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Your dynamics under the $T_{i-1}$-forward measure is wrong. Specifically, let $P_{i-1}$ and $P_i$ be, respectively, the $T_{i-1}$- and $T_i$-forward probability measures. Moreover, let $\Delta_i = T_i-T_{i-1}$. Then, for $0\le t \le T_{i-1}$, \begin{align*} \eta_t &\equiv \frac{dP_{i-1}}{dP_i}\big|_t \\ &= \frac{P_i(0, T_i)}{P_{i-1}(0, T_{i-1})}\...

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Thanks a lot for the above great answer:). Here I also added the link for Levy's Characterization of Brownian Motion: http://individual.utoronto.ca/normand/Documents/MATH5501/Project-3/Levy_characterization_of_Brownian_motion.pdf

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I have a take on the intuition part of the question. Isn't it a simple consequence of Jensen's inequality? Thus, assuming $r=0$ for simplicity, we have in the money market measure: $E(S_T)=S_t$, but then $E(1/S_T)>1/S_t$ by Jensen since $1/x$ is convex. Now in the stock measure, we must force $E_S (1/S_T)=1/S_t$ to create the correct martingale, but ...

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The drift is the expectation of the return over an infinitesimal interval. Let $Q$ be the risk-neutral measure and $Q^S$ be measure associated with the stock price numeraire defined by \begin{align*} \frac{dQ^S}{dQ}\big|_t = \frac{S_t}{B_t S_0}, \end{align*} where $B_t=e^{rt}$ is the value at time $t$ of the money-market account. Moreover, let $E$ and $E^S$ ...

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As a general principle, I would be wary of economic or financial interpretations of change of measure techniques. Changing numéraires is merely a mathematical tool to ease pricing, see for example the last part of this answer. Nevertheless, here’s my take on your question. Think of a numéraire as the basic financial asset of your economy, namely a store of ...

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