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I am trying to solve this question on stochastic processes which is to show that for $$I_0=0\\\\ I_n=\sum_{j=0}^{n-1}M_j(M_{j+1}-M_j), \quad n=0,1,2,... $$ the equation can be writen as; $$I_n=\frac{M_n^2}{2}-\frac{n}{2}$$ Where ${M_n}$ is a symmetric random walk. So far I have been able to show without too much trouble that; $$I_n=M_n^2-\sum_{j=1}^{n}X_jM_j$$ However I have been stuck ever since. I dont know if what I have done will be of any use but I am hoping you guys could help me out on this one.

Have a good day everybody.

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    $\begingroup$ You have not specified a relationship between $M_n$ and $X_n$. Nevertheless if you assume $M_0=0$ and $|M_{n+1}-M_n|=1$ for all $n$ then you can indeed derive the desired equality. Try writing an expression for $\frac{1}{2}(M_{j+1}-M_j)^2$ and expand and massage it to include the $M_j(M_{j+1}-M_j)$ expression. $\endgroup$ – Michael Apr 5 at 21:29
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    $\begingroup$ PS: The question can be solved with algebra and does not require any theory of stochastic processes. $\endgroup$ – Michael Apr 5 at 21:31

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