in
\eqref{b1}
results in
\begin{align*} 
\int_{0}^{t}	b_i(t) dt = \begin{cases}
0,  &t<t_i,\\
\int_{0}^{t_i} b_i(t)dt + \int_{t_i}^{t} b_i(t)dt, &t_i\le t<t_{i+1}, \\
\int_{0}^{t_i} b_i(t)dt + \int_{t_i}^{t_{i+1}} b_i(t)dt + \int_{t_{i+1}}^{t} b_i(t)dt,  &t_{i+1} \le t,
\end{cases}
\end{align*}
according the definition of the quasi Block-Pulse functions we have
\begin{align*} 
\int_{0}^{t}	b_i(t) dt = \begin{cases}
0,  &t<t_i,\\
t-t_i,  &t_i\le t<t_{i+1}, \\
\Delta_i t,   &t_{i+1} \le t.
\end{cases}
\end{align*}
Since
$t-t_i$ 
in the middle point of the 
$[t_i,t_{i+1})$
is
$\dfrac{\Delta_i t}{2}$,
we approximate the latter integral on
$[t_i,t_{i+1})$
with
$\dfrac{\Delta_i t}{2}$, therefore

\begin{equation*}
\int_{0}^{t} b_i(t) dt \approx  \begin{bmatrix} 0,\dots,0,\frac{\Delta_i t}{2},\Delta_{i}t,\dots,\Delta_i t \end{bmatrix}\begin{bmatrix}
b_1(t)\\ \vdots \\ b_{i-1}(t)\\b_i(t)\\b_{i+1}(t) \\ \vdots \\ b_m(t)
\end{bmatrix}.
\end{equation*}

Hence
\begin{equation}\label{opi}
\int_{0}^{t}B_m(s) ds \approx \textbf{P}_m B_m(t),
\end{equation}
where
\begin{equation*}
\textbf{P}_m = \begin{bmatrix}
\frac{\Delta_1 t}{2}& \Delta_1 t& \Delta_1 t&\dots&\Delta_1 t\\
&\frac{\Delta_2 t}{2}& \Delta_2 t&\dots&\Delta_2 t\\
& &\ddots & &
\\
& &  &\ddots & \\
\textbf{0}& & & &\frac{\Delta_m t}{2}
\end{bmatrix}.
\end{equation*}

\subsection{Stochastic Integration Operational Matrix  for the quasi Block-Pulse Functions}

By stochastic integrating on 
$[0,t]$ from functions in
\eqref{b1} we have
\begin{align*} 
\int_{0}^{t}	b_i(t) dW_t = \begin{cases}
0,  &t<t_i,\\
\int_{0}^{t_i} b_i(t)dW_t + \int_{t_i}^{t} b_i(t)dW_t, &t_i\le t<t_{i+1}, \\
\int_{0}^{t_i} b_i(t)dW_t + \int_{t_i}^{t_{i+1}} b_i(t)dW_t + \int_{t_{i+1}}^{t} b_i(t)dW_t,  &t_{i+1} \le t,
\end{cases}
\end{align*}
definition of the quasi Block-Pulse functions leads to
\begin{align*} 
\int_{0}^{t}	b_i(t) dW_t = \begin{cases}
0,  &t<t_i,\\
W(t)-W(t_i), &t_i\le t<t_{i+1}, \\
W(t_{i+1})-W(t_i),  &t_{i+1} \le t.
\end{cases}
\end{align*}
Since 
$W(t)-W(t_i)$ 
in the middle point of 
$[t_i,t_{i+1})$
is 
$W\Big(t_i + \dfrac{\Delta_i t}{2}\Big)-W(t_i) $,
we approximate the latter integral on
$[t_i,t_{i+1})$
with
$W\Big(t_i + \dfrac{\Delta_i t}{2}\Big)-W(t_i) $,
which equals to
$W\Big(\dfrac{t_i+t_{i+1}}{2}\Big)$. Therefore
\begin{equation}\label{ops}
\int_{0}^{t} \textbf{B}_m(s) dW_s \approx \textbf{R}_s  \textbf{B}_m(t),
\end{equation}
where
\begin{equation*}
\textbf{R}_s = \begin{bmatrix}
W\Big(\dfrac{t_1+t_2}{2}\Big)-W(t_1)& W(t_2)-W(t_1)& W(t_2)-W(t_1)&\dots&W(t_2)-W(t_1)\\
&W\Big(\dfrac{t_2+t_3}{2}\Big)-W(t_2) & W(t_3)-W(t_2) &\dots& W(t_3)-W(t_2)\\
& & \ddots &\vdots & \vdots\\
& & &\ddots &\vdots
\\
\textbf{0}& & & & W\Big(\dfrac{t_m+t_{m+1}}{2}\Big)-W(t_m)
\end{bmatrix}.
\end{equation*}


\section{Operational Matrix of the Scaling function of the Daubechies Wavelet}\label{OPD}

Consider vector of the scaling functions for the Duabechies wavelet
\begin{equation*}
\mathbf{\phi}_m(t)=  \begin{bmatrix}
\mathbf{\phi}_{j,k_1}(t),\mathbf{\phi}_{j,k_1+1}(t), \dots , \mathbf{\phi}_{j,k_2}(t)
\end{bmatrix}^T.
\end{equation*}
Let grid points in the following form
\begin{equation}\label{gp}
\textbf{T}=\bigg\{ t_i \vert i=1,\dots,m  \bigg\},
\end{equation}
we introduce matrix of the scaling functions of the Daubechies wavelet in the following form 
\begin{equation*}
\mathbf{\Phi}_m =[ \mathbf{\phi}_m(t_1), \mathbf{\phi}_m(t_2),\dots, \mathbf{\phi}_m(t_m)],
\end{equation*}
such that $\mathbf{\Phi}_m$ is a sparse matrix due to compactly support of the scaling function of the Daubechies wavelet. For grid points we have
\begin{equation}\label{rel}
\mathbf{\phi}_m(t) = \mathbf{\Phi}_m \textbf{B}_m(t),
\end{equation}
and hence
\begin{equation}\label{b}
\textbf{B}_m(t) =\Big(\mathbf{\Phi}_m\Big)^{-1}  \mathbf{\phi}_m(t).  
\end{equation}

