Consider the ODE
\[ \dot x_t = x_t(1-x_t)\]
and the SDE
\[dX_t = X_t(1-X_t) dt + \sqrt{X_t(1-X_t)} dW_t\]
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Posted in Boundary Behavior, SDE examples
Consider the following ODE and SDE:
\[\dot x_t = x^2_t \qquad x_0 >0\]
\[d X_t = X^2_t dt + \sigma |X_t|^\alpha dW_t\qquad X_0 >0\]
where \(\alpha >0\) and \(\sigma >0\).
[ From Klebaner, ex 6.12]
Comments Off on No Explosions from Diffusion
Posted in Explosions, SDE examples
Tagged JCM_math545_HW8_S14
The following model has SDE has been suggested as a model for interest rates:
\[ dr_t = a(b-r_t)dt + \sigma \sqrt{r_t} dW_t\]
for \(r_t \in \mathbf R\), \(r_0 >0\) and constants \(a\),\(b\), and \(\sigma\).
Comments Off on Cox–Ingersoll–Ross model
Posted in Boundary Behavior, SDE examples, Stochastic Calculus
Tagged JCM_math545_HW8_S14
Consider the SDE
\[
dX(t)=b(X(t))dt +\sigma(X(t))dB(t)
\]
with \(b(x)\to b_0 >0\) as \(x\to\infty\) and with \(\sigma\) bounded and positive. Suppose that \(b\) and \(\sigma\) are such that
\[\lim_{t\to\infty}X(t)=\infty\], with probability one for any starting point. Show that
\[
P_x\Big\{\lim_{t\to\infty}\frac{X(t)}{b_0 t}=1\Big\}=1 \ .
\]
From
\[
X(t)=x+\int_0^{t}b(X(s))ds +\int_0^{t}\sigma(X(s))dB(s)
\]
and the hypotheses, note that the result follows from showing that
\begin{align*}
\mathbf P_x\Big\{\lim_{t\to\infty}\frac{1}{t}\int_0^{t}\sigma(X(s))dB(s)=0\Big\}=1 \ .
\end{align*}
There are a number of ways of thinking about this. In the end they all come down to essentially the same calculations. One way is to show that for some fixed \(\delta \in(0,1)\) the following statement holds with probability one:
There exist a constants \(C(\omega)\) so that
\begin{align*}
\int_0^{t}\sigma(X(s))dB(s) \leq Ct^\delta
\end{align*}
for all \(t >0\).
To show this partition \([0,\infty]\) into blocks and use the Doob-Kolmogorov inequality to estimate the probability that the max of \( \int_0^{t}\sigma(X(s))ds\) on each block excess \(t^\delta\) on that block. Then use the Borel-Cantelli to show that this happens only a finite number of times.
A different way to organize the same calculation is to estimate
\[
\mathbf P_x\Big\{\sup_{t>a}\frac{1}{t}|\int_0^t \sigma(X(s))dB(s)|>\epsilon\Big\}
\]
by breaking the interval \(t>a\) into the union of intervals of the form \(a2^k <t\leq a2^{k+1}\) for \(k=0,1,\dots\) and using Doob-Kolmogorov Martingale inequality. Then let \(a\to\infty\).
Consider the following one dimensional SDE.
\begin{align*}
dX_t&= \cos( X_t )^\alpha dW(t)\\
X_0&=0
\end{align*}
Consider the equation for \(\alpha >0\). On what interval do you expect to find the solution at all times ? Classify the behavior at the boundaries.
For what values of \(\alpha < 0\) does it seem reasonable to define the process ? any ? justify your answer.
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Posted in Stochastic Calculus
Let \(\tau\) be the first time that a continuous martingale \(M_t\) starting from \(x\) exits the interval \((a,b)\), with \(a<x<b\). In all of the following, we assume that \(\mathbf P(\tau < \infty)=1\). Let \(p=\mathbf P_x\{M(\tau)=a\}\).
Find and analytic expression for \(p\) :
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Posted in Martingales, Stochastic Calculus
Tagged JCM_math545_HW4_S23, JCM_math545_HW5_S17, JCM_math545_HW8_S14
For fixed \(\alpha\) and \(\beta\) consider the stochastic differential equation
\[
dY(t)=\frac{\beta-Y(t)}{1-t} dt + dB(t) ~,~~ 0\leq t < 1 ~,~~Y(0)=\alpha.
\]
Verify that \(\lim_{t\to 1}Y(t)=\beta\) with probability one. ( This is called the Brownian bridge from \(\alpha\) to \(\beta\).)
Hint: In the problem “Solving a class of SDEs“, you found that this equation had the solution
\begin{equation*}
Y_t = a(1-t) + bt + (1-t)\int_0^t \frac{dB_s}{1-s} \quad 0 \leq t <1\; .
\end{equation*}
To answer the question show that
\begin{equation*}
\lim_{t \rightarrow 1^-} (1-t) \int_0^t\frac{dB_s}{1-s} =0 \quad \text{a.s.}
\end{equation*}
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Posted in Brownian Bridge, Stochastic Calculus
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