lower bounds for a stochastic integral

Question

for all $t \in [0,T]$, consider a stochastic integral as follows:

$\int_0^{min \{t^*,T \}} f(t,\omega) dt$

where $f \geq 0$ is a nonnegative stochastic process and $t^*$ is a random stopping time. I know such integral has a simple upper bound

$\int_0^{min \{t^*,T \}} f(t,\omega) dt \leq \int_0^T f(t,\omega) dt$

but I was wondering can I find a lower bound for this integral? not the trivial zero.

Answer 1

An obvious lower bound is $$\int_0^{t^{\ast}(\omega) \wedge T} f(t,\omega) \, dt \geq \inf_{t \in [0,t^{\ast}(\omega) \wedge T]} f(t,\omega) \cdot (t^{\ast}(\omega) \wedge T).$$ Without further assumptions on the stopping time $t^{\ast}$ or $f$, it will be difficult to get a better lower bound. Since the considered integral is defined pathwise you are basically looking for a lower bound for a deterministic integral of the form

$$\int_a^b f(t) \, dt.$$