\subsection{Integration Operational Matrix for the Scaling function of the Duabechies Wavelet}
Consider the integration operational matrix for the scaling functions of the Daubechies wavelet in the following form
\begin{equation}\label{opi2}
\int_{0}^{t} \mathbf{\phi}_m (s) ds \approx  \textbf{P}_D\mathbf{\phi}_m (t).
\end{equation}
In the grid points, from \eqref{rel} we have
\begin{equation*}
\int_{0}^{t} \mathbf{\phi}_m (s) ds =\int_{0}^{t} \mathbf{\Phi}_m \textbf{B}_m(s) ds = \mathbf{\Phi}_m  \int_{0}^{t}  \textbf{B}_m(s) ds.
\end{equation*}
From \eqref{opi} we can write
\begin{equation*}
\int_{0}^{t} \mathbf{\phi}_m (s) ds \approx 
\mathbf{\Phi}_m  \textbf{P}_m \textbf{B}_m(t),
\end{equation*}
and from \eqref{b} we have
\begin{equation*}
\int_{0}^{t} \mathbf{\phi}_m (s) ds \approx 
\mathbf{\Phi}_m  \textbf{P}_m \bigg( \mathbf{\Phi}_m  \bigg)^{-1} \mathbf{\phi}_m (t).
\end{equation*}
So we can conclude
\begin{equation*}
\textbf{P}_D=\mathbf{\Phi}_m  \textbf{P}_m \bigg( \mathbf{\Phi}_m  \bigg)^{-1}.
\end{equation*}

\subsection{Stochastic Integration Operational Matrix for the Scaling Function of the Daubechies Wavelet}
Assume stochastic integration operational matrix for the scaling functions of the Daubechies wavelet has the following form
\begin{equation}\label{ops2}
\int_{0}^{t} \mathbf{\phi}_m (s) dW_s \approx  \textbf{M}_s\mathbf{\phi}_m (t).
\end{equation}
In the grid points, from \eqref{rel} we can write
\begin{equation*}
\int_{0}^{t} \mathbf{\phi}_m (s) dW_s =\int_{0}^{t} \mathbf{\Phi}_m \textbf{B}_m(s) dW_s = \mathbf{\Phi}_m  \int_{0}^{t}  \textbf{B}_m(s) dW_s.
\end{equation*}
From \eqref{ops} we can get
\begin{equation*}
\int_{0}^{t} \mathbf{\phi}_m (s) dW_s \approx 
\mathbf{\Phi}_m  \textbf{R}_s \textbf{B}_m(t),
\end{equation*}
and  \eqref{b} results in
\begin{equation*}
\int_{0}^{t} \mathbf{\phi}_m (s) dW_s \approx 
\mathbf{\Phi}_m  \textbf{R}_s \bigg( \mathbf{\Phi}_m  \bigg)^{-1} \mathbf{\phi}_m (t).
\end{equation*}
So we have
\begin{equation*}
\textbf{M}_s=\mathbf{\Phi}_m  \textbf{R}_s \bigg( \mathbf{\Phi}_m  \bigg)^{-1}.
\end{equation*}

\section{Numerical Method}\label{nm}
If we let
$T=1$, from Eq. \eqref{equ183} we have 
\begin{equation}\label{e1}
X_t = X_0 + \int_{0}^{t} a(s,X_s) ds + \int_{0}^{t}  b(s,X_s) dW_s, \quad 0\le t < 1.
\end{equation}
We let the dyadic points in $[0,1)$ as the grid points in \eqref{gp}. 

\subsection{The Linear case}

\begin{theorem}\label{t3}
	Assume 
	$\textbf{X}_m$
	is a  vector with
	$m$ elements, then
	\begin{equation*}
	\mathbf{\phi}_m (t) \Big(   \mathbf{\phi}_m (t) \Big)^T \textbf{X}_m=
	\tilde{\textbf{Q}}_m \mathbf{\phi}_m (t),
	\end{equation*}
	such that
	\begin{equation*}
	\tilde{\textbf{Q}}_m = \text{diag} \Big( \big( \mathbf{\Phi}_m\big)^T \textbf{X}_m\Big).
	\end{equation*}
	
	\begin{proof}
		From \eqref{rel} we have
		\begin{equation}\label{p11}
		\mathbf{\phi}_m (t) \Big(   \mathbf{\phi}_m (t) \Big)^T \textbf{X}_m=
		\mathbf{\Phi}_m \textbf{B}_m(t)\Big( \textbf{B}_m(t)\Big)^T \textbf{Q}_m,
		\end{equation}
		where
		\begin{equation*}
		\textbf{Q}_m = \big( \mathbf{\Phi}_m \big)^T \textbf{X}_m.
		\end{equation*}
		Let 
		$\textbf{Q}_m = \begin{bmatrix}
		q_1,q_2,\dots,q_m
		\end{bmatrix}^T$, therefore we have
		\begin{align*}
		\textbf{B}_m(t)\Big( \textbf{B}_m(t)\Big)^T \textbf{Q}_m &= \begin{bmatrix}
		b_1(t)\\b_2(t)\\ \vdots \\ b_m(t)
		\end{bmatrix}\begin{bmatrix}
		b_1(t),b_2(t), \dots , b_m(t)
		\end{bmatrix}\begin{bmatrix}
		q_1\\q_2\\ \vdots \\ q_m
		\end{bmatrix}\\
		& =\begin{bmatrix}
		\Big(b_1(t)\Big)^2& b_1(t)b_2(t) & \dots& b_1(t)b_m(t)\\
		b_2(t)b_1(t)& \Big(b_2(t)\Big)^2 & \dots & b_2(t)b_m(t)\\
		\vdots &\vdots& \ddots& \vdots \\ b_m(t)b_1(t)& b_m(t)b_2(t)& \dots& \Big(b_m(t)\Big)^2
		\end{bmatrix}\begin{bmatrix}
		q_1\\q_2\\ \vdots \\ q_m
		\end{bmatrix}\\
		& = \begin{bmatrix}
		b_1(t)&0&\dots&0\\
		0&b_2(t)&\dots&0\\
		\vdots& \vdots&\ddots&\vdots \\
		0&0&\dots& b_m(t)
		\end{bmatrix}\begin{bmatrix}
		q_1\\q_2\\ \vdots \\ q_m
		\end{bmatrix}\\
		& = \begin{bmatrix}
		q_1&0&\dots&0\\
		0&q_2&\dots&0\\
		\vdots& \vdots&\ddots&\vdots \\
		0&0&\dots& q_m
		\end{bmatrix}\begin{bmatrix}
		b_1(t)\\b_2(t)\\ \vdots \\ b_m(t)
		\end{bmatrix}.
		\end{align*}
		So, we can write
		\begin{equation*}
		\textbf{B}_m(t)\Big( \textbf{B}_m(t)\Big)^T \textbf{Q}_m = \tilde{\textbf{Q}}_m \textbf{B}_m(t),
		\end{equation*}
		which
		$\tilde{\textbf{Q}}_m=\text{diag}\Big( \textbf{Q}_m \Big)$. 
		Now according to \eqref{p11} we have 
		\begin{equation*}
		\mathbf{\phi}_m (t) \Big(   \mathbf{\phi}_m (t) \Big)^T \textbf{X}_m=
		\mathbf{\Phi}_m\tilde{\textbf{Q}}_m \mathbf{B}_m (t).
		\end{equation*}
		Due to definition of quasi Block-Pulse functions and since $\tilde{\textbf{Q}}_m $ is a  diagonal matrix, in grid points we can write				\begin{equation*}
		\mathbf{\phi}_m (t) \Big(   \mathbf{\phi}_m (t) \Big)^T \textbf{X}_m=
		\tilde{\textbf{Q}}_m  \mathbf{\Phi}_m  \mathbf{B}_m (t),
		\end{equation*}
		and  \eqref{rel} leads to
		\begin{equation*}
		\mathbf{\phi}_m (t) \Big(   \mathbf{\phi}_m (t) \Big)^T \textbf{X}_m=
		\tilde{\textbf{Q}}_m  \mathbf{\phi}_m   (t).
		\end{equation*}
		It completes the proof.
	\end{proof}
\end{theorem}


