2454

Page 1 of 21

Journal for Studies in Management and Planning

Available at http://internationaljournalofresearch.org/index.php/JSMaP

e-ISSN: 2395-0463

Volume 01 Issue 07

August 2015

Available online: http://internationaljournalofresearch.org/ P a g e | 126

N-dimensional stochastic Runge-Kutta scheme for Stochastic

Differential Equation Model

1E.O.Ogbaji, 2E.S.Onah, 3T.Aboiyar and 4A.R.Kimbir

1Department of Mathematics and Statistics, Federal University Wukari

(Ogbajieka@yahoo.com)

2,3,4Department of Mathematics/Statistics/Computer Science University Of Agriculture, Makurdi

Abstract

We formulated a four-compartment stochastic

differential equation which follow geometric

Brownian motion model. We showed the

existence and uniqueness of the solution

.Stochastic Runge-Kutta scheme was used to

solve the model.N- dimension stochastic Rung- Kutta scheme ,existence and uniqueness of

solution of stochastic differential equation was

established.

Key Words: geometric Brownian motion

model, stochastic differential equation,

Stochastic Runge-Kutta scheme

1.0 Introduction

The concept of Stochastic Differential

Equation (SDE) has been initiated by Einstein

in 1905, [56]. In his article he presented a

mathematical connection between microscopic

random motion of particles and the

macroscopic diffusion equation. Later it has

been seen that the stochastic differential

equation (SDE) model plays a prominent role

in a range of application areas such as physics,

chemistry, mechanics, biology,

microelectronics, economics and finance.

Earlier the SDEs were solved by using Ito

integral as an exact method which is discussed

in, [41]. But using exact method it is noticed

that there occur some difficulty to study

nontrivial problems and hence approximation

methods are used. In this context various

authors have given their contribution in these

field but we have mentioned which are directly

related to this problem. In [55] defined general

Runge-Kutta approximations for the solution

of stochastic differential equations and there

was given an explicit form of the correction

term. This work was carried out and then [37]

are discussed about the numerical solutions of

stochastic differential equation in detail. [49]

added discrete time strong and weak

approximation methods for the numerical

methods to find the solution of stochastic

differential equations. Next, [29] gave a major

contribution in this field to solve the

approximate solutions of stochastic differential

Page 2 of 21

Journal for Studies in Management and Planning

Available at http://internationaljournalofresearch.org/index.php/JSMaP

e-ISSN: 2395-0463

Volume 01 Issue 07

August 2015

Available online: http://internationaljournalofresearch.org/ P a g e | 127

equations and discussed few problems. Further

[30] investigated nonlinear stochastic

differential equations numerically. They

presented two implicit methods for Ito

stochastic differential equations (SDEs) with

Poisson-driven jumps. The first method is a

split-step extension of the backward Euler

method and the second method arises from the

introduction of a compensated, martingale,

form of the Poisson process. In this context

different authors have tried for various other

diffusion and application based problems. [28]

solved stochastic point kinetic reactor

problem. They modelled the point stochastic

reactor problem into ordinary time dependent

stochastic differential equation and studied the

stochastic behaviour of the neutron flux.

2.0 Formulation of Stochastic Differential

Equation Model

We formulate a four-compartment stochastic

differential equation for stock markets model.

Stochastic differential equation consists of

deterministic differential equations and white

noise term ‘dwt’. The white noise term is

stochastic in nature. Our formulation follow

geometric Brownian motion model which is

one form of Ito stochastic differential

equation. We use Figure1 Schematic diagram

of stock market interaction for the formulation.

Definition of Variables and Parameters

St Stock price

Return on investment

Stock price volatility

Ut Return on investment volatility

μ Rate of return on investment with reinvestment,

X Mean reversion of Vt,

Y Volatility of Vt,

Page 3 of 21

Journal for Studies in Management and Planning

Available at http://internationaljournalofresearch.org/index.php/JSMaP

e-ISSN: 2395-0463

Volume 01 Issue 07

August 2015

Available online: http://internationaljournalofresearch.org/ P a g e | 128

J Mean reversion of Ut,

N Rate of return on investment without reinvestment

 X

Figure1 Schematic diagram of stock market interaction

The stochastic differential equation model with respect to the stock price compartment is given as,



St +

t Vt

ξ(t)

where

  0