The goal of this section is to recall notations and definitions of RBFs that are used for approximating the solutions of BOD equations. Also, applying the method for solving stochastic integral equations is described.

RBFs are mostly identified on the basis of smoothness. Some functions are infinitely smooth and some are piecewise smooth. Gaussian Function (GS), Multiquadric (MQ), Inverse Multi- quadric (IMQ) and Inverse quadric (IQ) are some example of infinitely smooth RBFs where as Thin Plate Spline (TPS) and Linear radial function (LR) are piecewise smooth RBFs. For infinitely smooth RBFs, there exists a free parameter called the shape parameter which controls the shape of RBF. The RBF become flat if shape parameter is closer to 0.

Therefore, the solution of the interpolation problem based on the extended expansion(1) reduces to the solution of a system of linear equations of the matrix form

## 2.1 Description of Method on stochastic Volterra integral equation

The nonlinear stochastic Volterra integral equation takes the following form:

$$u(t)={u_0}(t)+\int\limits_{0}^{t} {{k_1}(s,{\text{ }}t)u(s)ds} +\int\limits_{0}^{t} {{k_2}(s,t)u(s)dB(s)} \begin{array}{*{20}{c}} {}&{} \end{array}~t \in ~[0,T],$$

2

where, \(u\left(t\right), {\text{u}}_{0}\left(t\right), {k}_{1}(s, t)\) and \({k}_{2}(s, t),\)for \(s, t \in [0, T ),\)are the stochastic processes defined on the same probability space \(({\Omega }, F, P )\) with a filtration \(\{{F}_{t}, t \ge 0\)*}* that is increasing and right continuous and \({F}_{0}\) contains all P-null sets.\(u\left(t\right)\) is unknown random function and \(B\left(t\right)\) is a standard Brownian motion defined on the probability space and \(\int\limits_{0}^{t} {{k_2}(s,t)u(s)dB(s)}\) is the Itˆo integral.

Let’s approximate the function \(u\left(t\right)\) in terms of radial basis functions,\(\varphi \left(t\right),\) as follows

$$u(t) \simeq ~\sum\limits_{{i=1}}^{N} {{\lambda _i}\varphi (\parallel t - {t_i}\parallel )} ~~~~$$

3

Then, from substituting Eq. (3) into Eq. (2) we have,

\(\sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel t - {t_i}\parallel )={u_0}(t)+\int_{0}^{t} {{k_1}(s,t)~} \sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel s - {t_i}\parallel )ds+\)

\(\int_{0}^{t} {} {k_2}(s,{\text{ }}t)\sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel s - {t_i}\parallel )dB(s),\begin{array}{*{20}{c}} {}&{} \end{array}~t \in [0,T],~~~~~~~~~~~~~~~~~~~~~~~~~~\left( 4 \right)\)

Substituting the collocation points \(\begin{array}{*{20}{c}} {{t_j}} \end{array}_{{j=1}}^{N}\) into Eq. (4), we obtain:

\(\sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel {t_j} - {t_i}\parallel )={u_0}({t_j})+\int_{0}^{{{t_j}}} {{k_1}(s,{t_j})~} \sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel s - {t_i}\parallel )ds+\)

\(\int_{0}^{{{t_j}}} {} {k_2}(s,{\text{ }}{t_j})\sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel s - {t_i}\parallel )dB(s),\begin{array}{*{20}{c}} {}&{} \end{array}~t \in [0,T],~~~j=1,...,N.~~~~~~~~~~~~~~~~~\left( 5 \right)\)

In above equation, we let \(s=\frac{{t}_{j}}{2}\)*x+*\(\frac{{t}_{j}}{2}\). It reduces Eq. (5) to the following equation

