Let $B_t$ be a standard Brownian motion and let $I_t=\max_{0\leq s \leq t} B_s$. I need to prove that the following is a martingale: $$M_t = \int_0^t [-2(I_s-B_s) + 4B_s^3] dB_s$$ From a previous question, $M_t$ is a continuous local martingale so it is a martingale if $$\mathbb{E} (M_t^2) < \infty$$ So I need to show that $$\mathbb{E} \left( \int_0^t [4(I_s-B_s)^2 - 16B_s^3(I_s-B_s) + 16B_s^6] ds \right)< \infty$$ This can be separated to three expectations but I am not sure how to find $$4\mathbb{E} ((I_s-B_s)^2)$$ and $$16\mathbb{E} (B_s^3(I_s-B_s))$$
Proving a stochastic process is a martingale
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stochastic-processes
stochastic-calculus
brownian-motion
martingales