KARHUNEN-LOÉVE EXPANSION OF BROWNIAN MOTION FOR APPROXIMATE SOLUTIONS OF LINEAR STOCHASTIC DIFFERENTIAL MODELS USING PICARD ITERATION

This work presents an application of the Picard Iterative Method (PIM) to a class of Stochastic Differential Equations where the randomness in the equation is considered in terms of the Karhunen-Loéve Expansion finite series. Two applicable numerical examples are considered to illustrate the convergence of the approximate solutions to the exact solutions and also to check the efficiency of the method. The results obtained show clearly that accuracy will be more visible by increasing the number of terms in the iteration. Thus, it is recommended for nonlinear financial models of different classes of Stochastic Differential Equations.


INTRODUCTION
Stochastic Differential Equations (SDEs) are used to model random phenomena. Recently, SDEs have become a typical model for quantifying financial products like options, stocks, interest rates, 1713 KARHUNEN-LOÉVE EXPANSION OF BROWNIAN MOTION and some other underlying assets [1]. Merton's work brought phenomena advancement in the application of SDEs to financial problems [2]. Several other areas, such as population dynamic, option pricing, and so on have been discussed in [3]. The methods for solving different classes of stochastic differential equations are presented in [4,5], but the explicit solutions are limited for such equations [6]. Picard Iteration Method (PIM) is one of the effective numerical methods that can be used for solving differential equations. It transforms a differential equation into an integral equation. This method can be used to get the approximate analytical solutions of linear and nonlinear class of differential equations. Youssef, in [7] used Picard's Iterative Method with Gauss-Seidel Method to solve some IVP. Xiaohui et al., in [8] applied PIM to singular fractional differential equation. In [9], Edeki et al., used PIM and Differential Transforms Method to obtain the solutions of linear and nonlinear differential equations. Rach, in his research work, used Adomian Decomposition Method (ADM) and PIM [10]. ADM and PIM were compared by Bellomo and Sarafyan [11], Antonis et al., used Picard's for the approximation of stochastic differential equations by applying it to Libor models [12]. Little research has been done on the application of PIM to stochastic differential models. Therefore, in this paper, our concern is to modify this numerical method by considering the Stratonovich linear stochastic differential models where the random path of the SDEs are being taken care of by the Karhunen-Loeve expansion of Brownian Motion.
The rest of the work is organized as follows. In section 2, linear SDEs are discussed, the method of solution is discussed in section 3, two numerical examples are given in section 4, and lastly, a concluding remark is presented in section 5.

LINEAR STOCHASTIC DIFFERENTIAL MODEL
Considering the Stratonovich SDE of the form:  [13].
Basically, the SDE in (2.1) can be written in Itô form as: In integral form, the SDE in (2.1) is written as: 3) The first integral and second integral in (2.3) are known as the Riemann-Stiltjes and stochastic integral, respectively.

METHODS OF SOLUTION
This section presents the concepts of the Picard Iterative Method (PIM) and Karhunen-Loeve Expansion (K-LE).

Picard Iterative Method (PIM)
Let us consider the differential equation of this form: .
First order differential equations fall under this type of equation, and PIM is one of the suitable methods for handling this type of differential equation. Now, by integrating both sides of (3.1), xt is appearing on both sides of equation (3.3) for arbitrary t , we therefore adopt this iterative process by choosing an initial condition: In what follows, the approximation of (3.1) is: The existence and uniqueness of this method have been studied by [9,14].

Karhunen-Loeve Expansion (K-LE) finite series of Brownian Motion
The K-L expansion is an extension of the Fourier Transform (FT). Its analysis is from deterministic functions to probabilistic form. It is also referred to as a bi-orthogonal stochastic process expansion [15]. Stefano [16], in his work, used the K-L expansion to characterize the Brownian Motion (Wiener process). In this expansion [16], represents a stochastic process in terms of sequence of identically and independent sample variables   Therefore, The K-L expansion for Brownian Motion can be represented as: form a basis of orthogonal function [6,16]. Replacing the random path of SDE (2.1) with (3.7), we obtain: For the purpose of this research work, j z is generated using a mathematical computer software.

CONCLUDING REMARKS
In this research paper, we have considered a procedure for obtaining an approximate analytical solution of a class of SDEs. The approach is by Picard Iterative Method (PIM), which transforms the differential equation into an equivalent integral equation, provided that the Lipchitz condition is satisfied and also, by converting the entire random sample into piecewise-continuous polynomials. In comparing the results obtained, it showed that more accuracy would be attained with an increasing number of terms. So based on these, further investigation can be carried out on the higher order of different classes of SDEs.

ACKNOWLEDGEMENT
The support of Covenant University Centre for Research Innovation and Development (CUCRID) is deeply appreciated.