A computational method for solving stochastic Itô–Volterra integral equations based on stochastic operational matrix for generalized hat basis functions

doi:10.1016/j.jcp.2014.03.064

Journal of Computational Physics

Volume 270, 1 August 2014, Pages 402-415

https://doi.org/10.1016/j.jcp.2014.03.064 Get rights and content

Abstract

In this paper, a new computational method based on the generalized hat basis functions is proposed for solving stochastic Itô–Volterra integral equations. In this way, a new stochastic operational matrix for generalized hat functions on the finite interval $[0, T]$ is obtained. By using these basis functions and their stochastic operational matrix, such problems can be transformed into linear lower triangular systems of algebraic equations which can be directly solved by forward substitution. Also, the rate of convergence of the proposed method is considered and it has been shown that it is $O (\frac{1}{n^{2}})$ . Further, in order to show the accuracy and reliability of the proposed method, the new approach is compared with the block pulse functions method by some examples. The obtained results reveal that the proposed method is more accurate and efficient in comparison with the block pule functions method.

Introduction

Stochastic and deterministic functional equations are fundamental for modeling science and engineering phenomena. 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 will be very useful in application, because of the fact that they arise in many situations. For example, stochastic integral equations arise in the stochastic formulation of problems in reactor dynamics [1], [2], [3], in the study of the growth of biological populations [4], in the theory of automatic systems resulting in delay-differential equations [5], 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 applications and finance [6], [7], [8], [9], [10], [11], [12], [13]. These systems are often dependent on a noise source, like e.g. a Gaussian white noise, governed by certain probability laws, so that modeling such phenomena naturally requires the use of various stochastic differential equations [4], [14], [15], [16], [17], [18], [19], [20], or in more complicated cases, stochastic Volterra integral equations and stochastic integro-differential equations [21], [22], [23], [24], [25], [26], [27], [28]. 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 [1], [2], [4], [5], [6], [7], [8], [14], [23], [24], [25].

Babolian and Mordad [29] have used hat basis function for solving systems of linear and nonlinear integral equations of the second kind by the hat functions. In [30], Tripathi et al. have used generalized hat basis functions to obtain approximate solutions of linear and nonlinear fractional differential equations.

In this paper, generalized hat basis functions will be used to solve the following linear stochastic Volterra integral equation: $X (t) = f (t) + \int_{0}^{t} K_{1} (τ, t) X (τ) d τ + \int_{0}^{t} K_{2} (τ, t) X (τ) d B (τ), t \in [0, T],$ where $X (t)$ , $f (t)$ , $K_{1} (τ, t)$ and $K_{2} (τ, t)$ , for $t, τ \in [0, T]$ , are some stochastic processes defined on the same probability space $(Ω, F, P)$ , $X (t)$ is the unknown function to be found, $B (t)$ is a Brownian motion process and the second integral in (1) is the Itô integral [31], [32].

In order to compute the approximate solution on this equation, we first describe some properties of the generalized hat basis functions. Then, the new operational matrix of stochastic integration for the generalized hat functions is derived and applied to obtain approximate solution for the under study problem. Convergence and error analysis of the proposed method are also investigated and the efficiency of our method is shown on some concrete examples.

It is worth noting that, the main advantage of the proposed method is that it reduces the problem under consideration into solving a linear lower triangular system of algebraic equations by expanding the solution in generalized hat functions with unknown coefficients and using the operational matrices of integration and stochastic integration.

This paper is organized as follows: In Section 2, a brief review of the generalized hat functions and their properties is described. In Section 3, the stochastic integration operational matrix of generalized hat functions is obtained. In Section 4, the proposed method is described. In Section 5, convergence and error analysis of the proposed method are investigated. In Section 6, some numerical examples are presented. Finally a conclusion is drawn in Section 7, where a discussion on future applications is also given.

Section snippets

The generalized hat basis functions and their properties

The traditional hat basis functions are continuous functions, also called triangle, tent or triangular functions are defined on the interval $[0, 1]$ . They are fundamental functions in signal analysis, since their Fourier transform coincide with the square of the sinc-function. Sinc-function is the Fourier transform of the rectangular (or box) function. The translated and dilated instances of the sinc-function can be used to define the so-called Shannon wavelets, which have been shown to be an

The stochastic integration operational matrix

Theorem 3.1

Let $Φ (t)$ be the vector defined in (9). The Itô integral of $Φ (t)$ can be expressed as: $\int_{0}^{t} Φ (τ) d B (τ) ≃ P_{s} Φ (t),$ where the $(n + 1) \times (n + 1)$ stochastic operational matrix of integration is given by: $P_{s} = (\begin{matrix} 0 & α_{0} (h) & α_{0} (h) & 0 & \dots & α_{0} (h) & α_{0} (h) \\ 0 & B (h) + α_{1} (h) & β_{1} (h) & 0 & \dots & β_{1} (h) & β_{1} (h) \\ 0 & 0 & B (2 h) + α_{2} (h) & β_{2} (h) & \dots & β_{2} (h) & β_{2} (h) \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & B ((n - 1) h) + α_{n - 1} (h) & β_{n - 1} (h) \\ 0 & 0 & 0 & 0 & \dots & 0 & B (T) + α_{n} (h) \end{matrix}),$ and ${\begin{matrix} α_{0} (h) = \frac{1}{h} \int_{0}^{h} B (τ) d τ, \\ α_{i} (h) = - \frac{1}{h} \int_{(i - 1) h}^{i h} B (τ) d τ, & i = 1, 2, \dots, n, \\ β_{i} (h) = - \frac{1}{h} (\int_{(i - 1) h}^{i h} B (τ) d τ - \int_{i h}^{(i + 1) h} B (τ) d τ), & i = 1, 2, \dots, n - 1 . \end{matrix}$

