# When given expected value is finite?

I consider the stochastic process $$(X_t)_{t\geq 0}$$ that satisfies: $$dX_t = a(t, X_t) dt + b(t, X_t) dB_t.$$ We assume that solution of the above SDE is unique non-exploding solution, e.g. $$(X_t)_{t\geq 0}$$ can be a Brownian Motion $$\left(a(t, X_t)=0 \text{ and } b(t, X_t) = 1\right)$$ or geometric Brownian motion $$\left(a(t, X_t)=r X_t \text{ and } b(t, X_t) = \sigma X_t\right)$$.

Let us consider the expected value $$E[e^{-\int_t^T X_sds}].$$

I wonder when this expected value is finite. I suppose that for some functions $$a(t, X_t)$$ and $$b(t, X_t)$$ I can encounter inifite expected value (I suppose the problem arises when $$(X_t)_{t\geq 0}$$ will be 'very' negative). My question is: Are there any necessary conditions on $$a(t, X_t)$$ and $$b(t, X_t)$$ which guarantee that the above expectation is finite.