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Category Archives: Boundary Behavior
A modified Wright-Fisher Model
Consider the ODE
\[ \dot x_t = x_t(1-x_t)\]
and the SDE
\[dX_t = X_t(1-X_t) dt + \sqrt{X_t(1-X_t)} dW_t\]
- Argue that \(x_t\) can not leave the interval \([0,1]\) if \( x_0 \in (0,1)\).
- What is the behavior of \(x_t\) as \(t \rightarrow\infty\) if if \( x _0\in (0,1)\) ?
- Can the diffusion \(X_t\) exit the interval \( (0,1) \) ? Prove your claims.
- 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 \(\sigma\).
- Find a closed form expression for \(\mathbf E( r_t)\).
- Find a closed form expression for \(\mathrm{Var}(r_t)\).
- Characterize the values of parameters of \(a\), \(b\), and \(\sigma\) such that \(r=0\) is an absorbing point.
- What is the nature of the boundary at \(0\) for other values of the parameter ?