Proof

By considering definitions of $ϕ_{i} (t)$ $(i = 0, 1, \dots, n)$ , and

The proposed method using stochastic operational matrix

In this section, we apply the stochastic operational matrix for generalized hat basis functions to solve linear stochastic Volterra integral equation: $X (t) = f (t) + \int_{0}^{t} K_{1} (τ, t) X (τ) d τ + \int_{0}^{t} K_{2} (τ, t) X (τ) d B (τ), t \in [0, T],$ where $X (t)$ , $f (t)$ , $K_{1} (τ, t)$ and $K_{2} (τ, t)$ , for $τ, t \in [0, T]$ , are the stochastic processes defined on the same probability space $(Ω, F, P)$ , and $X (t)$ is an unknown function to be found. Also $B (t)$ is a Brownian motion process and the second integral in (33) is an Itô integral.

For solving this problem, we

Error analysis

In this section, we investigate the error analysis of the proposed method. From now on, ‖‖ denotes the sup-norm which for any continuous function $f (t)$ is defined on the interval $[0, T]$ by: $‖ f (t) ‖ = \sup_{t \in [0, T]} | f (t) | .$

Theorem 5.1

Suppose $f (t) \in C^{2} ([0, T])$ and $e_{n} (t) = f (t) - f_{n} (t)$ , $t \in I = [0, T]$ , where $f_{n} (t) = \sum_{i = 0}^{n} f (i h) ϕ_{i} (t)$ is the generalized hat functions expansion of $f (t)$ . Then we have: $‖ e_{n} (t) ‖ ⩽ \frac{T^{2}}{2 n^{2}} ‖ f^{″} (t) ‖,$ and so the convergence is of order two, that is $‖ e_{n} (t) ‖ = O (\frac{1}{n^{2}}) .$

Proof

Let $e_{n j} (t) = {\begin{matrix} f (t) - f_{n} (t), & t \in D_{j}, \\ 0, & t \in I - D_{j}, \end{matrix}$ where $D_{j} = {t | j h ⩽ t <$

Numerical examples

In this section, we consider some numerical examples which their exact solutions are available to illustrate the efficiency and reliability of the proposed method.

Note from now on, n is the number of basis functions and m is the number of iterations.

Example 1

Let us consider the following linear stochastic Volterra integral equation [6]: $X (t) = 1 + \int_{0}^{t} τ^{2} X (τ) d τ + \int_{0}^{t} τ X (τ) d B (τ), (τ, t) \in [0, 0.5],$ where $X (t)$ is a known stochastic process defined on the probability space $(Ω, F, P)$ , and $B (t)$ is a Brownian motion process

Applications in the mathematical finance

The market consists of a riskless cash bond, ${B (t)}_{t ⩾ 0}$ , and a single risky asset with price process ${S (t)}_{t ⩾ 0}$ governed by [6]: ${\begin{matrix} d B (t) = r (t) B (t) d t, B_{0} = 1, \\ d S (t) = μ (t) S (t) d t + σ (t) S (t) d W (t), \end{matrix}$ or ${\begin{matrix} d B (t) = r (t) B (t) d t, B_{0} = 1, \\ S (t) = S_{0} + \int_{0}^{t} μ (τ) S (τ) d τ + \int_{0}^{t} σ (τ) S (τ) d W (τ), \end{matrix}$ where ${W (t)}_{t ⩾ 0}$ is a P-Brownian motion generating the filtration ${F (t)}_{t ⩾ 0}$ and $r (t)$ , $μ (t)$ and $σ (t)$ are ${F (t)}_{t ⩾ 0}$ -predictable processes [6], [36].

A solution to these equations should take the form [6]: $B (t) = \exp (\int_{0}^{t} r (τ) d τ),$ $S (t) = S_{0} \exp (\int_{0}^{t} (μ (τ) - \frac{1}{2} σ_{s}^{2}) d τ + \int_{0}^{t} σ ($

Conclusion

Some stochastic differential equations can be written as stochastic Volterra integral equations given in (1). It may be impossible to find the exact solutions of such problems. So, it would be convenient to determine their numerical solutions using a stochastic numerical analysis. Employing generalized hat functions as basis functions to solve linear stochastic Volterra integral equations is very simple and effective. In this paper, the stochastic operational matrix of Itô-integration for

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A computational method for solving stochastic Itô–Volterra integral equations based on stochastic operational matrix for generalized hat basis functions

Abstract

Introduction

Section snippets

The generalized hat basis functions and their properties

The stochastic integration operational matrix

The proposed method using stochastic operational matrix

Error analysis

Numerical examples

Applications in the mathematical finance

Conclusion

Appl. Math. Model.

Math. Comput. Model.

Comput. Math. Appl.

Comput. Math. Appl.

Appl. Math. Model.

Math. Comput. Model.

Comput. Math. Appl.

J. Comput. Phys.

J. Differ. Equ.

Acta Math. Sci.

J. Funct. Anal.

Stoch. Process. Appl.

Comput. Math. Appl.

Commun. Nonlinear Sci. Numer. Simul.

On a system of integro-differential equations occurring inreactor dynamics

J. Math. Mech.

On a system of integro-differential equations occurring in reactor dynamics

SIAM J. Appl. Math.

Spatial Besov regularity for semilinear stochastic partial differential equations on bounded Lipschitz domains

Int. J. Comput. Math.

Time-Lag Control Systems