Let us consider the following SDE: $$dY_t=b(Y_t)dt+\sigma(Y_t)dW_t\tag{1}$$ with $b, \sigma: (l, r)\to\mathbb{R}$, $−\infty \leq l < r \leq \infty$ bounded functions on compact intervals of $(l, r)$.
In particular, $$b(Y_t)=(u-(u+i)Y_t)$$ $$\sigma(Y_t)=o\sqrt{(Y_t)(1-Y_t)}$$ with $u$, $i$ and $o$ arbitrary parameters.
Hence, focus will be on the following SDE: $$dY_t=(u-(u+i)Y_t)dt+o\sqrt{(Y_t)(1-Y_t)}dW_t\tag{2}$$ I must check whether the process $\{X_t\}$ remains within the interval $(l,r)$ or not for each $0\leq t\leq T$.

To this, I use the Feller test for explosions. Such a test requires that the following two integrals must be defined and computed: $$p(x)=\int_c^x \exp\bigg\{-2\int_c^{\xi}\frac{b(\zeta)}{\sigma^2(\zeta)}d\zeta\bigg\}d\xi\tag{3}$$ $$v(x)=\int_c^x\frac{2(p(x)-p(y))}{p\hspace{0.1cm}'(y)\sigma^2(y)}dy\tag{4}$$ with $c\in(l,r)$.
According to Feller test, probability that the process at least touches the bounds of interval $I$ equals $1$ or is less than $1$ according to whether $v(l+)=v(r-)=\infty$ or not. Let us fix $(l,r)=(0,1)$ and $c=\frac{1}{2}$.

I would like to study the asymptotic behaviour of the integral $(4)$ with $c=\frac{1}{2}$ at bounds $l=0$ and $r=1$, but I have not any experience with analyses like that. Is there a good standard method or is it just a matter of manipulation? Could you please help me understand how could I study asymptotic behaviour of $(4)$?


1 Answer 1


We follow the Feller test exposition in Shreve-Karatzas 5.5.

We indeed have to compute the scale function

$$p(x)=\int_c^x \exp\bigg\{-2\int_c^{\xi}\frac{b(\zeta)}{\sigma^2(\zeta)}d\zeta\bigg\}d\xi$$ which solves the ode $y'=-\frac{2(c_{1}+c_{2}x)}{x(1-x)}y(x)$ and so $p(x) = c x^{-2 c_{1}} (1 - x)^{2 (c_{1} +c_{2})}.$

So we see that if $c_{1}>0, c_{1}+c_{2}<0$, then $p(0)=p(1)=\infty$ and so according to 5.22, the process is trapped in $(0,1)$.


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