Legendre wavelets Galerkin method for solving nonlinear stochastic integral equations
Abstract
In this paper, an efficient and accurate computational method based on the Legendre wavelets (LWs) together with the Galerkin method is proposed for solving a class of nonlinear stochastic Itô–Volterra integral equations. For this purpose, a new stochastic operational matrix (SOM) for LWs is derived. A collocation method based on hat functions (HFs) is employed to derive a general procedure for forming this matrix. The LWs and their operational matrices of integration and stochastic Itô-integration and also some useful properties of these basis functions are used to transform such problems into corresponding nonlinear systems of algebraic equations, which can be simply solved to achieve the solution of such problems. Moreover, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient. Furthermore as some useful applications, the proposed method is applied to obtain approximate solutions for some stochastic problems in the mathematics finance, biology, physics and chemistry.
Keywords
Legendre wavelets (LWs) Stochastic operational matrix (SOM) Nonlinear stochastic Itô–Volterra integral equations Brownian motion process Stochastic volatility models Stochastic Lotka–Volterra model Duffing–Van der Pol Oscillator Stochastic Brusselator problem1 Introduction
Approximation by orthogonal families of basis functions has found wide applications in sciences and engineering [1]. The main idea of using an orthogonal basis is that the problem under consideration is reduced into solving a system of algebraic equations which can be simply solved to achieve the solution of the problem under study. This can be done by truncated series of orthogonal basis functions for the solution of the problem and using the operational matrices of these basis functions [1]. Depending on their structure, the orthogonal functions may be mainly classified into three families [2]. The first family includes sets of piecewise constant orthogonal functions such as the Walsh functions, block pulse functions, etc. The second family consists of sets of orthogonal polynomials such as Laguerre, Legendre, Chebyshev, etc., and the third family is the widely used sets of sine-cosine functions. It is worth noting that approximating a continuous function with piecewise constant basis functions results in an approximation that is piecewise constant. On the other hand if a discontinuous function is approximated with continuous basis functions, the resulting approximation is continuous and can not properly model the discontinuities. In remote sensing, images often have properties that vary continuously in some regions and discontinuously in others. Thus, in order to properly approximate these spatially varying properties, it is necessary to use approximating functions that can accurately model both continuous and discontinuous phenomena. Therefore, neither continuous basis functions nor piecewise constant basis functions taken alone can efficiently and accurately model these spatially varying properties. However, wavelets basis functions are another basis set which offers considerable advantages over alternative basis sets and allows us to attack problems not accessible with conventional numerical methods. Their main advantages are [1]: the basis set can be improved in a systematic way, different resolutions can be used in different regions of space, the coupling between different resolution levels is easy, there are few topological constraints for increased resolution regions, the Laplace operator is diagonally dominant in an appropriate wavelet basis, the matrix elements of the Laplace operator are very easy to calculate and the numerical effort scales linearly with respect to the system size.
It is also well known that we can approximate any smooth function by the eigenfunctions of certain singular Sturm–Liouville problems such as Laguerre, Legendre or Chebyshev orthogonal polynomials. In this manner, the truncation error approaches zero faster than any negative power of the number of basis functions used in the approximation [3]. This phenomenon is usually referred to as “The spectral accuracy” [3]. But, in the case that the function under approximation is not analytic, these basis functions do not work well and therefore spectral accuracy does not happen. For these situations, wavelet functions will be more effective. In this communication, it is worth mentioning that the LWs have mutually spectral accuracy, orthogonality and other useful properties of wavelets.
Nonlinear stochastic functional equations have been extensively studied over a long period of time since they are fundamental for modeling science and engineering phenomena [4, 5, 6, 7, 8]. As the computational power increases, it becomes feasible to use more accurate functional equation models and solve more demanding problems. Moreover, the study of stochastic or random functional equations can be very useful in application, due to the fact that they arise in many situations. For example, stochastic integral equations arise in a wide range of problems such as the stochastic formulation of problems in reactor dynamics [9, 10], the study of the growth of biological populations [11], the theory of automatic systems resulting in delay-differential equations [12], and in many other problems occurring in the general areas of biology, physics and engineering. Also, nowadays, there is an increasing demand to investigate the behavior of even more sophisticated dynamical systems in physical, medical, engineering and financial applications [13, 14, 15, 16, 17, 18, 19]. These systems often depend on a noise source, like a Gaussian white noise, governed by certain probability laws, so that modeling such phenomena naturally involves the use of various stochastic differential equations (SDEs) [11, 20, 21, 22, 23, 24, 25, 26], or in more complicated cases, stochastic Volterra integral equations and stochastic integro-differential equations [27, 28, 29, 30, 31, 32, 33, 34]. In most cases it is difficult to solve such problems explicitly. Therefore, it is necessary to obtain their approximate solutions by using some numerical methods [9, 10, 11, 12, 13, 14, 15, 20, 29, 30, 31].
