Data Analysis

STOCHASTIC DIFFERENTIAL EQUATIONS

Let us start our consideration with the case of linear stochastic differential equation. Let us consider the following first order, homogeneous differential equation

\frac{d}{d t} Y (t, ω) + η Y (t, ω) = 0

(6.1)

Looking at the realizations it is a deterministic differential equation

\frac{d}{d t} y (t) + η y (t) = 0

(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 $m = 0$ and variance equal to $σ^{2}$ . The solution of the differential equation is

Y (t, ω) = A (ω) e^{- η t}

(6.3)

The process is Gaussian. The mean value and correlation functions are

m (t) = 0

R (t_{1}, t_{2}) = σ^{2} e^{- η (t_{1} + t_{2})}

(6.4)

The variance is

P (t, t) = R (t, t) = σ^{2} e^{- 2 η t}

(6.5)

As a second example let us consider the following set of two differential equations

\begin{matrix} \frac{d}{d t} Y_{0} (t, ω) + η Y_{0} (t, ω) = 0 \\ \frac{d}{d t} Y_{1} (t, ω) + η Y_{1} (t, ω) = η Y_{0} (t, ω) \end{matrix}

(6.6)

The randomness is in the initial conditions. Let us assume that the initial condition for $Y_{0} (0, ω) = A_{0} (ω)$ is a normally distributed random variable with mean value $m_{0} = 0$ and variance equal to $σ_{0}^{2}$ . The initial conditions for $Y_{1} (0, ω) = A_{1} (ω)$ are independent, normally distributed; the mean value is $m_{1} = 0$ , and the variance is $σ_{1}^{2}$ . The solution of the differential equation is

\begin{matrix} Y_{0} (t, ω) = A_{0} (ω) e^{- η t} \\ Y_{1} (t, ω) = η [A_{1} (ω) + t A_{0} (ω)] e^{- η t} \end{matrix}

(6.7)

The mean values of the stochastic processes are

m_{0} (t) = 0

m_{1} (t) = 0

(6.8)

and the elements of the correlation matrix are

$R_{Y_{0} Y_{0}} (t_{1}, t_{2}) = σ_{0}^{2} e^{- η (t_{1} + t_{2})}$	(6.9)
$R_{Y_{0} Y_{1}} (t_{1}, t_{2}) = σ_{0}^{2} t_{1} e^{- η (t_{1} + t_{2})}$
$R_{Y_{1} Y_{0}} (t_{1}, t_{2}) = σ_{0}^{2} t_{2} e^{- η (t_{1} + t_{2})}$
$R_{Y_{1} Y_{1}} (t_{1}, t_{2}) = (σ_{1}^{2} + t_{1} t_{2} σ_{0}^{2}) e^{- η (t_{1} + t_{2})}$

Let us consider the following first order, not homogeneous differential equation with the initial value $Y (0, ω) = 0$ :

\frac{d}{d t} Y (t, ω) + η Y (t, ω) = A (ω) + t D (ω)

(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 $σ_{A}^{2}$ and. $σ_{D}^{2}$ The realizations have to be computed from the corresponding deterministic equation:

\frac{d}{d t} y (t) + η y (t) = a + t d

y (0) = 0

(6.11)

The solution for a realization is

y (t) = \frac{a}{η} - \frac{d}{η^{2}} + t \frac{d}{η}

(6.12)

The solution for the random case (for the family of realizations) is

Y (t, ω) = \frac{1}{η} A (ω) + \frac{1}{η} (- \frac{1}{η} + t) D (ω)

(6.13)

The mean value and the correlation functions are

$m_{Y} (t) = 0$	(6.14)
$R_{Y Y} (t_{1}, t_{2}) = \frac{1}{η^{2}} σ_{A}^{2} + \frac{1}{η^{4}} (- 1 + η t_{1}) (- 1 + η t_{2}) σ_{D}^{2}$

The stochastic process is continuous and differentiable. The second mixed partial derivative is a constant, thus all derivatives exist.