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Let $(X_t, t ≥ 0)$ be a 1-dimensional diffusion process with generator $Af(x) =\frac{1}{2}a(x)f''(x)+b(x)f'(x), \mathcal{D}(A)=C^2({\mathbb{R}})$ where $b$ and $a=\sigma^2$ are continuous functions of $x$ and $\sigma(x)> 0$ for all $x$. Let $\tau_0=\inf\{t\ge 0: X_t=0\}.$

1) Suppose that for some $x_0$, $b(x)\leq 0,$ for all $x\ge x_0$. Show that for any $x> 0$: $\mathcal{P}^{x}[\tau_0< \infty]=1.$

2) Suppose that for some $x_0,$ $\frac{b(x)}{a(x)}>\epsilon> 0,$ for all $x\ge x_0.$ Show that for any $x> 0$: $\mathcal{P}^x[\tau_0< \infty]<1$

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This is Exercise 3.1 (page 221) in Rick Durrett's Stochastic Calculus: A Practical Introduction. There is a solution in the back.

You use the fact that ${\cal P}^x(\tau_0<\infty)=1$ if and only if $\varphi(\infty)=\infty$ where $\varphi$ is the natural scale function $$\varphi(x)=\int_0^x\exp\left(\int_0^y-{2b(z)\over a(z) }dz \right)dy.$$

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Wow! If there are any extension of this criteria? –  Ilya Apr 17 '11 at 19:37
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