- (i)
\(B(0)=0\) (with the probability 1).
- (ii)
For \(0 \le s < t \le 1\) the random variable given by the increment \(B(t)-B(s)\) is normally distributed with mean zero and variance \(t-s\); equivalently, \(B(t)-B(s)\sim \sqrt{t-s}~{\mathcal {N}}(0,1)\), where \({\mathcal {N}}(0,1)\) denotes a normally distributed random variable with zero mean and unit variance.
- (iii)
For \(0\le s< t < u < v \le 1\) the increments \(B(t)-B(s)\) and \(B(v)-B(u)\) are independent.
The proposed method is based on reducing the problem under study to a system of nonlinear algebraic equations by expanding the solution as LWs with unknown coefficients and using the operational matrices of integration and stochastic integration. Moreover, a new technique for computation of the nonlinear terms in such equations is presented.
This paper is organized as follows: In Sect. 2, the LWs and their properties are described. In Sect. 3, the proposed method is described for solving nonlinear stochastic Itô–Volterra integral equations. In Sect. 4, the proposed method is applied for solving some numerical examples. In Sect. 5, some applications of the proposed computational method are described. Finally, a conclusion is drawn in Sect. 6.
2 The LWs and their properties
In this section, we briefly review the LWs and their properties which are used further in this paper.
2.1 Wavelets and the LWs
2.2 Function approximation
2.3 Operational matrix of stochastic Itô-integration
Theorem 2.1
Proof
2.4 Operational matrix of integration
Remark 1
2.5 Some new useful results for the LWs
In this section, we obtain some new useful results for the LWs which will be used further in this paper.
Lemma 2.2
Proof
Corollary 2.3
Proof
By considering Lemma 2.2, the proof will be straightforward. \(\square \)
Theorem 2.4
Theorem 2.5
Proof
Corollary 2.6
3 Description of the proposed computational method
Finally, by solving this system for the unknown vector X, we obtain an approximate solution for the problem by substituting X in Eq. (42).
Algorithm 1 |
Input:\(M,\,N\in {\mathbb {N}},\,k\in {\mathbb {Z}}^{+}\); Brownian motion process B(t); the functions \(h,\,f,\, g\in L^{2}\left[ 0,1\right] \) and \(\mu ,\,\sigma \in C^{\infty }\left[ 0,1\right] \). |
Step 1: Define the LWs \(\psi _{nm}(t)\) from Eq. (5). |
Step 2: Construct the LWs vector \(\Psi (t)\) from Eq. (11). |
Step 3: Compute the LWs matrix \(\Phi _{\hat{m}\times \hat{m}}\triangleq \left[ \Psi (0),\Psi \left( \frac{1}{\hat{m}-1}\right) ,\ldots ,\Psi (1)\right] \). |
Step 4: Compute the integration operational matrix P using Eqs. (31)–(33) and SOM \(\hat{P}_s\) using Eq. (25). |
Step 5: Compute the LWs stochastic operational matrix \(P_s=\Phi _{\hat{m}\times \hat{m}}\hat{P}_s\Phi ^{-1}_{\hat{m}\times \hat{m}}\). |
Step 6: Compute the vectors \(H,\,C\) and D in Eqs. (43) and (44) using Eq. (8). |
Step 7: Compute the vectors \(\widetilde{C}^{T}=C^{T}\Phi _{\hat{m}\times \hat{m}}\) and \(\widetilde{D}^{T}=D^{T}\Phi _{\hat{m}\times \hat{m}}\). |
Step 8: Put \(R(t)=\left( X^{T}- H^{T}-\left( \widetilde{C}^{T}\odot \mu \left( \widetilde{X}^{T}\right) \right) \Phi _{\hat{m}\times \hat{m}}^{-1}P\right. \)\( \left. - \left( \widetilde{D}^{T}\odot \sigma \left( \widetilde{X}^{T}\right) \right) \Phi _{\hat{m}\times \hat{m}}^{-1}P_{s}\right) \Psi (t)\). |