In the linear case we consider
$a(s,X_s)$
and
$b(s,X_s)$
in the form of
\begin{equation}\label{abl}
a(s,X_s)=f_1(s)X_s,\\ \quad
b(s,X_s)=f_2(s)X_s,
\end{equation}
such that  the functions
$f_1$ and $f_2$
are real-valued functions. Consider approximation of functions 
$f_1$ and $f_2$ by using the scaling function of the Daubechies wavelet in the following form
\begin{equation}\label{e2}
f_1(s) = \textbf{D}_1^T \mathbf{\phi}_m   (s), \quad f_2(s) = \textbf{D}_2^T \mathbf{\phi}_m  (s),
\end{equation}
where 
$\textbf{D}_1$
and
$\textbf{D}_2$
are column vectors with  
$m$ 
elements. In continue assume
\begin{equation}\label{e3}
X_s = \Big( \mathbf{\phi}_m   (s)\Big)^T \textbf{X}_m = {\textbf{X}_m}^T \mathbf{\phi}_m   (s),
\end{equation}
which
$\textbf{X}_m$
is the unknowns column vector. By substituting \eqref{e2} and \eqref{e3} in Eq. \eqref{e1} we get
\begin{equation*}
{\textbf{X}_m}^T \mathbf{\phi}_m   (t) = X_0 + {\textbf{D}_1}^T \int_{0}^{t}  \mathbf{\phi}_m   (s) \Big( \mathbf{\phi}_m   (s)\Big)^T \textbf{X}_m ds +  {\textbf{D}_2}^T\int_{0}^{t} \mathbf{\phi}_m   (s) \Big( \mathbf{\phi}_m   (s)\Big)^T \textbf{X}_m  dW_s.
\end{equation*}
From Theorem  \ref{t3} we have
\begin{equation*}
{\textbf{X}_m}^T \mathbf{\phi}_m   (t) = X_0 + {\textbf{D}_1}^T \tilde{\textbf{Q}}_m \int_{0}^{t}  \mathbf{\phi}_m   (s)  ds +  {\textbf{D}_2}^T  \tilde{\textbf{Q}}_m  \int_{0}^{t} \mathbf{\phi}_m   (s) dW_s,
\end{equation*}
where
\begin{equation*}
\tilde{\textbf{Q}}_m =\text{diag} \bigg( \Big( \mathbf{\Phi}_m \Big) ^T  \textbf{X}_m \bigg).
\end{equation*}
Now from \eqref{rel}, \eqref{opi2} and \eqref{ops2}, in grid points we have
\begin{equation*}
{\textbf{X}_m}^T \mathbf{\Phi}_m   \textbf{B}_m(t) = [X_0,X_0,\dots,X_0] \textbf{B}_m(t) + {\textbf{D}_1}^T \tilde{\textbf{Q}}_m 
\textbf{P}_D
\mathbf{\Phi}_m 
\textbf{B}_m(t)+  {\textbf{D}_2}^T \tilde{\textbf{Q}}_m  \textbf{M}_s \mathbf{\Phi}_m \textbf{B}_m(t).
\end{equation*}
Finally, we can have a linear system of equations 
\begin{equation}\label{l}
{\textbf{X}_m}^T \mathbf{\Phi}_m   = [X_0,X_0,\dots,X_0]  + {\textbf{D}_1}^T \tilde{\textbf{Q}}_m 
\textbf{P}_D
\mathbf{\Phi}_m
+  {\textbf{D}_2}^T \tilde{\textbf{Q}}_m  \textbf{M}_s \mathbf{\Phi}_m. 
\end{equation}
As it is seen \eqref{l} is a  system of linear algebraic equations included $m$ equations and $m$ unknowns. By solving this system  $\textbf{X}_m$ is determined and then by substituting in \eqref{e3} the solution of Eq. \eqref{e1} with coefficient functions in \eqref{abl} is obtained.
\subsection{The nonlinear case}

\begin{theorem}\label{t4}
	Assume that
	$V$
	is a row vector with 
	$m$ 
	elements in the form of
	$V=[v_1,v_2,\dots,v_m]$, then
	\begin{equation*}
	\mathbf{B}_m (t) V  \mathbf{B}_m (t)=
	\text{diag} (V) \mathbf{B}_m (t).
	\end{equation*}
	
	\begin{proof}
		We have
		\begin{align*}
		\mathbf{B}_m (t) V \mathbf{B}_m (t)=
		\mathbf{B}_m (t) [v_1,v_2,\dots,v_m] \mathbf{B}_m (t) =&
		\begin{bmatrix}
		b_1(t)v_1& b_1(t)v_2&\dots& b_1(t)v_m \\b_2(t)v_1& b_2(t)v_2&\dots& b_2(t)v_m\\ \vdots& \vdots& \vdots& \vdots\\ b_m(t)v_1& b_m(t)v_2&\dots& b_m(t)v_m 
		\end{bmatrix}
		\begin{bmatrix}
		b_1(t)\\b_2(t)\\ \vdots \\ b_m(t)
		\end{bmatrix}\\
		=&\begin{bmatrix}
		v_1 b_1(t)\\v_2 b_2(t)\\ \vdots \\ v_m b_m(t)
		\end{bmatrix}\\
		=&\begin{bmatrix}
		v_1 & & &\textbf{0}\\&v_2&& \\ &&\ddots& \\ \textbf{0}&&&v_m 
		\end{bmatrix}\begin{bmatrix}
		b_1(t)\\ b_2(t)\\ \vdots \\  b_m(t)
		\end{bmatrix},
		\end{align*}
		and proof is completed.
	\end{proof}
\end{theorem}

