Example of a regular strong solution of an SDE, which doesn't satisfy a Lyapunov condition?

Example of a regular strong solution of an SDE, which doesn't satisfy a
Lyapunov condition?

Let $$dX_t = a(t,X_t) dt + b(t, X_t) dW_t, \quad t \in [0,T]$$ be a
stochastic differential equation, where $W$ is an $m$-dimensional Brownian
motion, $X_0 = x \in \mathbb{R}^d$, and the coefficients are given by
$a(t,x) : [0,T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ and $b(t,x)
: [0,T] \times \mathbb{R}^{d \times m} \rightarrow \mathbb{R}^d$.
Furthermore the coefficients satisfy some local Lipschitz and linear
growth condition, i.e. if $B_R := \{x \in \mathbb{R}^d : |x| < R \}$, then
for every cylinder $[0,T] \times B_R$ there exists a constant $B$ s.t.
$$|a(s,x) - a(s,y)| + \sum_{i=1}^m |b_i(s,x) - b_i(s,y)| \leq B|x-y|$$ and
$$|b(s,x)| + \sum_{i=1}^m |b_i(s,x)| \leq B(1+|x|)$$ for every
$(s,x),(s,y) \in [0,T] \times B_R$.
There is a Lyapunov-criterion, which guarantees the existence of a strong
solution, which is also regular, i.e. there are no explosions in $[0,T]$.
E.g. Theorem 3.5 in Khasminskii's book "Stochastic Stability of
differential equations" states the following: If there exists a
Lyapunov-function $V: [0,T] \times \mathbb{R}^d \rightarrow \mathbb{R}$,
which is two times continuous differentiable w.r.t. $x$ and once w.r.t.
$t$, $$\lim_{R \rightarrow \infty} \inf_{t \in [0,T],|x| > R} V(t,x) =
\infty,$$ and there exists a constant $c>0$, s.t. $$LV(t,x) \leq
cV(t,x),$$ where $L$ is the generator of the SDE, then there exists a
strong regular solution.
Finally my question: Is there any example of an SDE, which possesses a
regular strong solution, but doesn't satisfy the above Lyapunov condition?
Or does even the other direction hold, i.e. if we have a strong regular
solution, such a Lyapunov-function exists?
Thank you very much.