Functional Integral Approach to the Solution of a System of Stochastic Differential Equations

A new method for the evaluation of the characteristics of the solution of a system of stochastic differential equations is presented. This method is based on the representation of a probability density function p through a functional integral. The functional integral representation is obtained by means of the Onsager-Machlup functional technique for a special case when the diffusion matrix for the SDE system defines a Riemannian space with zero curvature.


Introduction
Stochastic differential equations (SDE) are very popular in physics, chemistry, biology and etc. [1,2].There are many publications devoted to different aspects of SDE theory [1][2][3].Except for some very special cases, SDE cannot be solved analytically and numerical methods are needed.The most common approach to the numerical solution of SDE is based on the discretization of a time interval and numerical simulation of SDE solutions at discrete times [4][5][6][7].In this work a new method for evaluating of the characteristics of a SDE solution is presented.This method is based on the representation of the probability density function (PDF) p through a functional integral.The Onsager-Machlup functional technique is used to get the functional integral representation [8][9][10][11].For one dimension case this approach was considered in [12].In this work this approach is given for a system of SDE.Here we consider the Onsager-Machlup functional technique only for the flat space when the diffusion matrix for a SDE system defines a Riemannian space with zero curvature.
In section 2 the representation of PDF p through a functional integral is discussed.In section 3 the method of approximate evaluation of functional integrals is presented.Using the minimal action principle [13], we can distinguish among all trajectories the classical trajectory for which the action S takes the extremal value.The classical trajectory is found as the solution of the multidimensional Euler-Lagrange equation.Further, we use the expansion of the action with respect to the classical trajectory to compute the integral.In section 4 approximate and exact values for the expectations of some functionals of solution of specific system of SDE are presented.

Representation of characteristics through functional integral
We consider the SDE system (the Ito interpretation) where the solution x(t) satisfies the initial condition x(t 0 ) = x 0 , w(t) is a d-dimensional Wiener process.By means of the Onsager-Machlup functional technique the PDF can be represented in a form of the functional integral where G denotes a matrix of elements

Evaluation of functional integrals
The quantity L 0 ( x(τ), ẋ(τ)) in ( 2) can be considered as the Lagrangian of the system, and the quantity S = t t 0 L 0 ( x(τ), ẋ(τ)) dτ can be considered as its action.Using the minimal action principle [13], we can distinguish among all trajectories the classical trajectory x cl for which the action S takes the extremal value.The classical trajectory is found as the solution of the multidimensional Euler-Lagrange equation Further, to compute the integral, we can use the decomposition of the action with respect to the classical trajectory The second order variation δ 2 S [ x cl (τ)] can be written in the form δx i Λ i j δx j dτ, After these transformations the formula ( 2) is written in the form δx i Λ i j δx j dτ .
By evaluating the functional integral from the exponential function with the quadratic form in exponent we obtain result for the PDF p.

Numerical results
In this section we consider an application of the proposed method for the evaluation of the expectation of some functionals of the SDE system solutions x 1 (t 0 ) = x 10 , x 2 (t 0 ) = x 20 .
The approximate and exact values of the expectation of some functionals of the SDE system solution (4) for concrete coefficients and initial conditions are presented in Tables 1 and 2. The form of the functionals is chosen in such way that we can calculate the exact values of the expectations.To compute the function x cl approximately we use the partitioning of the interval [t 0 , t] into n parts.

Conclusion
A new method for evaluating of the characteristics of a SDE solution is proposed.From results presented in Tables 1 and 2 one can see that the increase of the parameter n leads to the decreasing differences between exact and approximate values.

Table 1 .
Approximate and exact values for expectations for a 1