In the nonlinear case we consider the functions
$a(s,X_s)$
and
$b(s,X_s)$
which have the following form 
\begin{equation}\label{abnl}
a(s,X_s)=f_1(s){X_s}^\alpha,\\ \quad
b(s,X_s)=f_2(s){X_s}^\beta,
\end{equation}
such that
$\alpha,\beta\ge 2$
and belong $\mathbb{N}$. The functions
$f_1$ and $f_2$ 
are real-valued functions. Again we
consider approximation of functions 
$f_1$ and $f_2$ by using the scaling function of Daubechies wavelet in the form of \eqref{e2}. Again let  
\begin{equation}\label{x2}
X_s = {\textbf{X}_m}^T \mathbf{\phi}_m   (s),
\end{equation}
from \eqref{rel} we can have the following result 
\begin{equation}\label{x}
X_s = {\textbf{X}_m}^T \mathbf{\Phi}_m   \textbf{B}_m(s).
\end{equation}
It is clear that
${\textbf{X}_m}^T \mathbf{\Phi}_m$
is a row vector with $m$ elements. Let
\begin{equation*}
{\textbf{X}_m}^T \mathbf{\Phi}_m =  [d_1,d_2,\dots,d_m],
\end{equation*}
therefore
\begin{equation*}
X_s =  [d_1,d_2,\dots,d_m]\textbf{B}_m(s).
\end{equation*}
Now due to definition and properties of the quasi Block-Pulse functions, we can write
\begin{equation}\label{alp1}
{X_s}^\alpha = \begin{bmatrix}
{d_1}^\alpha,{d_2}^\alpha,\dots,{d_m}^\alpha
\end{bmatrix} \textbf{B}_m(s),
\end{equation}
and
\begin{equation}\label{bet1}
{X_s}^\beta = \begin{bmatrix}
{d_1}^\beta,{d_2}^\beta,\dots,{d_m}^\beta
\end{bmatrix} \textbf{B}_m(s).
\end{equation}
By substituting \eqref{rel}, \eqref{x}, \eqref{alp1} and \eqref{bet1}  in Eq. \eqref{e1}, in grid points we get
\begin{align*}
{\textbf{X}_m}^T \mathbf{\Phi}_m  \textbf{B}_m (t) =& [X_0,X_0,\dots,X_0]\textbf{B}_m (t) \\&+ {\textbf{D}_1}^T
\mathbf{\Phi}_m 
\int_{0}^{t}      \textbf{B}_m(s)
\begin{bmatrix}
{d_1}^\alpha,{d_2}^\alpha,\dots,{d_m}^\alpha
\end{bmatrix}
\textbf{B}_m(s)ds \\&+ {\textbf{D}_2}^T
\mathbf{\Phi}_m 
\int_{0}^{t}      \textbf{B}_m(s)
\begin{bmatrix}
{d_1}^\beta,{d_2}^\beta,\dots,{d_m}^\beta
\end{bmatrix}
\textbf{B}_m(s)dW_s. 
\end{align*}
Now, Theorem \ref{t4} leads to
\begin{align*}
{\textbf{X}_m}^T \mathbf{\Phi}_m  \textbf{B}_m (t) =& [X_0,X_0,\dots,X_0]\textbf{B}_m (t) \\&+ {\textbf{D}_1}^T
\mathbf{\Phi}_m 
\mathbf{V}_1 
\int_{0}^{t}      \textbf{B}_m(s)
ds \\&+ {\textbf{D}_2}^T
\mathbf{\Phi}_m 
\mathbf{V}_2
\int_{0}^{t}      \textbf{B}_m(s)
dW_s,
\end{align*}
where
\begin{align*}
\mathbf{V}_1 = \text{diag}\Big( \begin{bmatrix}
{d_1}^\alpha,{d_2}^\alpha,\dots,{d_m}^\alpha 
\end{bmatrix}\Big),\\
\mathbf{V}_2 = \text{diag}\Big(
\begin{bmatrix}
{d_1}^\beta,{d_2}^\beta,\dots,{d_m}^\beta
\end{bmatrix}\Big).
\end{align*}
Finally \eqref{opi} and \eqref{ops} results in
\begin{align*}
{\textbf{X}_m}^T \mathbf{\Phi}_m  \textbf{B}_m (t) =& [X_0,X_0,\dots,X_0]\textbf{B}_m (t) \\&+ {\textbf{D}_1}^T
\mathbf{\Phi}_m 
\mathbf{V}_1 
\mathbf{P}_m     \textbf{B}_m(t)
\\&+ {\textbf{D}_2}^T
\mathbf{\Phi}_m 
\mathbf{V}_2
\mathbf{R}_s
\textbf{B}_m(t). 
\end{align*}
Hence it is concluded that
\begin{equation}\label{nl}
{\textbf{X}_m}^T \mathbf{\Phi}_m   = [X_0,X_0,\dots,X_0]+ {\textbf{D}_1}^T
\mathbf{\Phi}_m 
\mathbf{V}_1 
\mathbf{P}_m     + {\textbf{D}_2}^T
\mathbf{\Phi}_m 
\mathbf{V}_2
\mathbf{R}_s. 
\end{equation}

It is easy to see that \eqref{nl} is a  system of nonlinear algebraic equations with $m$ equations and $m$ unknowns. By finding  $\textbf{X}_m$ and then  substituting it in \eqref{x2} the solution of Eq. \eqref{e1} with coefficient functions defined in \eqref{abnl} is obtained.

