Martingales which are not Markov chains – Libres pensées d'un mathématicien ordinaire

Yesterday, a colleague of mine asked during a dinner ``is there an elementary way to construct martingales which are not Markov chains?'' Let us show that the answer is positive, by using a recursive recipe. Let \( {{(f_n)}_{n\geq1}} \) be a sequence of functions where \( {f_{n+1}:\mathbb{R}^{n+1}\rightarrow\mathbb{R}} \). Let \( {{(\varepsilon_n)}_{n\geq1}} \) be a sequence of i.i.d. real random variables…

Martingales which are not Markov chains Published 2012-01-20 Yesterday, a colleague of mine asked during a dinner `` is there an elementary way to construct martingales which are not Markov chains? '' Let us show that the answer is positive, by using a recursive recipe. Let \( {{(f_n)}_{n\geq1}} \) be a sequence of functions where \( {f_{n+1}:\mathbb{R}^{n+1}\rightarrow\mathbb{R}} \). Let \( {{(\varepsilon_n)}_{n\geq1}} \) be a sequence of i.i.d. real random variables of zero mean, independent of a real random variable \( {X_0} \). We now define the sequence \( {{(X_n)}_{n\geq0}} \) by setting

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