A very important special case is the linear stochastic differential equation
(7.1) |
where and are deterministic functions and is the differential of the Brownian motion process. This is a first order differential equation with an additive noise. A more general case is when the function is random , but then we have a product of random functions. The integral form of the differential equation is
(7.2) |
where the first integral is the Riemann stochastic integral and the second in general is an Itô stochastic integral and in the above case is a Wiener stochastic integral.
Let us introduce the Itô stochastic integral by plausible arguments. Rigorous proves need a lot of advanced mathematics. The stress will be on applications and the use of numerical methods. We are going to discuss the Itô stochastic integral
(7.3) |
Assume , is a scalar Brownian motion process with variance parameter and the function is defined on . Let us introduce a partition , and a step function
(7.4) |
where is independent of , and . The Itô integral is defined by
(7.5) |
In view of the independence condition the expected value is zero
(7.6) |
Let us take another function and the corresponding step function and consider the expected value
(7.7) |
where . The last expression results from the independence conditions.
Now let us consider a sequence of step functions , converging to the random function in the sense that
(7.8) |
Finally the following theorem is true:
Let the random functions and satisfy the condition that they are independent of , for all and the conditions
Then their Itô integrals are well defined as
(7.9) |
with basic properties
Let us consider the following linear stochastic differential equation with constant coefficients
(7.10) |
where and are constants and is the differential of the Brownian motion process with variance parameter equal to one. This is a first order differential equation with an additive noise. The general solution of the homogeneous differential equation is
(7.11) |
where is the random initial value for . Addition of a particular solution of the non homogeneous equation yields the general solution
(7.12) |
where the integral on the right side is an Itô (Wiener) integral. Let us assume that the initial value is normally distributed with a mean value equal to zero. The mean value of the Itô integral is zero. Thus the mean value of the random function is zero. The correlation (covariance) function is
(7.13) |
From the basic formula for the Itô integrals it follows
(7.14) |
Upon integration it follows
(7.15) |
where .
When , tends to infinity in such a way that is constant, then in the limit
(7.16) |
The process is asymptotically stationary in the wide sense and because it is Gaussian it is strictly asymptotically too. If the variance of the initial condition is equal to the process is strictly stationary.
In general the process is not stationary. The variance of the process is
(7.17) |
The variance changes from at the time to the asymptotic value when tends to infinity. The derivative with respect to time is
(7.18) |
It is easy to show that the differential equation for the evolution of the variance in our example is
(7.19) |