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As in title, I have a process $$X_{t}=\int_{0}^{t}s^{2}dB_{s}$$ I found here a sufficient condition for such integral to be a martingale on the interval. But I am asked if it is a martingale, not martingale restricted to the finite interval. Besides, I couldn't find anything more in web about such sufficent condition. Is it a well known theorem? If it is not, i would appreaciate some details how to proceed.

I'm sorry if it is a basic question but I'm at the "applied an Ito formula for the first time in his life half an hour ago" level, and in the venture of rapid and chaotic acquiring of the knowledge from the stochastic calculus :)

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It is true. A theorem (often just called the martingale property) says that $\int_0^T Y_t dB_t$ is a martingale provided $\mathbb{E}[\int_0^T Y_t^2 dt] < \infty$. Your integral, having a deterministic integrand, clearly satisfies this. Good luck on your venture.

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