The Analytic solution for some non-linear stochastic differential equation by linearization (Linear-transform)

Abstract


Inroduction:
The stochastic differential equations are increasingly used in many scientific and industrial fields which demonstrate the importance of stochastic modeling, since they can be used to any address problem caused by noise, accidents, etc. [1] Sunday Fadugba and Ado Ekiti (2013) [2], study and explain the convergence of the Euler-Maruyama method and Milstein scheme for the solution of stochastic differential equations, Elborala M.M. and M.I. Youssef (2019) [3] extends some results about the existences of solution for a functional nonlocal SDEs under suitable conditions, Xu, J., et.al (2012) [4] explain and study Uniqueness and explosion time of solutions of stochastic differential equations driven by fractional Brownian motion, D. Kim and D. Stanescu, (2008) [5] used the Runge-Kutta methods for Low-storage stochastic differential equations. In this paper, we study the linearization to the non-linear SDEs in order to find the analytic solution to the obtained linear stochastic differential equations, there are several examples with numerical solution are used to validate the results. Let X(t) satisfied the following stochastic differential equation Where W(t) denote wiener process , ( ( )) and ( ( )) are deterministic non-linear functions and ( ) represent the differential form of ( ), equation (1) Equation (2) is called (Ito-formula). [ 6] 2: Preliminaries and Results: We introduce some fundamental and necessary concept steps for the transformation of non-linear SDEs to reduce to linear and then find the analytic solution.

(Stochastic integral) [8]
A stochastic integral is an integral which is defined as a sum more than integration and it is increased by the rise in time on the Wiener process trajectory. That is:

Definition2.3. (The analytic solution of linear SDE) [9]
Let we have the general linear stochastic differential equation given by ( ( ) ( )) ( ( ) ( )) … (4) Then the solution will be in the form: 3. Linearization method for non-linear (SDEs): [10] The linearization method means to find a suitable function which reduce such nonlinear stochastic differential equation to linear in order to find their analytic solution, we explain it by the following main steps: Step 1.
Suppose we have the following nonlinear stochastic differential equation: ( ) ( ( )) ( ( )) ( ) ; y( )= … (7) Here f(t, y) and g(t, y) are real non-linear functions, Suppose that the function ( ( )) smooth and has continuous derivatives , that reduce equation (7) to some linear stochastic differential equation. Which we want to find it. By using Ito formula to ( ( )) , we get Step 2. suppose that equation (8) transformed to linear stochastic differential equation of the form: .. (9) Step3. Determined the parameters ( , , , ) for the linearity by using the comparison of eq. (8) and eq. (9), then we have By solving equation (10.1) and equation (10.2), we get the values of the parameter of the linear equation (9) Step4. The analytic solution for the obtained linear stochastic differential equation has the form: By using transformation ( ) we obtain the solution of eq. (7).

3.1.The method of finding the analytic solution:
In this paragraph, we explain the main step of finding the suitable function which transformed the nonlinear stochastic differential equation to linear stochastic differential equation. From eq(10.2), let ( ) and ( ) , then we get The derivative of equation (11) is equal to zero, i.e.
From equation (4) we obtain the general solution of the transformed linear stochastic differential equation by the transformed function rewrite y(t)= ( ) . [11]

4.Numerical methods for solving nonlinear SDEs:
Since many stochastic differential equations have unknown solution, so it is necessary to derive numerical methods to generate approximations to the exact solution.

Euler -Maruyama method: [12] [13]
Euler-Maruyama method is similar to Euler method for solving ordinary differential equations that are presented from the point of view of Taylor's algorithm which greatly simplifies accurate analysis.
Euler approximation is one of the simplest discrete time estimates for Ito-Taylor expansion .let be an Ito process on , satisfying the stochastic differential equation

Milstein's method: [12] [13]
The Milstein's method obtained by add the following second order term for Ito integral to the Euler-Maruyama scheme From the Ito-Taylor expansion, we obtain (Milstein formula) given Eq. (21) is called Milstein's formula.

Examples:
In this paragraph we give some examples in order to explain the methods (analytically and numerically).

CONCLUSION AND FUTURE WORKS:
Through our study we applied the linearization method (linear -transform) for the nonlinear stochastic differential equation(SDES) by applying Ito formula. After we obtained a suitable function ( ) for the reducible linear stochastic differential equation , we find the analytic solution. lastly we compare the exact solution with the numerical solution (Euler-Maruyama and milestone) methods by several examples, in the numerical solution we see that the Milstein method better than Euler-Maruyama in convergence with the exact solution for the first example while in the square root example it seems that they are the same.
As a future studies one can study the linearization of some nonlinear(harmonic) stochastic differential equation by using stratonovich formula for their solution and compare it with Ito formula.