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Let $(B_t)$ the standard Brownian Motion and $(H_t)$ be an adapted continuous process. Show that $$\frac{1}{B_t}\int _0^tH_sdB_s $$ converge in probability.

I guess that the limit is $H_0$ but I don't know how to prove this statement. How can we show that $$\mathbb{P}(|\frac{1}{B_t}\int _0^tH_s-H_0dB_s|>\epsilon )\to 0$$ when $ t\to 0 $ ?

I have tried to use Markov inequality then Cauchy Schartz inequality in order to make appear $\mathbb{E}((\int H_s-H_0dB_s)^2)$ but this didn't work because of the term $\frac{1}{B_t}$.

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marked as duplicate by saz, Community Jun 20 '15 at 11:06

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