\section{Convergence of Method }\label{error}


Here, first we declare   some  theorems that we need them to prove  convergence of the method introduced in the previous section, then we prove our theorem.


\begin{theorem}\label{th2}
	Assume $f(t)$ is $L^\infty$ on a finite interval $I$, it means $f(t)$ is bounded on $I$ if there is a number $M>0$ such that $\vert f(t)\vert \le M$ for all $t \in I$,  then  $f(t)$ is $L^1$ on $I$.
	\begin{proof}
		A complete proof can be found in \cite{Walnut2002}.	 
	\end{proof}	
\end{theorem}



\begin{theorem}\label{th3}
	Any function that is  $L^\infty$ and $L^1$  on any interval $I$, is $L^2$ on $I$.
	\begin{proof}
		The proof is available in		\cite{Walnut2002}.
	\end{proof}	
\end{theorem}

\begin{lemma}{(Gronwall's lemma)}\label{lemma1}
	Let $f$ and $g$ be nonnegative, continuous functions defined on $0\le t \le T$, and let  $\delta,\epsilon \ge 0 $ are constants. If
	\[
	f(t) \le \delta + \epsilon \int_{0}^{t} f(s)g(s) ds, \quad 0\le t \le T,
	\]
	then
	\[
	f(t) \le \delta  \exp\Big( {  \epsilon \int_{0}^{t} g(s) ds}\Big), \quad 0\le t \le T.
	\]
\end{lemma}

\begin{theorem}
	Let 
	$X_t$ and
	$\tilde{X}_t$
	be the exact and numerical solution by  the present method explained in Section \ref{nm} by using the scaling function of the Daubechies wavelet with $N$ vanishing moments in space $V_j$ for Eq. \eqref{e1}. Assume that the conditions of Theorem \ref{th1} are satisfied, then $\tilde{X}_t$ converges to $X_t$.
	\begin{proof}
		We consider functions
		$z_1(t)=a(t,X_t)$
		and
		$z_2(t)=b(t,X_t)$
		and their approximations by using scaling functions of Daubechies wavelet,  
		$\tilde{z}_{1}(t)=\tilde{ a}(t,\tilde{X}_t)$ 
		and
		$\tilde{z}_{2}(t)=\tilde{b}(t,\tilde{X}_t)$, respectively.
		Also, we define
		$\hat{z}_{1}(t)$
		and
		$\hat{z}_{2}(t)$ as follows
		\begin{align*}
		\hat{z}_{1}(t)={ a}(t,\tilde{X}_t),
		\\
		\hat{z}_{2}(t)={b}(t,\tilde{X}_t).
		\end{align*}
		Now we let error function as 
		\[
		e(t)=\mathbb{E}{\vert X_t -
			\tilde{X}_t \vert}^2.
		\]
		We can write
		\begin{align*}
		e(t)&=\mathbb{E}{\Big\vert \int_{0}^{t} \big(z_1(s)- \tilde{z}_{1}(s)\big)ds
			+
			\int_{0}^{t} \big(z_2(s)- \tilde{z}_{2}(s)\big)dW_s
			\Big\vert}^2 \\
		&\le 2 \mathbb{E} \bigg(
		{\Big(\int_{0}^{t} \big(z_1(s)- \tilde{z}_{1}(s)\big)ds \Big)}^2
		+
		{\Big(\int_{0}^{t} \big(z_2(s)- \tilde{z}_{2}(s)\big)dW_s \Big)}^2
		\bigg).
		\end{align*}
		But we can rewrite
		\begin{align*}
		\mathbb{E} 
		{\Big(\int_{0}^{t} \big(z_1(s)- \tilde{z}_{1}(s)\big)ds \Big)}^2 &\le 
		\mathbb{E} 
		{\Big(\int_{0}^{t} {\big(z_1(s)- \tilde{z}_{1}(s)\big)}^2ds \Big)} \\
		& \le 
		\int_{0}^{t}
		\mathbb{E} 
		{\big(z_1(s)- \tilde{z}_{1}(s)\big)}^2ds, 
		\end{align*}
		and by using of  \eqref{Iso}  we can have
		\begin{align*}
		\mathbb{E} 
		{\Big(\int_{0}^{t} \big(z_2(s)- \tilde{z}_{2}(s)\big)dW_s \Big)}^2 =  
		{\Big(\int_{0}^{t} \mathbb{E}{\big(z_2(s)- \tilde{z}_{2}(s)\big)}^2ds \Big)}. 
		\end{align*}
		Therefore we can continue
		\begin{align*}
		e(t)
		&\le 2  \bigg(
		{\int_{0}^{t} \mathbb{E}{\big(z_1(s)- \tilde{z}_{1}(s)\big)}^2ds }
		+
		\int_{0}^{t} \mathbb{E}{\big(z_2(s)- \tilde{z}_{2}(s)\big)}^2ds 
		\bigg)\\
		&\le
		4  \bigg(
		{\int_{0}^{t} \mathbb{E}{\big(z_1(s)- \hat{z}_{1}(s)\big)}^2ds }
		+
		{\int_{0}^{t} \mathbb{E}{\big(\hat{z}_1(s)- \tilde{z}_{1}(s)\big)}^2ds }\bigg)\\& 
		+
		4  \bigg(
		{\int_{0}^{t} \mathbb{E}{\big(z_2(s)- \hat{z}_{2}(s)\big)}^2ds }
		+
		{\int_{0}^{t} \mathbb{E}{\big(\hat{z}_2(s)- \tilde{z}_{2}(s)\big)}^2ds }\bigg)
		\\&=
		4  \bigg(
		{\int_{0}^{t} \mathbb{E}{\big(z_1(s)- \hat{z}_{1}(s)\big)}^2ds }
		+\mathbb{E}\bigg[
		{\int_{0}^{t} {\big(\hat{z}_1(s)- \tilde{z}_{1}(s)\big)}^2ds }\bigg]\bigg)\\& 
		+
		4  \bigg(
		{\int_{0}^{t} \mathbb{E}{\big(z_2(s)- \hat{z}_{2}(s)\big)}^2ds }
		+
		\mathbb{E}\bigg[
		{\int_{0}^{t} {\big(\hat{z}_2(s)- \tilde{z}_{2}(s)\big)}^2ds }\bigg]\bigg).
		\end{align*}
		In addition,  Theorem \ref{thD} about approximation of functions by using the scaling function of Daubechies wavelet  leads to
		\begin{align*}
		e(t)
		& 
		\le 
		4  \bigg(
		{\int_{0}^{t} \mathbb{E}{\big(z_1(s)- \hat{z}_{1}(s)\big)}^2ds }
		+
		\mathbb{E}{\big(C_1 2^{-jN}\big)}^2\bigg) \\& 
		+ 
		4  \bigg(
		{\int_{0}^{t} \mathbb{E}{\big(z_2(s)- \hat{z}_{2}(s)\big)}^2ds }
		+
		\mathbb{E}{\big(C_2 2^{-jN}\big)}^2 \bigg)
		\\& 
		\le 
		4  \bigg(
		{\int_{0}^{t} \mathbb{E}{\big(z_1(s)- \hat{z}_{1}(s)\big)}^2ds }
		+
		{\int_{0}^{t} \mathbb{E}{\big(z_2(s)- \hat{z}_{2}(s)\big)}^2ds }
		\bigg)
		+
		8C^2 2^{-2jN},
		\end{align*}
		where $C=\max\Big\{C_1,C_2\Big\}$. We can  conclude from Theorems \ref{th2} and \ref{th3} that $X_t$ is $L^2([0,1])$. Finally, the Lipschitz condition	results in
		\begin{align*}	
		e(t)
		&  
		\le 
		8 {L_{Lip}}^2 \bigg(
		{\int_{0}^{t} \mathbb{E}{\big\vert
				X_s-\tilde{X}_s	
				\big\vert}^2ds }
		\bigg)+
		8C^2 2^{-2jN},
		\end{align*}
		and hence
		\[
		e(t)
		\le 
		8C^2 2^{-2jN}+8 {L_{Lip}}^2 \bigg(
		\int_{0}^{t} e(s)ds
		\bigg).
		\]
		In conclusion Lemma \ref{lemma1} results in
		\[
		e(t)
		\le 
		8C^2 2^{-2jN}\exp\Big(
		{8 {L_{Lip}}^2}\Big).
		\]	
		The latter relation proves convergence as $j\to \infty$.
	\end{proof}	
\end{theorem}



