Let us start our consideration with the case of linear stochastic differential equation. Let us consider the following first order, homogeneous differential equation
(6.1) |
Looking at the realizations it is a deterministic differential equation
(6.2) |
The randomness is in the initial condition. Let us assume that the initial condition is a normally distributed random variable with mean value and variance equal to . The solution of the differential equation is
(6.3) |
The process is Gaussian. The mean value and correlation functions are
(6.4) |
The variance is
(6.5) |
As a second example let us consider the following set of two differential equations
(6.6) |
The randomness is in the initial conditions. Let us assume that the initial condition for is a normally distributed random variable with mean value and variance equal to . The initial conditions for are independent, normally distributed; the mean value is , and the variance is . The solution of the differential equation is
(6.7) |
The mean values of the stochastic processes are
(6.8) |
and the elements of the correlation matrix are
(6.9) | |
Let us consider the following first order, not homogeneous differential equation with the initial value :
(6.10) |
On the right side is a linear function in time with random independent coefficients that are normally distributed with zero mean values and variances and. The realizations have to be computed from the corresponding deterministic equation:
(6.11) |
The solution for a realization is
(6.12) |
The solution for the random case (for the family of realizations) is
(6.13) |
The mean value and the correlation functions are
(6.14) | |
The stochastic process is continuous and differentiable. The second mixed partial derivative is a constant, thus all derivatives exist.