Step 9: Construct the nonlinear system of algebraic equations: |
\(\qquad \displaystyle \int _{0}^{1}R(t)\psi _{j}(t)\mathrm{d}t=0, \quad j=1,2,\ldots ,\hat{m}.\) |
Step 10: Solve the nonlinear system of algebraic equations in Step 9 and obtain the unknown vector X. |
Output: The approximate solution: \(X(t)\simeq X^T\Psi (t)\). |
4 Illustrative test problems
Example 1
The absolute errors of the approximate solution at some different points for Example 1
\(\hat{m}\) | \(t=0.2\) | \(t=0.4\) | \(t=0.6\) | \(t=0.8\) | \(t=1.0\) |
---|---|---|---|---|---|
24 | 2.0645E\(-\)5 | 1.8513E\(-\)5 | 6.3995E\(-\)6 | 3.7607E\(-\)6 | 3.4588E\(-\)7 |
48 | 6.8033E\(-\)6 | 1.0172E\(-\)6 | 6.3954E\(-\)7 | 4.7170E\(-\)6 | 3.1923E\(-\)7 |
96 | 5.7369E\(-\)8 | 1.1786E\(-\)7 | 1.7839E\(-\)7 | 2.3998E\(-\)7 | 3.0300E\(-\)7 |
Example 2
The absolute errors of the approximate solution at some different points for Example 2
\(\hat{m}\) | \(t=0.2\) | \(t=0.4\) | \(t=0.6\) | \(t=0.8\) | \(t=1.0\) |
---|---|---|---|---|---|
24 | 3.0807E\(-\)4 | 1.5827E\(-\)3 | 5.8391E\(-\)3 | 2.7070E\(-\)3 | 9.6720E\(-\)6 |
48 | 6.8570E\(-\)4 | 1.0130E\(-\)3 | 1.3281E\(-\)3 | 1.5258E\(-\)3 | 7.8065E\(-\)6 |
96 | 8.4116E\(-\)7 | 1.7767E\(-\)6 | 4.7233E\(-\)6 | 6.6628E\(-\)6 | 8.4139E\(-\)6 |
Example 3
The absolute errors of the approximate solution at some different points for Example 3
\(\hat{m}\) | \(t=0.2\) | \(t=0.4\) | \(t=0.6\) | \(t=0.8\) | \(t=1.0\) |
---|---|---|---|---|---|
24 | 1.4887E\(-\)5 | 2.0779E\(-\)4 | 2.4626E\(-\)4 | 1.0138E\(-\)3 | 1.3854E\(-\)4 |
48 | 1.4257E\(-\)4 | 2.2798E\(-\)5 | 4.8847E\(-\)5 | 3.6948E\(-\)4 | 1.3799E\(-\)4 |
96 | 2.5441E\(-\)5 | 5.2877E\(-\)5 | 7.7811E\(-\)5 | 1.0434E\(-\)4 | 1.3902E\(-\)4 |
Example 4
The absolute errors of the approximate solution at some different points for Example 4
\(\hat{m}\) | \(t=0.2\) | \(t=0.4\) | \(t=0.6\) | \(t=0.8\) | \(t=1.0\) |
---|---|---|---|---|---|
24 | 3.0697E\(-\)4 | 5.7637E\(-\)5 | 1.2910E\(-\)3 | 3.7701E\(-\)3 | 5.9236E\(-\)6 |
48 | 1.4019E\(-\)3 | 1.1434E\(-\)3 | 5.7682E\(-\)4 | 1.9447E\(-\)3 | 8.6169E\(-\)6 |
96 | 1.2937E\(-\)7 | 1.2398E\(-\)6 | 1.1533E\(-\)6 | 5.6874E\(-\)6 | 7.6646E\(-\)6 |
5 Some applications of the proposed method
This section deals with the proposed computational method in Sect. 3, to obtain approximate solutions for some practical stochastic problems.
5.1 The mathematical finance
It is worth mentioning that stochastic volatility models have become popular for derivative pricing and hedging in the last decade as the existence of a non-flat implied volatility surface (or term-structure) has been noticed and become more pronounced, especially since the 1987 crash. This phenomenon, which is well-documented [48, 49], stands in empirical contradiction to the consistent use of a classical Black–Scholes (constant volatility) approach to pricing options and similar securities. However, it is clearly desirable to maintain as many of the features as possible that have contributed to this model’s popularity and longevity, and the natural extension pursued both in the literature and in practice has been to modify the specification of volatility in the stochastic dynamics of the underlying asset price model.
