# Tag Archives: JCM_math545_HW8_S14

## A modified Wright-Fisher Model

Consider the ODE

$\dot x_t = x_t(1-x_t)$

and the SDE

$dX_t = X_t(1-X_t) dt + \sqrt{X_t(1-X_t)} dW_t$

1. Argue that $$x_t$$ can not leave the interval $$[0,1]$$ if $$x_0 \in (0,1)$$.
2. What is the behavior of $$x_t$$ as $$t \rightarrow\infty$$ if if $$x _0\in (0,1)$$ ?
3. Can the diffusion $$X_t$$ exit the interval $$(0,1)$$ ? Prove your claims.
4. What do you think happens to $$X_t$$ as $$t \rightarrow \infty$$ ? Argue as best you can to support your claim.

## No Explosions from Diffusion

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 $$\sigma >0$$.

1. Show that $$x_t$$ blows up in finite time.
2. Find the values of  $$\sigma$$ and $$\alpha$$ so that $$X_t$$ does not explode (off to infinity).

[ From Klebaner, ex 6.12]

## Cox–Ingersoll–Ross model

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 $$\sigma$$.

1. Find a closed form expression for $$\mathbf E( r_t)$$.
2. Find a closed form expression  for $$\mathrm{Var}(r_t)$$.
3. Characterize the values of parameters of $$a$$, $$b$$, and $$\sigma$$ such that $$r=0$$ is an absorbing point.
4. What is the nature of the boundary at $$0$$ for other values of the parameter ?

## Ballistic Growth

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 $$a\to\infty$$.

## Entry and Exit through boundaries

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.

## Martingale Exit from an Interval – I

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 $$p$$ :

1. For this part assume that $$M_t$$ is the solution to a time homogeneous SDE. That is that $dM_t=\sigma(M_t)dB_t.$ (with $$\sigma$$ bounded and smooth.) What PDE should you solve to find $$p$$ ? with what boundary data ? Assume for a moment that $$M_t$$ is standard Brownian Motion ($$\sigma=1$$). Solve the PDE you mentioned above in this case.
2. A probabilistic way of thinking: Return to a general martingale $$M_t$$. Let us assume that $$dM_t=\sigma(t,\omega)dB_t$$ again with $$\sigma$$ smooth and uniformly bounded  from above and away from zero. Assume that $$\tau < \infty$$ almost surely and notice that $\mathbf E_x M(\tau)=a \mathbf P_x\{M_\tau=a\} + b \mathbf P_x\{M_\tau=b.\}$ Of course the process has to exit through one side or the other, so $\mathbf P_x\{M_\tau=a\} = 1 – \mathbf P_x\{M_\tau=b\}$. Use all of these facts and the Optimal Stopping Theorem to derive the equation for $$p$$.
3. Return to the case when $dM_t=\sigma(M_t)dB_t$. (with $$\sigma$$ bounded and smooth.) Write down the equations that $$v(x)= \mathbf E_x\{\tau\}$$, $$w(x,t)=\mathbf P_x\{ \tau >t\}$$, and $$u(x)=\mathbf E_x\{e^{-\lambda\tau}\}$$ with $$\lambda > 0$$ satisfy. ( For extra credit: Solve them for $$M_t=B_t$$ in this one dimensional setting and see what happens as $$b \rightarrow \infty$$.)

## Probability Bridge

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*}