Yes, square the processes then $$ dX_t^2 = 2X_t dX_t + (dX_2)^2 = 2|X_t| dt + 1 + 2X_t dB_t$$ and similarly for the other. Then you see the the drift on this process must be bigger that that on the $Y_t$ process at the same level. Then you can finish it off with a comparison theorem that says $X_t^2$ will be stochastically larger than $Y_t^2$ and the same is therefore true about the absolute value as well. There is a chapter on the comparison thoerems in Ikeda and Watanabe , I don't know where else they can be found. If this is true, the I acknowledge that I learned this trick from Ionnis Karatzis about 1986, but if I'm muddled, i's just me.

Yes, square the processes then $$ dX_t^2 = 2X_t dX_t + (dX_t)^2 = 2|X_t| dt + 1 + 2X_t dB_t$$ and similarly for the other. Then you see the the drift on this process must be bigger than that of the $Y_t$ process at the same level. Then you can finish it off with a comparison theorem that says $X_t^2$ will be stochastically larger than $Y_t^2$ and the same is therefore true about the absolute value as well. There is a chapter on the comparison theorems in Ikeda and Watanabe , I don't know where else they can be found. If this is true, the I acknowledge that I learned this trick from Ionnis Karatzis about 1986, but if I'm muddled, i's just me.