5.2 The biological systems
Finally, by solving system (60) for the unknown vectors \(N_{1}\) and \(N_{2}\), we obtain the approximate solutions of the problem as \(N_{1}(t)\simeq N_{1}^{T}\Psi (t)\) and \(N_{2}(t)\simeq N_{2}^{T}\Psi (t)\).
Algorithm 2 |
Input:\(M\in {\mathbb {N}},\,k,N\in {\mathbb {Z}}^{+}\); Brownian motion processes \(B_{1}(t)\) and \(B_{2}(t)\); \(a_{i},\,b_{i},\,\bar{\sigma }_{i},\,N_{i0}\) for \(i=1,2\). |
Step 1: Define the LWs \(\psi _{nm}(t)\) from Eq. (5). |
Step 2: Construct the LWs vector \(\Psi (t)\) from Eq. (11). |
Step 3: Compute the LWs matrix \(\Phi _{\hat{m}\times \hat{m}}\triangleq \left[ \Psi (0),\Psi \left( \frac{1}{\hat{m}-1}\right) ,\ldots ,\Psi (1)\right] \). |
Step 4: Compute the integration operational matrix P using Eqs. (31)–(33) and SOM \(\hat{P}_s\) using Eq. (25). |
Step 5: Compute the LWs stochastic operational matrix \(P_s=\Phi _{\hat{m}\times \hat{m}}\hat{P}_s\Phi ^{-1}_{\hat{m}\times \hat{m}}\). |
Step 6: Compute the vectors \(e^{T}\) using Eq. (8). |
Step 7: Compute the vectors \(\widetilde{N}_{1}^{T}=N_{1}^{T}\Phi _{\hat{m}\times \hat{m}}\) and \(\widetilde{N}_{2}^{T}=N_{2}^{T}\Phi _{\hat{m}\times \hat{m}}\). |
Step 8: Put \(\displaystyle \left\{ \begin{array}{l} \displaystyle R_{1}(t) \!=\!\left( N_{1}^{T}-N_{10}e^{T}\!-\!\left( b_{1}N_{1}^{T}\right. \right. \\ \left. \left. -a_{1}\left( \widetilde{N}_{1}^{T}\odot \widetilde{N}_{2}^{T}\right) \Phi _{\hat{m}\times \hat{m}}^{-1}\right) P\!-\!\bar{\sigma }_{1}N_{1}^{T}P_{s}\right) \Psi (t),\\ \displaystyle R_{2}(t)\!=\!\left( N_{2}^{T}\!-\!N_{20}e^{T}\!-\!\left( b_{2}N_{2}^{T}\right. \right. \\ \left. \left. -a_{2}\left( \widetilde{N}_{1}^{T}\odot \widetilde{N}_{2}^{T}\right) \Phi _{\hat{m}\times \hat{m}}^{-1}\right) P\!-\!\bar{\sigma }_{2}N_{2}^{T}P_{s}\right) \Psi (t). \end{array}\right. \). |
Step 9: Construct the nonlinear system of algebraic equations: |
\(\displaystyle \left\{ \begin{array}{l} \displaystyle \left( R_{1}(t),\psi _{j}(t)\right) \!=\!\int _{0}^{1}R_{1}(t)\psi _{j}(t)\mathrm{d}t\!=\!0, \quad j\!=\!1,2,\ldots ,\hat{m}, \\ \displaystyle \left( R_{2}(t),\psi _{j}(t)\right) \!=\!\int _{0}^{1}R_{2}(t)\psi _{j}(t)\mathrm{d}t\!=\!0, \quad j\!=\!1,2,\ldots ,\hat{m}, \end{array}\right. \) |
Step 10: Solve the nonlinear system of algebraic equations in Step 9 and obtain the unknown vectors \(N_{1}\) and \(N_{2}\). |
Output: The approximate solutions: \(N_{1}(t)\simeq N_{1}^T\Psi (t)\) and \(N_{2}(t)\simeq N_{2}^T\Psi (t)\). |
As a numerical example, we consider the nonlinear system of stochastic integral equations (54) with \(a_{1}=0.3,\,a_{2}=0.1, \,b_{1}=2.0,\,b_{2} = 1.5,\,\bar{\sigma }_{1}=0.2, \,\bar{\sigma }_{2}=0.4,\,N_{10}\) and \(N_{20}=1.0\). This problem is also solved by the proposed method for \(\hat{m}=48\) and \(N=80\). The behavior of the numerical solutions is shown in Fig 6.