\section{Numerical Examples}\label{examples}

In order to show  application of the  present method in practice, we provide five examples. In all examples we use the scaling function of the Daubechies wavelet of order $2$. We present some errors such as: absolute error $(e)$ and the Root-Mean-Square error (RMSE). Let $X$ and ${x_j}$ as the exact and approximate solution,  respectively. Then

\[
e_j(t_i) =\vert X(t_i)- x_j(t_i)\vert,\quad t_i \in [0,1), \quad  i =1,2,\dots, m,
\]

\[
RMSE =\sqrt{\frac{\sum_{i=1}^{m} {{e_j}^{2}(t_i)}}{m}}.
\]
All codes are written in MATLAB R2018b, on a pc system with an 
Intel® Pentium(R) CPU G4400 @ 3.30GHz $\times$ 2, 7.7GB(RAM), 64-bit under Ubuntu 20.04.1 LTS.

\subsection{Linear case}
\begin{example}\label{ex1}
	We consider the linear It\^{o} stochastic differential equation
	\begin{equation}\label{exam1}
	X_t = X_0 + c \int_{0}^{t} X_s d s + \sigma \int_{0}^{t} X_s d W_s, \quad 0\leq s,t < 1,
	\end{equation}
	where $c=0.01$, $\sigma=0.01$, and the initial condition is $X_0=\frac{1}{2}$. In the stochastic  integral Brownian motion and the process $X$ are together. It is said Eq. \eqref{exam1} has \textit{multiplicative noise}.  
	The exact solution for Eq. \eqref{exam1} is \cite{Mikosch2000}:
	\begin{equation}\label{GBM}
	X_t = X_0 \exp \Big({(c-0.5\sigma ^2)t + \sigma W_t}\Big).
	\end{equation}
	$X_t$ in 
	\eqref{GBM} is known as the geometric Brownian motion.
	The RMS error for several sample paths is shown in Table \ref{tab11} for some different $j$. As we can see RMS error decreases when $j$ increases which means convergence. Table \ref{tab12} demonstrates numerical results obtained by the present method compared to the exact solution for the first path and $j=7$. Also, graph of the absolute error for the first path and $j=7$ is shown in Figure \ref{fig12}.
	
	\begin{table}[!ht]
		\centering
		\caption{ The RMS error for $N=2$,  Example \ref{ex1}. }
		\label{tab11}
		\begin{tabular}{cccc} \hline\noalign{\smallskip}
			Path &j= 3&j= 5  &j= 7 \\
			\noalign{\smallskip}\hline\noalign{\smallskip}
			$\omega_1$&1.189724e-03  & 5.598874e-04  &   3.420675e-04 \\
			$\omega_2$&1.233056e-03 &7.253622e-04 &  3.083418e-04  \\
			$\omega_3$	&1.529944e-03 &  6.141261e-04    &  3.330084e-04  \\
			$\omega_4$& 1.818255e-03 &6.425052e-04 &   3.254339e-04 \\
			$\omega_5$&1.176162e-03     &  6.467252e-04&  3.210786e-04 \\
			\noalign{\smallskip}\hline		
		\end{tabular}
	\end{table}	
	
	\begin{table}[!ht]
		\centering
		\caption{ Numerical solution based on the present method and the exact solution for $\omega_1 $, $N=2$ and $j=7 $,  Example \ref{ex1}. }
		\label{tab12}
		\begin{tabular}{cccc} \hline\noalign{\smallskip}
			Time &Present Method&Exact   &Absolute Error \\
			\noalign{\smallskip}\hline\noalign{\smallskip}
			0.00& 0.499822    & 0.500000   &   1.783313e-04    \\
			0.25& 0.504242  & 0.504309 & 6.728393e-05  \\
			0.50& 0.503772   &     0.503986 & 2.144939e-04     \\
			0.75&  0.503696    &   0.503909&  2.125174e-04   \\
			1.00& 0.503122  & 0.503047    &  7.499398e-05     \\
			\noalign{\smallskip}\hline		
		\end{tabular}
	\end{table}	
	
	\begin{figure}[!ht]
		{\centering
			\resizebox*{11cm}{!}{\includegraphics{fig12.eps}}}
		\caption{Absolute error  for  $\omega_1$,   $N=2$ and $j=7$,  Example \ref{ex1}.}  \label{fig12}
	\end{figure}
	
\end{example}	

\begin{example}\label{ex3}
	Consider the following linear It\^{o}  stochastic differential equation 
	\[
	X_t = X_0 + \int_{0}^{t} \mathrm{ln} (1+s) X_s ds + \int_{0}^{t} s X_s dW_s, \quad 0\leq s,t < 1,
	\]
	with $X_0 = \frac{1}{75}$. The exact solution is \cite{Rostami201639}:
	\[
	X_t = \frac{1}{75} \exp \Big({(1+t)\mathrm{ln}(1+t)-t-\frac{t^3}{6}+  \int_{0}^{t} s dW_s    }\Big). 
	\]
	The RMS error for some sample paths is shown in Table \ref{tab31} for some values of  $j$. We can find the RMS error decreases as $j$ increases. Table \ref{tab32} indicates numerical results by the present method and the exact solution for the fifth path and $j=8$. Figure \ref{fig3} shows the numerical solution based on the present method compared to the exact solution for the fifth path and $j=8$.
	
