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$dX_t=\left(\sqrt{1+X_t^2}+\frac{1}{2}X_t\right)dt+\sqrt{1+X_t^2}dW_t, X_0=0$, where $W_t$ is brownian. I tried using $X_t=\sinh(W_t)$ but then when I apply Ito's lemma to it, I can't get the first sqrt term.

any suggestion as to what I should try? I think it has to be some trigonometric functions.... I even tried $\operatorname{csch}$

is the solution $X_t=\sinh(W_t+t)$ a submartingale

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solution is $X_t=sinh(W_t+t)$ –  Glenn K. Nov 14 '11 at 3:21
but is $X_t$ a submartingale? –  Glenn K. Nov 14 '11 at 3:31
see this question of yours –  Ilya Nov 14 '11 at 7:49

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