Solution of Stochastic Non-Homogeneous Linear First-Order Difference Equations

In this paper, the closed form solution of the non-homogeneous linear first-order difference equation is given. The studied equation is in the form: xn = x0 + bn, where the initial value x0 and b, are random variables.


Introduction
A difference equation or dynamical system describes the evolution of some variable over time.The value of this variable in period t is denoted by x n .The time index n takes on discrete values and typically runs over all integer numbers Z, e.g.t = ⋅⋅⋅, −2, −1, 0, 1, 2,⋅⋅⋅.By interpreting t as the time index, we have automatically introduced the notion of past, present and future [1].
Many problems in Probability give rise to difference equations.Difference equations relate to differential equations as discrete mathematics relates to continuous mathematics.Anyone who has made a study of differential equations will know that even supposedly elementary examples can be hard to solve.By contrast, elementary difference equations are relatively easy to deal with.
Aside from Probability, Computer Scientists take an interest in difference equations for a number of reasons.For example, difference equations frequently arise while determining the cost of an algorithm in big-O notation [2].In engineering, difference equations arise in control engineering, digital signal processing, electrical networks, etc.In social sciences, difference equations arise to study the national income of a country and then its variation with time, Cobweb phenomenon in economics [3], etc.Similar to differential equation, difference equation is the most powerful instrument for the treatment of discrete processes.
Stochastic difference equations are difference equations with random parameters.Such equations arise in various disciplines, for example economics [4], physics, nuclear technology, biology and sociology.In this paper, we present a new technique based on the direct transformation technique to express analytically the probability density function of the general solution of stochastic linear 1 st order difference equations.

Linear 1st Order Difference Equations
A typical linear non-homogeneous first order difference equation is given by: 1 0 , , 0 where 0 n a ≠ , n a and n b are real-valued functions.The unique solution of (1) may be found as follows: ( ) Now by induction, we can show: To establish this, assume that (2) holds for order n.Then from (1) , hence (2) gives: There is a special case of (1) that is important in many engineering and business applications, is given by: This case is studied throughout this paper where x 0 and b, are random variables.

Direct Transformation Technique
There are several versions of the Direct Transformation Technique in the literature [5]- [9]; we will provide a simple one for 1-dimension variable then for N-dimension variables.

1-dimension variable
Let X be a continuous random variable with probability density function (pdf), ( ) X f x and defined over an interval I. Let g: I R → be a continuous one-to-one function, i.e. its inverse h: J I → ( ) Example: X has probability density function given by ( ) ( ) .

N-dimension variables
Let ( ) , , , n X X X  n continuous random variable with given joint probability density function ( ) . Consider a one-to-one transformation function :  is continuous with respect to every variable and its partial derivatives is continuous as well.Then the joint probability density function of ( ) is given by: ( ) ( ) ( ) (5)

Application
In this section, we will solve the stochastic difference equation .The joint probability density function of 1 y and 2 y (Figure 1) is given by: ( ) Now to find the probability density function of n x , i.e. 2 y we must find the marginal probability density function of Y 2 (Figure 2) by integrating out of Y 1 : We can show easily that ( ) x, α, and β are strictly positive real numbers).Let 2e x Y = − , we want to find the probability density function of Y, (3)), where 0 x and b are random variables with the known joint probability density function, by finding the probability density function of n x using(5).So in this stochastic difference equation we have two random variables ( 0 x , b ) as input and one random variable ( ) n x as output.For example, let ( )