Data Analysis

ITÔ DIFFERENTIAL EQUATION

A very important special case is the linear stochastic differential equation

d Y (t, ω) = f (t) Y (t, ω) d t + g (t) d B (t, ω)

(7.1)

where $f (t)$ and $g (t)$ are deterministic functions and $d B (t, ω)$ 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 $g (t)$ is random $G (t, ω)$ , but then we have a product of random functions. The integral form of the differential equation is

Y (t, ω) - Y (t_{0}, ω) = \int_{t_{0}}^{t} f (τ) Y (τ, ω) d τ + \int_{t_{0}}^{t} g (τ) d B (τ, ω),

(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 $G (t, ω) \to g (t)$ 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

I (ω) = \int_{a}^{b} G (τ, ω) d τ

(7.3)

Assume $T = [a, b]$ , $\{B (t, ω), t \in T\}$ is a scalar Brownian motion process with variance parameter $σ^{2}$ and the function $G (t, ω)$ is defined on $t \in T$ . Let us introduce a partition $a = t_{0} < t_{1} < \dots < t_{n} = b$ , and a step function

G_{i} (ω) = {\begin{cases} 0, & t < t_{0} \\ G_{i} (ω), & t_{i} \leq t \leq t_{i + 1} \\ 0, & t \geq t_{n} \end{cases}

(7.4)

where $G_{i} (ω)$ is independent of $\{B (t_{k}, ω) - B (t_{l}, ω)\} : t_{i} \leq t_{l} \leq t_{k} \leq b$ , and $E \{{| G_{i} (ω) |}^{2}\} < \infty$ . The Itô integral is defined by

\int_{T} G (t, ω) d B (t, ω) = \sum_{i = 0}^{n - 1} G_{i} (ω) [B (t_{i + 1}, ω) - B (t_{i}, ω)]

(7.5)

In view of the independence condition the expected value is zero

E \{\int_{T} G (t, ω) d B (t, ω)\} = 0

(7.6)

Let us take another function $F (t, ω)$ and the corresponding step function $F_{i} (ω)$ and consider the expected value

E \{I_{G} (ω) I_{F} (ω)\} = E \{[\sum_{i = 0}^{n - 1} G_{i} (ω) Δ B] [\sum_{i = 0}^{n - 1} F_{i} (ω) Δ B]\} = σ^{2} \sum_{i = 0}^{n - 1} E \{G_{i} (ω) F_{i} (ω)\} (t_{i + 1} - t_{i})

(7.7)

where $B (t_{i + 1}, ω) - B (t_{i}, ω)$ . The last expression results from the independence conditions.

Now let us consider a sequence of step functions $G_{i}^{n} (ω)$ , $F_{i}^{n} (ω)$ converging to the random function $G (t, ω)$ in the sense that

\int_{T} E \{| G (t, ω) - G_{i}^{n} (ω) |\} d t \to 0, n \to \infty

(7.8)

Finally the following theorem is true:

Let the random functions $G (t, ω)$ and $F (t, ω)$ satisfy the condition that they are independent of $\{B (t_{k}, ω) - B (t_{l}, ω)\} : t \leq t_{l} \leq t_{k} \leq b$ , for all $t \in T$ and the conditions

\int_{T} E \{{| G (t, ω) |}^{2}\} d t < \infty

\int_{T} E \{{| F (t, ω) |}^{2}\} d t < \infty

Then their Itô integrals are well defined as

\int_{T} G (t, ω) d B (t, ω) = \underset{n \to \infty}{l.i.m.} \int_{T} G_{t}^{n} d B (t, ω)

(7.9)

with basic properties

E \{\int_{T} G (t, ω) d B (t, ω)\} = 0,

E \{\int_{T} G (t, ω) d B (t, ω) \int_{T} F (t, ω) d B (t, ω)\} = σ^{2} \int_{T} E \{G (t, ω) F (t, ω)\} d t .

Let us consider the following linear stochastic differential equation with constant coefficients

d A_{0} (t, ω) - η A_{0} (t, ω) d t = α d B (t, ω)

(7.10)

where $η$ and $α$ are constants and $d B (t, ω)$ is the differential of the Brownian motion process with variance parameter $σ^{2}$ equal to one. This is a first order differential equation with an additive noise. The general solution of the homogeneous differential equation is

A_{0} (t, ω) = A_{0} (0, ω) e^{- η t}

(7.11)

where $A_{0} (0, ω)$ is the random initial value for $t = 0$ . Addition of a particular solution of the non homogeneous equation yields the general solution

A_{0} (t, ω) = A_{0} (0, ω) e^{- η t} + α \int_{0}^{t} e^{- η u} d B (u)

(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 $A_{0} (t, ω)$ is zero. The correlation (covariance) function is

C_{A_{0} A_{0}} (t_{1}, t_{2}) = E \{[A_{0} (t_{2}) - m (t_{2})] [A_{0} (t_{1}) - m (t_{1})]\} = e^{- η (t_{2} - t_{1})} E {[A_{0} (0)]}^{2} + α^{2} E \{\int_{0}^{t_{2}} e^{- η (t_{2} - u)} d B (u) \int_{0}^{t_{1}} e^{- η (t_{1} - ν)} d B (ν)\}

(7.13)

From the basic formula for the Itô integrals it follows

C (t_{1}, t_{2}) = C (0) e^{- η (t_{1} + t_{2})} + α^{2} e^{- η (t_{1} + t_{2})} \int_{0}^{t_{1}} e^{2 η u} d u .

(7.14)

Upon integration it follows

C (t_{1}, t_{2}) = [C (0) - P] e^{- η (t_{1} + t_{2})} + P e^{- η (t_{2} - t_{1})},

(7.15)

where $P = α^{2} / 2 η$ .

When $t_{1}$ , $t_{2}$ tends to infinity in such a way that $t_{2} - t_{1} = τ$ is constant, then in the limit

C (t_{1}, t_{2}) \to P e^{- η (t_{2} - t_{1})} = P e^{- η τ}

(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 $C (0)$ is equal to $P$ the process is strictly stationary.

In general the process is not stationary. The variance of the process is

P (t) = C (t, t) = [C (0) - P] e^{- 2 η t} + P

(7.17)

The variance changes from $C (0)$ at the time $t = 0$ to the asymptotic value $P$ when $t$ tends to infinity. The derivative with respect to time is

\frac{d P (t)}{d t} = - 2 η [C (0) - P] e^{- 2 η t}

(7.18)

It is easy to show that the differential equation for the evolution of the variance in our example is

\frac{d P (t)}{d t} = - 2 η P (t) + α^{2}

(7.19)