Martingales – Problem (27/365)

I am still unable to follow the martingales based proof for the Ballot Theorem, so I’ll work out a few more problems first. A problem asks the following. Let \mathcal{D}_0 \le \dots \le \mathcal{D}_n be a sequence of decompositions with \mathcal{D}_0 = \{ \Omega \}, and let \eta_k be \mathcal{D}_k-measurable variables. Show that the sequence \theta = (\theta_k, \mathcal{D}_k) with

\displaystyle  \theta_k = \sum_{l=1}^{k+1}\left(\eta_{l}-E(\eta_{l}|\mathcal{D}_{l-1})\right)

is a martingale.

To show this we need to show that (1) \theta_k is \mathcal{D}_k-measurable. This is clear because \theta_k takes on a single value (the expectation) conditioned on each D \in \mathcal{D}_k. Next, we need to show that (2) E(\theta_{k+1} | \mathcal{D}_k) = \theta_k.

\displaystyle  E(\theta_{k+1}|\mathcal{D}_{k})	\\ = E\left[\sum_{l=1}^{k+1}\left(\eta_{l}-E(\eta_{l}|\mathcal{D}_{l-1})\right)|\mathcal{D}_{k}\right] \\ = \sum_{l=1}^{k+1}E\left[\eta_{l}-E(\eta_{l}|\mathcal{D}_{l-1})|\mathcal{D}_{k}\right] \\ = E(\eta_{k+1}|\mathcal{D}_{k})-E\left[E(\eta_{k+1}|\mathcal{D}_{k})|\mathcal{D}_{k}\right]+\sum_{l=1}^{k}E(\eta_{l}|\mathcal{D}_{k})-E\left[E(\eta_{l}|\mathcal{D}_{l-1})|\mathcal{D}_{k}\right] \\ = E(\eta_{k+1}|\mathcal{D}_{k})-E\left[E(\eta_{k+1}|\mathcal{D}_{k})|\mathcal{D}_{k}\right]+\sum_{l=1}^{k}\eta_{l}-E\left[E(\eta_{l}|\mathcal{D}_{l-1})|\mathcal{D}_{k}\right] \\ \mbox{ since }\eta_{l}\mbox{ is }\mathcal{D}_{i\ge l}\mbox{-measurable} \\ = E(\eta_{k+1}|\mathcal{D}_{k})-E\left[E(\eta_{k+1}|\mathcal{D}_{k})|\mathcal{D}_{k}\right]+\sum_{l=1}^{k}\eta_{l}-E(\eta_{l}|\mathcal{D}_{l-1}) \\ \mbox{ since }\mathcal{D}_{l-1}\le\mathcal{D}_{k}\mbox{ and }E(\eta_{l}|\mathcal{D}_{l-1})\mbox{ is }\mathcal{D}_{l-1}\mbox{-measurable} \\ = E(\eta_{k+1}|\mathcal{D}_{k})-E(\eta_{k+1}|\mathcal{D}_{k})+\sum_{l=1}^{k}\eta_{l}-E(\eta_{l}|\mathcal{D}_{l-1}) \\ \mbox{ since }E(\eta_{k+1}|\mathcal{D}_{k})\mbox{ is }\mathcal{D}_{k}\mbox{-measurable} \\ = \theta_{k}

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