# Lyapunov function of exponential growth for existence of a solution of an SDE

Let $$dX_t = a(X_t) dt + b(X_t) dW_t$$ be a one-dimensional 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: $$|a(x) - a(y)| + |b(x) - b(y)| \leq K_R|x-y|$$ 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?

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Intuitively, the coefficients have to grow very fast, say, exponentially. But still, Khasminskii's conditions on non-explosion or existence of a solution have to be satisfied. –  epsilon Aug 27 '13 at 3:11