	\begin{table}[!ht]
		\centering
		\caption{The RMS error for $N=2$, Example \ref{ex3}. }
		\label{tab31}
		\begin{tabular}{cccc} \hline\noalign{\smallskip}
			Path &j=4&j=6  &j=8 \\
			\noalign{\smallskip}\hline\noalign{\smallskip}
			$\omega_1$&1.840026e-03&1.891139e-03 &   9.125506e-04  \\
			$\omega_2$&4.147880e-03&3.154048e-03&   1.276819e-03\\
			$\omega_3$	&4.274300e-03&  8.463968e-04  &  6.089297e-04 \\
			$\omega_4$&7.260763e-03 &2.499652e-03 &   7.411351e-04\\
			$\omega_5$&9.040620e-04 &2.942558e-03  &  1.037556e-03  \\
			\hline\noalign{\smallskip}			
		\end{tabular}
	\end{table}	
	
	\begin{table}[!ht]
		\centering
		\caption{ Numerical solution based on the peresent method and the exact solution for $\omega_5$, $N=2$ and $j=8$, Example \ref{ex3}. }
		\label{tab32}
		\begin{tabular}{cccc} \hline\noalign{\smallskip}
			Time &Present Method&Exact   &Absolute Error \\
			\noalign{\smallskip}\hline\noalign{\smallskip}
			0.00&0.013333&  0.013333  &  1.166817e-10    \\
			0.25&0.011574&0.011501 & 7.302481e-05   \\
			0.50&0.011996 & 0.011866  &1.298661e-04    \\
			0.75& 0.018150  & 0.016157 &1.993857e-03  \\
			1.00&0.010601  & 0.008044 &  2.556790e-03   \\
			\noalign{\smallskip}\hline	
		\end{tabular}
	\end{table}	
	
	\begin{figure}[!ht]
		{\centering
			\resizebox*{11cm}{!}{\includegraphics{fig3.eps}}}
		\caption{   Numerical solution based on the present method and the exact solution for $\omega_5$,   $N=2$ and $j=8$, Example \ref{ex3}.}  \label{fig3}
	\end{figure}
	
	
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{example}\label{ex4}
	
	Let us consider another linear It\^{o}  stochastic differential equation 
	\[
	X_t = X_0 + \int_{0}^{t} \mathrm{cos}(s) X_s ds + \int_{0}^{t} \mathrm{sin}(s) X_s dW_s, \quad 0 \leq s,t < 1,
	\]
	with the initial condition  $X_0 = \frac{1}{120}$.  The exact solution is \cite{Rostami201639}:
	\[
	X_t = \frac{1}{120} \exp \Big({-\frac{t}{4}+\mathrm{sin}(t)+\frac{\mathrm{sin}(2t)}{8}  \int_{0}^{t} \mathrm{sin}(s) dW_s    }\Big). 
	\]
	
	Table \ref{tab41} indicates the RMS error for different $j$ for several sample paths. Numerical results by using the present method and the exact solution for the forth path and $j=7$ is presented in Table \ref{tab42}. Figure \ref{fig4} shows the numerical solution based on the present method compared to the exact solution for the same path and $j$.
	
	\begin{table}[!ht]
		\centering
		\caption{ The RMS error for $N=2$, Example \ref{ex4}. }
		\label{tab41}
		\begin{tabular}{cccc} \hline\noalign{\smallskip}	
			Path &j=5&j=6  &j=7 \\
			\noalign{\smallskip}\hline\noalign{\smallskip}
			$\omega_1$& 3.340622e-03&1.666418e-03 &  8.682893e-04    \\
			$\omega_2$&9.789401e-04&2.461135e-03&  1.261553e-03 \\
			$\omega_3$	&1.832066e-03&   1.809826e-03  & 9.267302e-04  \\
			$\omega_4$&2.774917e-03 &9.783834e-04& 7.145506e-04  \\
			$\omega_5$&3.059496e-03 & 7.803249e-04& 9.578801e-04  \\
			\noalign{\smallskip}\hline		
		\end{tabular}
	\end{table}	
	
	\begin{table}[!ht]
		\centering
		\caption{Numerical solution based on the present method and the exact solution for $\omega_4$, $N=2$ and $j=7$,  Example \ref{ex4}. }
		\label{tab42}
		\begin{tabular}{cccc} \hline\noalign{\smallskip}
			Time &Present Method&Exact   &Absolute Error \\
			\noalign{\smallskip}\hline\noalign{\smallskip}
			0.00&0.008350&  0.008333  &1.639841e-05     \\
			0.25&0.009634&0.009487&1.467543e-04    \\
			0.50&0.008743 & 0.008574    & 1.696685e-04    \\
			0.75& 0.011009   & 0.010215  & 7.931460e-04  \\
			1.00& 0.007040  & 0.005704  &  1.336274e-03   \\
			\noalign{\smallskip}\hline	
		\end{tabular}
	\end{table}	
	
	\begin{figure}[!ht]
		{\centering
			\resizebox*{11cm}{!}{\includegraphics{fig4.eps}}}
		\caption{   Numerical solution based on the present method and the exact solution for $\omega_4$,   $N=2$ and $j=7$, Example \ref{ex4}.}  \label{fig4}
	\end{figure}
	
	
\end{example}

\subsection{Nonlinear case}	
%%%%%%%%%%%%%%%%%%%
\begin{example}\label{ex1nl}
	
	We want to solve the nonlinear It\^{o}  stochastic differential equation 
	\[
	dX_t = \frac{1}{2} a^2 n_1  {X_t}^{2n_1-1} dt+ a{X_t}^{n_1} dW_t, \quad 0 \leq t \leq 1, \quad n_1 \neq 1.
	\]
	Here the initial condition is $X_0 = \frac{1}{2}$, $a=0.1$, $n_1=2$ and  the exact solution is \cite{Kloeden1995}:
	\[
	X_t =  \Big( {X_0}^{1-n_1} -a(n_1-1)W_t \Big)^{\dfrac{1}{1-n_1}}.
	\]
	Table \ref{tab11nl} shows the RMS error for some $j$ for five sample paths. Numerical results based on  the present method and the exact solution for the first path and $j=7$ is demonstrated in Table \ref{tab12nl}. Figure \ref{fig1nl} indicates the absolute error for the first path and $j=7$.
	
