Is this process strictly positive? Let $W_t$ is standard Brownian motion under probability measure $P$.
Consider 1-D stochastic differential equation
$$ dY_t = dt + \sigma(Y_t) dW_t, \ Y_0 = y\ge 0.$$
We assume $\sigma(0) = 0$, and $\sigma(x)$ is locally Holder-1/2, i.e.
for any bounded subinterval $I\subset [0,\infty)$,
we assume there exists constant $K_I$ s.t.
$$|\sigma(x_1)- \sigma(x_2)| \le K_I |x_1 - x_2|^{1/2}, \ \forall x_1, x_2 \in I.$$
Note that, the above SDE has strong non-negative solution by comparison with
$d X_t = \sigma(X_t) dW_t$.
[Q1] Define $\tau = \inf[t>0: Y_t = 0]$. Is $\tau>0$ almost surely?
[Q2] Can one show that $Y_t > 0$ almost surely for arbitrary given $t>0$?
In fact, it's enough to show the above results with $y= 0$. 
My guess is that, [Q2] is too strong to be true, but [Q1] is correct.
It will be helpful to get a proof of [Q1] at least.
 A: First by the Yamada–Watanabe theorem, we have existence and uniqueness at least locally.

Second, we indeed show that the solution cannot be negative. Here we can follow a similar argument as in this answer. Since we restricted to $[0,\infty)$ we can consider the sde
$$dZ_{t}=dt+\sigma(Z_{t}\vee 0)dW_{t}, Z_{0}=z_{0}\geq 0$$
for fixed $\epsilon>0$, the hitting time $\tau_{-\epsilon}:=\inf\{t>0: Z_{t}=-\epsilon\}$. We want to show that $P[\tau_{-\epsilon}=\infty]=1$. Suppose otherwise, then there is a short enough interval $I=(\tau_{-\epsilon}(\omega)-u,\tau_{-\epsilon}(\omega))$ so that for $t\in I$ we have $Z_{t}<0$ and
$$dZ_{t}=dt+\sigma(0)dW_{t}=dt\Rightarrow Z_{t}=t+z_{0}>0,$$
which is a contradiction.
Finally, indeed as mentioned in the comments the claim $P[\tau_{0}=\infty]=1$ or $<1$ depends on the volatility versus drift relationship. For example, in Bessel processes $dX_{t}=ndt+2\sqrt{X_{t}}dW_{t}$ we have

In terms of positive-direction results,

*

*In "An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process" for the CIR process $dXt = \kappa(\lambda − X_t) dt + \sigma \sqrt{X_t} dW_t , X_0 = x_0$, a main idea is using the Feller-explosion test in Karatzas-Shreeve theorem 5.29



*

*see here "Fractional Cox–Ingersoll–Ross process with non-zero «mean»" for a nice approach with last sup time.

*For a result in positive direction see here

$$dX_t = f(X_t)dt + \sigma(X_t)dW_t,\quad X_0 = 1,$$
where $W_t$ is a Brownian motion under the measure $\mathbb{P}$, in its natural filtration. Suppose that $f(x)-\sigma^2(x)/2x=0$ and $\sigma^2(x)\le x^2$. Show that $X_t>0$ almost surely.

