# Tagged Questions

For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.

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### Prove $M_{S(k) \wedge n}$ is bounded in $\mathscr L^2$

Probability with Martingales: To prove $$\sup E[M_{S(k) \wedge n}^2] < \infty,$$ how can we use 12.12c? There aren't any stopping times there.
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### Prove $S_k$ is a stopping based on $A$ being previsible

Probability with Martingales: It looks like we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ ...
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### showing a process martingle from ito's lemma.. [closed]

**by ito formula, Ft: filtratoin $M(t) = (aB(t) - t) \exp( 2B(t) - 2t )$ find constant a for $M(t)$ to be a martingale plz help!**
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### Why is $M$ bounded in $\mathscr L^2$ iff $E[\lim A_n] < \infty$?

Probability with Martingales: About $(c)$ My understanding is that $(c)$ is equivalent to: $$\sup E[A_n] < \infty \iff E[\lim A_n] < \infty$$ Why is that so?
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### Preservation of ui condition [closed]

I have a stopping time $\tau_n$ with $\mathbb{P}(\tau_n=\infty)\rightarrow 1$ for $n \to \infty$. With this stoppingtime $M^{\tau_n}$ is a uniformly integrable martingale. I deduced that $M$ is a ...
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### Example of a semimartingale with special properties

I have to find an example of a semimartingale X such that $\lim_{t \rightarrow \infty} X_t$ exists a.s. and $X$ is not a semimartingale up to infinity. I think it could be a deterministic function ...
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### Show that $W^2 _t - t$ is a $\mathbb{P}$-martingale.

Claim: $V_t = W^2 _t - t$ is a $\mathbb{P}$-martingale. I have shown via Ito's formula, that $dV_t = 2 W_t \, dW_t$. For reference, I will list this "Proposition": If $X$ is a stochastic process ...
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### Find constants such that transformed simple symmetric random walk is martingale

Let $$S_0 :=0, \quad S_n = X_1 + ... + X_n \quad \forall n \in \mathbb{N}$$ be the simple symmetric random walk on $\mathbb{Z}$, i.e. the $X_i$ are i.i.d. with $$P[X_i = +1] = P[X_i = -1] = 1/2.$$ ...