	\begin{table}[!ht]
		\centering
		\caption{ The RMSE error for $N=2$,  Example \ref{ex1nl}. }
		\label{tab11nl}
		\begin{tabular}{cccc} \hline\noalign{\smallskip}
			Path &j=3&j= 5 &j=7 \\
			\noalign{\smallskip}\hline\noalign{\smallskip}
			$\omega_1$&8.317664e-03  & 2.710919e-03& 1.465871e-03    \\
			$\omega_2$&6.592069e-03&2.806349e-03&  2.053098e-03 \\
			$\omega_3$	&6.103472e-03&  3.229574e-03  &  1.889319e-03 \\
			$\omega_4$&7.648277e-03 &3.363171e-03&  1.928895e-03 \\
			$\omega_5$&8.969578e-03 & 3.206449e-03&  1.514159e-03 \\
			\noalign{\smallskip}\hline		
		\end{tabular}
	\end{table}	
	
	\begin{table}[!ht]
		\centering
		\caption{ Numerical solution based on the present method and the exact solution for $\omega_1$, $N=2$ and $j=7$,  Example \ref{ex1nl}. }
		\label{tab12nl}
		\begin{tabular}{cccc} \hline\noalign{\smallskip}
			Time &Present Method&Exact   &Absolute Error \\
			\noalign{\smallskip}\hline\noalign{\smallskip}
			0.00&0.500704 & 0.500000    & 7.041799e-04    \\
			0.25&0.499755 &0.496571& 3.183721e-03   \\
			0.50&0.494611  &0.495278     &  6.675488e-04   \\
			0.75&0.491089  & 0.491157  &  6.726415e-05  \\
			1.00& 0.494719  & 0.493867  &  8.514789e-04    \\
			\noalign{\smallskip}\hline	
		\end{tabular}
	\end{table}	
	
	\begin{figure}[!ht]
		{\centering
			\resizebox*{11cm}{!}{\includegraphics{fig12nl.eps}}}
		\caption{   Absolute error for $\omega_1$,   $N=2$ and $j=7$, Example \ref{ex1nl}.}  \label{fig1nl}
	\end{figure}
	
	
\end{example}

%%%%%%%%%%%%%%%%%%%%%%
\begin{example}\label{ex2nl}
	We solve
	another nonlinear It\^{o}  stochastic differential equation with the initial condition 
	\[
	dX_t = \frac{1}{2} a(a-1)  {X_t}^{1-\frac{2}{a}} dt+ a{X_t}^{1-\frac{1}{a}} dW_t, \quad 0 \leq t < 1.
	\]
	The initial condition is $X_0 = \frac{1}{2}$, $a=-\frac{1}{2}$ and  the exact solution is \cite{Kloeden1995}:
	\[
	X_t =  \Big( {X_0}^{\frac{1}{a}} +W_t \Big)^a.
	\]
	The RMS error for some values of $j$ for five sample paths is shown in Table \ref{tab21nl}. Numerical results obtained by  the present method and the exact solution for the third path and $j=7$ is presented in Table \ref{tab22nl}. Figure \ref{fig21nl} shows the numerical solution based on the present method compared to the exact solution for the third path and $j=7$. The absolute error for the same data is shown in Figure \ref{fig22nl}. 
	
	\begin{table}[!ht]
		\centering
		\caption{ The RMS error for $N=2$,  Example \ref{ex2nl}. }
		\label{tab21nl}
		\begin{tabular}{cccc} \hline\noalign{\smallskip}
			Path &j=5&j=6  &j=7 \\
			\noalign{\smallskip}\hline\noalign{\smallskip}
			$\omega_1$&  2.324759e-02& 1.872409e-02&  7.824655e-03   \\
			$\omega_2$&3.610773e-02&1.020272e-02&  8.480478e-03 \\
			$\omega_3$	&5.352925e-02&9.096077e-03    &  5.213692e-03  \\
			$\omega_4$& 3.375873e-02&1.324295e-02&  5.581646e-03 \\
			$\omega_5$& 9.625489e-03&1.077801e-02 &  7.724419e-03 \\
			\noalign{\smallskip}\hline		
		\end{tabular}
	\end{table}	
	
	\begin{table}[!ht]
		\centering
		\caption{ Numerical solutions based on the present method and the exact solution for $\omega_3$, $N=2$ and $j=7$,  Example \ref{ex2nl}. }
		\label{tab22nl}
		\begin{tabular}{cccc} \hline\noalign{\smallskip}
			Time &Present Method&Exact   &Absolute Error \\
			\noalign{\smallskip}\hline\noalign{\smallskip}
			0.00&0.494968  &0.500000     &  5.032328e-03   \\
			0.25&0.489550 &0.486787 &  2.763175e-03  \\
			0.50&0.460865& 0.458502     &   2.363386e-03   \\
			0.75&0.460560  & 0.459845  &  7.152582e-04 \\
			1.00& 0.425205   & 0.418582   & 6.623136e-03     \\
			\noalign{\smallskip}\hline	
		\end{tabular}
	\end{table}	
	
	\begin{figure}[!ht]
		{\centering
			\resizebox*{11cm}{!}{\includegraphics{fig21nl.eps}}}
		\caption{Numerical solution based on the present method and the exact solution for   $\omega_3$,   $N=2$ and $j=7$, Example \ref{ex2nl}.}  \label{fig21nl}
	\end{figure}	
	
	
	
	\begin{figure}[!ht]
		{\centering
			\resizebox*{11cm}{!}{\includegraphics{fig22nl.eps}}}
		\caption{   Absolute error for $\omega_3$,   $N=2$ and $j=7$, Example \ref{ex2nl}.}  \label{fig22nl}
	\end{figure}
	
	
\end{example}

\newpage
\section{Conclusion}\label{Conclusion}

In this work, by using the quasi Block-Pulse functions, we introduce integration and stochastic integration operational matrices based on the scaling function of the Daubechies wavelet to solve some linear and nonlinear stochastic differential problems. According to this method, the problems solve easily and faster.

These operational matrices have good properties due to properties of the Daubechies wavelet such as orthogonality, compactly support and vanishing moments. Accuracy of the proposed method proved by some numerical examples.

\section*{Declarations}

\begin{itemize}
	\item Funding "Not Applicable"
	\item Conflict of interest/Competing interests "Not Applicable"
	\item Ethics approval 
	\item Consent to participate "Not Applicable"
	\item Consent for publication
	\item Availability of data and materials 
	\item Code availability "Not Applicable"
	\item Authors' contributions: All sections was written by the first author and all the authors reviewed  paper.  
\end{itemize}

\noindent


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