\(\sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel {t_j} - {t_i}\parallel )={u_0}({t_j})+\frac{{{t_j}}}{2}\int_{{ - 1}}^{1} {{k_1}(\frac{{{t_j}}}{2}x+\frac{{{t_j}}}{2},{t_j})~} \sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel \frac{{{t_j}}}{2}x+\frac{{{t_j}}}{2} - {t_i}\parallel )dx+\)

\(\int_{0}^{{{t_j}}} {} {k_2}(s,{\text{ }}{t_j})\sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel s - {t_i}\parallel )dB(s),\begin{array}{*{20}{c}} {}&{} \end{array}~t \in [0,T],~~~j=1,...,N.~~~~~~~~~~~~~~~~~~~~~~~\left( 6 \right)\)

For computing or approximating the first integral, several methods can be used. Now, by applying Legendre-Gauss-Lobatto integration formula, we approximate the first integral of Eq. (6) as follow

\(\sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel {t_j} - {t_i}\parallel )={u_0}({t_j})+\frac{{{t_j}}}{2}\sum\limits_{{k=1}}^{N} {{w_k}{k_1}(\frac{{{t_j}}}{2}{x_k}+\frac{{{t_j}}}{2},{t_j})} \sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel \frac{{{t_j}}}{2}{x_k}+\frac{{{t_j}}}{2} - {t_i}\parallel )+\)

\(\int_{0}^{{{t_j}}} {} {k_2}(s,{\text{ }}{t_j})\sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel s - {t_i}\parallel )dB(s),\begin{array}{*{20}{c}} {}&{} \end{array}~t \in [0,T],~~~j=1,...,N.~~~~~~~~~~~~~~~~~~~~~~~\left( 7 \right)\)

where \({w}_{k}, {x}_{k}, k = 1, ..., N.\)are weights and nodes of the integration rule respectively.

For approximating the Itˆo integral, let \(0={s_0}<{s_1}<...<{s_m}=T\). Given a suitable function , the integral

\(\int\limits_{0}^{T} {f(t)dt} ,\)

may be approximated by the Riemann sum

$$\sum\limits_{{j=0}}^{{N - 1}} {f({s_j}} )({s_{j+1}} - {s_j})$$

8

where the discrete points \({s_j}=j\delta t\) Indeed, the integral may be defined by taking the limit \(\delta t \to 0\) in (8). In a similar way, we may consider a sum of the form

\(\sum\limits_{{j=0}}^{{N - 1}} {f({s_j}} )(B({s_{j+1}}) - B({s_j})),\)

which, by analogy with (8), may be regarded as an approximation to a stochastic integral

$$\int\limits_{0}^{T} {f(t)dB(t)=} \sum\limits_{{j=0}}^{{N - 1}} {f({s_j}} )(B({s_{j+1}}) - B({s_j})),$$

9

Here, we are integrating with respect to Brownian motion.

In Eq. (6) for Itˆo integral, let \(0={s_0}<{s_1}<...<{s_m}={t_0}\). So, achieved

\(\sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel {t_j} - {t_i}\parallel )={u_0}({t_j})+\frac{{{t_j}}}{2}\sum\limits_{{k=1}}^{N} {{w_k}{k_1}(\frac{{{t_j}}}{2}{x_k}+\frac{{{t_j}}}{2},{t_j})} \sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel \frac{{{t_j}}}{2}{x_k}+\frac{{{t_j}}}{2} - {t_i}\parallel )+\)

\(\sum\limits_{{k=0}}^{{m - 1}} {} {k_2}({s_k},{\text{ }}{t_j})\sum\limits_{{i=1}}^{N} {} {\lambda _i}\varphi (\parallel {s_k} - {t_i}\parallel )\left[ {B({s_{k+1}}) - B({s_k})} \right],\begin{array}{*{20}{c}} {}&{} \end{array}~t \in [0,T],~~~j=1,...,N.~~~~~~~~~~\left( {10} \right)\)

We have a system of equations that can be solved by mathematical software for the unknowns vector *λ*. By computing that, unknown function \(u\left(t\right)\) can be approximated.