5.3 The Duffing–Van der Pol Oscillator
Algorithm 3 |
Input:\(M\in {\mathbb {N}},\,k,N\in {\mathbb {Z}}^{+}\); Brownian motion process B(t); \(\alpha ,\,\bar{\sigma },\,X_{10}\) and \(X_{20}\). |
Step 1: Define the LWs \(\psi _{nm}(t)\) from Eq. (5). |
Step 2: Construct the LWs vector \(\Psi (t)\) from Eq. (11). |
Step 3: Compute the LWs matrix \(\Phi _{\hat{m}\times \hat{m}}\triangleq \left[ \Psi (0),\Psi \left( \frac{1}{\hat{m}-1}\right) ,\ldots ,\Psi (1)\right] \). |
Step 4: Compute the integration operational matrix P using Eqs. (31)–(33) and SOM \(\hat{P}_s\) using Eq. (25). |
Step 5: Compute the LWs stochastic operational matrix \(P_s=\Phi _{\hat{m}\times \hat{m}}\hat{P}_s\Phi ^{-1}_{\hat{m}\times \hat{m}}\). |
Step 6: Compute the vectors \(e^{T}\) using Eq. (8). |
Step 7: Compute the vector \(\widetilde{X}_{1}^{T}=X_{1}^{T}\Phi _{\hat{m}\times \hat{m}}\). |
Step 8: Put \(\displaystyle \left\{ \begin{array}{l} \displaystyle R_{1}(t)= \left( X_{1}^{T}-X_{10}e^{T}-X_{2}^{T}P\right) \Psi (t), \\ \displaystyle R_{2}(t)= \left( X_{2}^{T}-X_{20}e^{T}-\left( \alpha X_{1}^{T}\right. \right. \\ \left. \left. -\left( \widetilde{X}_{1}^{T}\right) ^{3}\Phi _{\hat{m}\times \hat{m}}^{-1}-X_{2}^{T}\right) P-\bar{\sigma } X_{1}^{T}P_{s}\right) \Psi (t) . \end{array}\right. \). |
Step 9: Construct the nonlinear system of algebraic equations: |
\(\displaystyle \left\{ \begin{array}{c} \displaystyle \left( R_{1}(t),\psi _{j}(t)\right) =\int _{0}^{1}R_{1}(t)\psi _{j}(t)\mathrm{d}t=0, \quad j=1,2,\ldots ,\hat{m}, \\ \displaystyle \left( R_{2}(t),\psi _{j}(t)\right) =\int _{0}^{1}R_{2}(t)\psi _{j}(t)\mathrm{d}t=0, \quad j=1,2,\ldots ,\hat{m}, \end{array}\right. \) |
Step 10: Solve the nonlinear system of algebraic equations in Step 9 and obtain the unknown vectors \(X_{1}\) and \(X_{2}\). |
Output: The approximate solutions: \(X_{1}(t)\simeq X_{1}^T\Psi (t)\) and \(X_{2}(t)\simeq X_{2}^T\Psi (t)\). |
5.4 Stochastic Brusselator problem
The deterministic Brusselator (\(\alpha =0\)) equation was developed at the occasion of a scientific congress in Brussels, Belgium, to develop a simple model for bifurcations in chemical reactions [51].
