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The proof uses the martingale property of the Ito integral. For an adapted stochastic process $X_t$ such that $$\mathbb{E}\int_0^{t}|X_s|^2ds <\infty$$ we have $$\mathbb{E}\int_0^{t}X_sdW_s =0$$ Now your result follows by setting $$X_t=\frac{1}{W_t^2+1}.$$ To see that the square integrability condition is satisfied note \mathbb{E}\int_0^{t}\frac{1}{(W_s^...