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3. Stochastic Differential Equations
3.1 Examples and Some Solution Methods
We have questions about a given stochastic differential equations from two aspects: (i) The theoretical one: Can we obtain existence and uniqueness theorems for such equations? What are the properties of the solutions? (ii) How can we solve a given such equation?
The following some examples may give us some intuitive hints.
Example 3.1.1:
Consider a population growth model described as
\[
\frac{\rmd}{\rmd t} N_t = (r_t+\alpha W_t)N_t\ ,
\]
where \(W_t\) is white noise and \(\alpha\) is a constant. Assume that \(r_t = r\) is also a constant. Then
\[
\rmd N_t = r N_t\ \rmd t + \alpha N_t\ \rmd B_t
\]
or equivalently
\[
\frac{\rmd N_t}{N_t} = r\ \rmd t + \alpha\ \rmd B_t\ .
\]
There must be some relationship between \(\ln N_t\) and \({\rmd N_t}/{N_t}\). So we put \(g(t,x) = \ln x\), hence we have
\[
\begin{aligned}
&&\rmd (\ln N_t) =&\ \frac{1}{N_t}\ \rmd N_t - \frac{1}{2{N_t}^2}(\rmd N_t)^2 \\
&&=& \ \frac{\rmd N_t}{N_t} - \frac{\alpha^2}{2}\ \rmd t\\
&\Leftrightarrow& \quad \frac{\rmd N_t}{N_t} = &\ \rmd(\ln N_t) + \frac{\alpha^2}{2} \rmd t\\
&\Leftrightarrow& \quad r\ \rmd t+\alpha\ \rmd B_t =&\ \rmd(\ln N_t) +\frac{\alpha^2}{2}\ \rmd t\\
&\Leftrightarrow& \quad \rmd (\ln N_t) =&\ \lr({r-\frac{\alpha^2}{2}})\rmd t + \alpha\ \rmd B_t\\
&\Leftrightarrow& \quad \ln N_t-\ln N_0 =&\ \lr({r-\frac{\alpha^2}{2}}) t + \alpha B_t \\
&\Leftrightarrow& \quad N_t = & \ N_0\exp\lr[{\lr({r-\frac{\alpha^2}{2}})t+\alpha B_t}]\ .
\end{aligned}
\]
We say a process \(X_t\) is Geometric Brownian Motion if \(X_t\) is of the form
\[
X_t = X_0 \exp (\mu t+\alpha B_t) \quad \mu,\alpha \text{ const.}
\]
Proposition 3.1.2:
Suppose \(X_t\) is a geometric Brownian motion described by
\[
X_t = X_0 \exp (\mu t + \alpha B_t)\ .
\]
If \(B_t\) is independent of \(X_0\) then
\[
\Exp[X_t] = \Exp[X_0]\cdot \exp\lr({\mu t+\frac{1}{2}\alpha^2 t})\ ,
\]
which is the same as when there is no noise term.
Proof: Let \(Y_t = \exp(\mu t+\alpha B_t)\) and apply Itô formula:
\[
\rmd Y_t =\mu\cdot Y_t\ \rmd t + \alpha\cdot Y_t\ \rmd B_t + \frac{1}{2}\alpha^2 Y_t\ \rmd t
\]
or in integral form
\[
Y_t = Y_0 + \mu \int_0^t Y_s\ \rmd s + \alpha \int_0^t Y_s\ \rmd B_s + \frac{\alpha^2}{2}\int_0^t Y_s\ \rmd s\ .
\]
By theorem 1.2.1,
\[
\Exp \lr[{ \int_0^t Y_s\ \rmd B_s }] = 0\ .
\]
Thus we get
\[
\begin{aligned}
&&\Exp[Y_t] =&\ \Exp[Y_0] + \lr({\mu + \frac{\alpha^2}{2}})\int_0^t E[Y_s]\ \rmd s\\
&\Leftrightarrow& \frac{\rmd }{\rmd t} \Exp[Y_t]
= &\ \lr({\mu + \frac{\alpha^2}{2}})\Exp[Y_t]\\
&\Leftrightarrow& \Exp[Y_t] =&\ \exp\lr({\mu t+\frac{1}{2}\alpha^2 t})\ .
\end{aligned}
\]
Since \(B_t\) is independent of \(X_0\) we have
\[
\Exp[X_t] = \Exp[X_0\cdot Y_t] = \Exp[X_0] \cdot \Exp[Y_t] = \Exp[X_0]\cdot \exp\lr({\mu t+\frac{1}{2}\alpha^2 t})\ .
\]
Theorem 5.1.2 (The Law of Iterated Logarithm):
\[
\underset{t\to\infty}{\lim \sup} \frac{B_t}{\sqrt{2t\log\log t}} = 1 \quad \text{a.s.}
\]