Let us consider the stochastic process $(X_t)_{t\geq0}$ that can be described by the following SDE: $$ dX_t = \alpha(X_t, t) dt + \sigma(X_t, t) dB_t $$
Now I consider the following expected value: $$ \mathbb{E}[e^{-\int_t^T X_s ds}]. $$ My question is:
What conditions should be fulfilled by the functions $\alpha(x,t)$ and $\sigma(x,t)$ to make the above expectation finite?
Intuitively, I think that boundedness of $\sigma(x, t)$ for negative $x$ is enough ensure finiteness of the above expected value, however I do not know how to prove it formally or even strengthen (or weaken) this condition.