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### If $\frac{dX_{t}}{X_{t}} = dL_{t}$, where $L_{t}$ is a local martingale, then is $X_{t}$ a local martingale?

There is essentially only one process $X$ satisfying $dX_t = X_t dL_t$, namely $X_t := C\exp(L_t - \frac 12 \langle L,L\rangle_t)$ where $C \in \mathbb{R}$ can be arbitrary. More precisely, if $Y$ is ...
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### Lévy's characterization of Brownian motion: right-continuous processes

Cool question. Proposition Let $(X_u)_{u}$ be right-continuous martingale with $X_0=0$, such that $(X^2_u-u)_u,(X_u^3-3uX_u)_u,(X_u^4-6uX_u^2+3u^2)_u$ are martingales. Then for every integer $M \ge 1$,...

### Two-dimensional random walk

Fleshing my comment out into an answer: this is not an extension of the 1D problem; it is, in fact, precisely the same as the 1D problem. You are considering a process $(R_n, S_n)$, where $R_n, S_n$ ...
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### Proof checking on the usage of the optional martingale theorem

Ok, I look at the book to see if I can see something useful, and indeed I find it. The exercise seems to rely in the definition of stopping time given in the book, what says that $T$ is a stopping ...

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