# Category Archives: Boundary Behavior

## 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.

## 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 ?