I was just wondering, if we let $(Z_n)_{n\geq 0}$be a martingale with $Z_0=0$, is it true then $$ E[Z_n^2]= \sum_{i=1}^n E[(Z_i-Z_{i-1})^2] $$ Please let me know and if it is true, can someone show me how?
Thanks
2026-03-29 08:57:54.1774774674
Show that $E[Z_n^2]= \sum_{i=1}^n E[(Z_i-Z_{i-1})^2] $ for a martingale with $Z_0=0$
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Hint: $$ {\rm E}[(Z_i-Z_{i-1})^2]={\rm E}[Z_i^2+Z_{i-1}^2-2Z_iZ_{i-1}],\quad 1\leq i\leq n. $$ Then use the tower property to deduce that $$ {\rm E}[Z_iZ_{i-1}]={\rm E}[{\rm E}[Z_iZ_{i-1}\mid\mathcal{F}_{i-1}]]={\rm E}[Z_{i-1}^2],\quad 1\leq i\leq n. $$ Conclude.