# Tag Info

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• 13.2k
Accepted

### Is a stopped martingale always a martingale?

Yes, if $(M_t)_{t\in\mathbb R_+}$ is a right-continuous martingale and $T$ is a stopping time, then $(M_{t\land T})_{t\in\mathbb R_+}$ also is a martingale, see for instance Corollary 3.24 in Le Gall, ...
• 13.4k
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### Why are there different definitions of admissibility in the literature, and why do we need admissibility?

Why do we need admissibility? The problem is that you need extra conditions for the first fundamental theorem of asset pricing to hold; the existence of an equivalent martingale measure is not enough ...
• 12.3k
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### Dense subset of $L^2$ in the integral representation of $X \in L^2(\Omega)$

The set $\mathcal U$ is not dense in $L^2$ as it contains only non-negative random variables with unit expectation. However, this shows that $\text{span}(\mathcal U)$ is dense in $L^2$, or ...
• 13.1k
Accepted

• 316
1 vote
Accepted

### Stopping time of a martingale $Y_n=e^{S_n}$

If I understand correctly, you assume that $q=\frac{1}{1+e}$ so $Y_n$ is a martingale? Then, note that $T = \inf\{n : Y_n > 100\} = \inf\{ n : Y_n = e^5\}$. Write $T_j = \min\{j,T\}$ and note that ...
• 8,096
1 vote

### Help me find the formula for this problem please (probably easy!)

It doesn't matter what the odds are. You will always eventually win and if the pay out is 1 to 1 if you double the bet it will work. If the odds are against you that just means it will take longer. ...
• 124k

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