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I am currently reading Financial Calculus of Baxter and Rennie and have one question regarding proof of Binomial Representation Theorem. In the text, we assume that increments of $N$ martingale are $$\Delta N_i = \Phi_i \Delta S_i + k_i$$ My question is: why do we assume the linear relation here between increments of these two martingales? I couldn't find any justification for that.

Thank you in advance for your help.

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    $\begingroup$ Is it not the case that $\Delta N_i $ and $\Delta S_i$ can take on only two values? $\endgroup$ – Bob Jansen Jun 10 '18 at 14:49

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