Let $$dX_t = a(X_t) dt + b(X_t) dW_t$$ be a onedimensional stochastic differential equation, where the coefficients $a,b: \mathbb{R} \rightarrow \mathbb{R}$ satisfy for every ball $B_R$ the following local Lipschitz and monotonicity conditions: $$a(x)  a(y) + b(x)  b(y) \leq K_Rxy$$ and $$a(x) + b(x) \leq K_R ( 1 + x).$$ In "Stochastic Stability of Differential Equations" by Khasminskii there is a theorem about the existence of a solution of the SDE in terms of a Lyapunov condition: Let $L$ be the generator of the SDE. If there exists a function $V \in C^2$ and a constant $c > 0$ s.t. $$LV \leq cV$$ and $$\inf_{x > R} V(x) \rightarrow \infty$$ as $R \rightarrow \infty$, then there exists solution of the SDE. The problem I can't figure out is the following: Is there a choice of the coefficients $a$ and $b$, s.t. $V$ has to be of exponential growth (i.e. something like $V(x) = \exp(x^2)$, and it doesn't suffice to choose a $V$ of polynomial growth?
