# Tag Info

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### An example such that $(X_n)$ and $(Y_n)$ are martingales with respect to their natural filtration and $(X_n +Y_n)$ is no martingale

Let $(X_n)$ and $(Y_n)$ be two simple random walks defined on the same probability space, generated by a sequence of i.i.d. fair coin tosses $\omega_1,\omega_2,\dots$ with $\omega_i\sim U\{-1,1\}$ ...
Accepted

### Counterexample that $(X_n)_n$ is not a martingale

A simple counterexample: $X_n=n$.
• 7,027
Accepted

• 15.6k
1 vote
Accepted

### Can we always find a martingale part in a càdlàg supermartingale?

Example: If $B_t$ is a 3-dimensional Brownian motion started at some point other than the origin, then $Z_t:=\|B_t\|^{-1}$ is a positive supermartingale and a (continuous) local martingale. The ...
• 26.5k
1 vote

### Quadratic covariation of martingale transforms to simple processes.

Consider stopping times $T>S$ and stopping times $T'>S'$. Moreover, $\xi_T$ shall be the coefficient for $(S,T]$ and $\varphi_{T'}$ the coefficient for $(S',T']$. Using Proposition 4.10, you ...
• 2,785
1 vote

### How can I show that $\mathbb{E}(T)=\frac{a^2}{\sigma^2}$?

There are other justifications that have to be made if you want to do it rigorously. For example, what is allowing you to conclude that $E(T\wedge t)\xrightarrow{t\to\infty}=E(T)$? What is allowing ...
• 14.4k
1 vote
Accepted

### Prove that for symmetric random walk $S_n$ it holds that $S_{min\{n,\tau\}}^2 - \min\{n,\tau\}$ is a uniform integrable martingale

Here's a way to do this by doing a bit of a roundabout argument. I'll denote $\min(a,b)$ by $a\wedge b$ for ease of writing. First see that $\tau\wedge n$ is an increasing sequence of random variables....
• 14.4k

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