Consider the SDE (stochastic differential equation) as follows:

$$dX_t=b(X_t)dt+a(X_t)dW_t$$

where $b,a:\mathbb R^d\to\mathbb R^d$ are Lipschitz and $W$ is a real-valued Brownian motion. Under which conditions on $\bar x\in\mathbb R^d, r>0, \theta>0$ and $b,a$, one has

$$\mathbb P\big[\exists T>0 \mbox{ such that } |X_t-\bar x|<r \mbox{ for all } t\in [T,T+\theta]\big]=1?$$

PS : To illustrate this claim, we may take $d=1$ and $b\equiv 1 \equiv a$. Does this result hold?

Consider the SDE (stochastic differential equation) as follows:

$$dX_t=X_t\big(b(X_t)dt+a(X_t)dW_t\big)$$

where $b,a:\mathbb R\to\mathbb R$ are Lipschitz and bounded and $W$ is a real-valued Brownian motion. Under which conditions on $r>0, \theta>0$ and $b,a$, one has

$$\mathbb P\big[\exists T>0 \mbox{ such that } |X_t|<r \mbox{ for all } t\in [T,T+\theta]\big]=1?$$

Consider the SDE (stochastic differential equation) as follows:

$$dX_t=b(X_t)dt+a(X_t)dW_t$$

where $b,a:\mathbb R^d\to\mathbb R^d$ are Lipschitz and $W$ is a real-valued Brownian motion. Under which conditions on $\bar x\in\mathbb R^d, r>0, \theta>0$ and $b,a$, one has

$$\mathbb P\big[\exists T>0 \mbox{ such that } |X_t-\bar x|<r \mbox{ for all } t\in [T,T+\theta]\big]=1?$$

Consider the SDE (stochastic differential equation) as follows:

$$dX_t=b(X_t)dt+a(X_t)dW_t$$

where $b,a:\mathbb R^d\to\mathbb R^d$ are Lipschitz and $W$ is a real-valued Brownian motion. Under which conditions on $\bar x\in\mathbb R^d, r>0, \theta>0$ and $b,a$, one has

$$\mathbb P\big[\exists T>0 \mbox{ such that } |X_t-\bar x|<r \mbox{ for all } t\in [T,T+\theta]\big]=1?$$

PS : To illustrate this claim, we may take $d=1$ and $b\equiv 1 \equiv a$. Does this result hold?