Algorithm 4 |
Input:\(M\in {\mathbb {N}},\,k,N\in {\mathbb {Z}}^{+}\); Brownian motion process B(t); \(\alpha ,\,\beta ,\,X_{0}\) and \(Y_{0}\). |
Step 1: Define the LWs \(\psi _{nm}(t)\) from Eq. (5). |
Step 2: Construct the LWs vector \(\Psi (t)\) from Eq. (11). |
Step 3: Compute the LWs matrix \(\Phi _{\hat{m}\times \hat{m}}\triangleq \left[ \Psi (0),\Psi \left( \frac{1}{\hat{m}-1}\right) ,\ldots ,\Psi (1)\right] \). |
Step 4: Compute the integration operational matrix P using Eqs. (31)–(33) and SOM \(\hat{P}_s\) using Eq. (25). |
Step 5: Compute the LWs stochastic operational matrix \(P_s=\Phi _{\hat{m}\times \hat{m}}\hat{P}_s\Phi ^{-1}_{\hat{m}\times \hat{m}}\). |
Step 6: Compute the vectors \(e^{T}\) using Eq. (8). |
Step 7: Compute the vectors \(\widetilde{X}^{T}=X^{T}\Phi _{\hat{m}\times \hat{m}}\) and \(\widetilde{Y}^{T}=Y^{T}\Phi _{\hat{m}\times \hat{m}}\). |
Step 8: Put \(\displaystyle \left\{ \begin{array}{l} \displaystyle R_{1}(t)=\left( X^{T}-X_{0}e^{T}-\left\{ \left( \beta -1\right) X^{T}\right. \right. \\ \left. \left. \!+\!\left[ \left( \left( \widetilde{X}^{T}\right) ^{2}\odot \widetilde{Y}^{T}\right) \!+\!2\left( \widetilde{X}^{T}\odot \widetilde{Y}^{T}\right) \right] \Phi _{\hat{m}\times \hat{m}}^{-1}\!+\!Y^{T}\right\} P\right. \\ \left. \qquad -\alpha \left\{ X^{T}+\left( \widetilde{X}^{T}\right) ^{2}\Phi _{\hat{m}\times \hat{m}}^{-1}\right\} P_{s}\right) \Psi (t),\\ \displaystyle R_{2}(t)\!=\! \left( Y^{T}-Y_{0}e^{T}\!+\!\left\{ \beta X^{T}\!+\!\left[ \left( \left( \widetilde{X}^{T}\right) ^{2}\odot \widetilde{Y}^{T}\right) \right. \right. \right. \\ \left. \left. \left. +2\left( \widetilde{X}^{T}\odot \widetilde{Y}^{T}\right) \right] \Phi _{\hat{m}\times \hat{m}}^{-1}+Y^{T}\right\} P\right. \\ \left. \qquad +\alpha \left\{ X^{T}+\left( \widetilde{X}^{T}\right) ^{2}\Phi _{\hat{m}\times \hat{m}}^{-1}\right\} P_{s}\right) \Psi (t). \end{array}\right. \). |
Step 9: Construct the nonlinear system of algebraic equations: |
\(\displaystyle \left\{ \begin{array}{c} \displaystyle \left( R_{1}(t),\psi _{j}(t)\right) =\int _{0}^{1}R_{1}(t)\psi _{j}(t)\mathrm{d}t=0, \quad j=1,2,\ldots ,\hat{m}, \\ \displaystyle \left( R_{2}(t),\psi _{j}(t)\right) =\int _{0}^{1}R_{2}(t)\psi _{j}(t)\mathrm{d}t=0, \quad j=1,2,\ldots ,\hat{m}, \end{array}\right. .\) |
Step 10: Solve the nonlinear system of algebraic equations in Step 9 and obtain the unknown vectors X and Y. |
Output: The approximate solutions: \(X(t)\simeq X^{T}\Psi (t)\) and \(Y(t)\simeq Y^T\Psi (t)\). |
As a numerical example, we consider the stochastic Brusselator problem (70) with \(\beta =2\) and some different values \(\alpha \) over the interval [0, 6.5], starting at \(\left( X_{0},Y_{0}\right) =(-0.1,0.0)\). This problem is also solved by the proposed method for \(\hat{m}=80\) and \(N=35\). The behavior of the numerical solutions for the stochastic Brusselator problem in the phase space are shown in Figs. 11 and 12. The non-noisy curve is the corresponding deterministic limit cycle.
6 Conclusion
Some SDEs can be written as nonlinear stochastic Volterra integral equations given in (1). It may be impossible to find exact solutions of such problems. So, it would be convenient to determine their numerical solutions using a stochastic numerical method. In this paper, the SOM of Itô-integration for the LWs was derived and applied for solving nonlinear stochastic Itô–Volterra integral equations. In the proposed method, a new technique for commuting nonlinear terms in problems under study was presented. Also some useful properties of the LWs were derived and used to solve problems under consideration. Applicability and accuracy of the proposed method were checked on some examples. Moreover, the results of the proposed method were in a good agreement with the exact solutions. Furthermore, as some applications, the proposed computational method was applied to obtain approximate solutions for some stochastic problems in the mathematics finance, biology, physics and chemistry.
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