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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.

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