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I previously asked whether $X_t=\sin B_t$ is a martingale, where $B_t$ is a standard Brownian motion and the underlined filtration is the canonical one.

As it can be seen in the answer provided, this turned out to be false. The answer consisted of an application of Ito's Lemma, and as I understood from the fact that an Ito Process with a drift can not be a martingale.

I would like now to see if I can check whether $X_t=\sqrt{t}\sin(B_t)$ is a martingale. I have been given the hint that it is, but I am not sure why is this the case. Following the notation of this page: we have that our function $f(w,t)=\sqrt{t} \sin(w)$ so that $\frac{\partial f }{\partial t}=\frac{1}{2 \sqrt{t}}\sin(w)$, $f'(w,t) = \sqrt{t} \cos (w)$ and $f''(w,t) = -\sqrt{t}\sin w$.

Since $dW(t)=dB(t)$ and $dB(t)^2=dt$ we would then have that:

$$dX_t = \frac{\partial f }{\partial t}(B_t,t)dt + f'(B_t,t) dB_t + \frac{1}{2}f''(B_t,t)dt = \sqrt{t}\cos (B_t)dB_t + \\ \sin(B_t)dt\bigg(\frac{1}{2 \sqrt{t}} - \sqrt{t}\frac{1}{2}\bigg)$$

Here it seems that again we have a drift, which would indicate that the process is not a martingale. Where is my mistake?

Jose Avilez
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Barreto
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  • Seems like I made an algebraic mistake when answering your comment on the linked post. Your calculation is correct, showing that $X$ is not a martingale. – Jose Avilez Apr 28 '22 at 00:04
  • Oh. Thank you! One more question. Suppose $f(t,x)=\sqrt{e^t} \sin(w)$, which does make the term cancel each other. Would that be a martingale? I mean, what other conditions do we need to conclude that it is a martingale? @JoseAvilez – Barreto Apr 28 '22 at 00:18
  • When you have a stochastic integral $X_t = \int_0^t f(s, B_s) dB_s$, $X_t$ will be a local martingale. For it to be a true martingale you could check that $\mathbb{E}\left[ \int_0^t f^2(s,B_s) ds \right] < \infty$ – Jose Avilez Apr 28 '22 at 00:23
  • Awesome! Then it is! Can you provide some reference please of your last statement? and the expectation needs to be finite for all finite $t$ I suppose @JoseAvilez – Barreto Apr 28 '22 at 00:26
  • You can find it on any stochastic calculus textbook. E.g. Try Kuo's Introduction to Stochastic Integration – Jose Avilez Apr 28 '22 at 00:38

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