# Questions tagged [martingales]

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

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### Does this stopped process converges to the original process ucp (uniformly on compacts in probability)?

Let $M$ be a continuous local martingale with null at zero. Let $T(n)=\inf(t:|M_t|>n)$ be a stopping time. Then, the stopped process $M^{T(n)}$ is a bounded martingale (Lemma 29.1, Rojas and ...
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### If $X$ is a continuous supermartingale, why is, for every $N$, $(X_s)_{s\leq N}$ uniformly integrable?

If $X$ is a continuous supermartingale, why is, for every $N$, $(X_s)_{s\leq N}$ uniformly integrable? This statement is used, without proof, in another statement. I haven't been able to prove it. Is ...
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### Derive the 0-1 law from a functional equation of a martingale limit

I am trying to understand a proof given by Jabbour-Hattab in the paper "Martingales and Large Deviations for Binary Search Trees". In this paper, we consider the limit of a non-negative ...
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### Choquet-Deny theorem

Problem (Choquet-Deny theorem): Let $\sigma$ be a probability measure on $\mathbb R$, and let $f : \mathbb R \to \mathbb R$ be a bounded (Borel) measurable function. We will prove the $f$ satisfies ...
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### Constant in the Banach valued Marcinkiewicz-Zygmund inequality

Let $(\xi_i)_{i=1}^n$ be a sequence of real-valued independent random variables with zero mean. The classical Marcinkiewicz-Zygmund inequality is the following \begin{align} (E(\sum_{i=1}^n\xi_i)^p)^{...
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### Exercise 4.3.11 of Durrett's Probability: Theory and Examples

I am trying to solve Exercise 4.3.11 from Durrett's book. The question is: Show that if $P(\lim Z_n/\mu^n = 0)<1$ then it is equal to $\rho = P(Z_n = 0 \text{ for some n}$) where $Z_n$ is the ...
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### Quadratic variation of continuous local martingales

Dear all, I hope you can help me with the proof of the following result: Fact If $X$ is a continuous local martingale, then $[X]_t < \infty$ a.s. for every $t \geq 0$, where $[X]$ denote the ...
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### Skorokhod Embedding theorem proof

It occurs in Durrett's proof of Skorokhod embedding that he needs the following. Suppose that we have a Brownian motion $B_t$ in $1$-d that starts at $0$. (To be clear, it is not necessarily ...
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### Calculate $\mathbb{E}(\exp (- \lambda T_x ))$ for $\lambda > 0$ where $X$ is a Brownian Motion with drift

In a course I am studying on Stochastic Processes, I encountered the following exercise: Let $X_t = B_t + ct$ for some $c \in \mathbb{R}$ and where $B$ is a standard Brownian Motion. Now define $T_x$ ...
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Define: For $n \geq 0$, on note $X_n=(n+1) \mathbb{1}_{[n+1,+\infty}$, and $\mathcal{F}_n=\sigma(\{1\},\{2\}, \ldots,\{n\},[n+$ $1,+\infty[)$ and $\forall k \in \mathbb{N}^*, \mathbb{P}(\{k\})=\frac{1}... 2 votes 1 answer 59 views ### Stirling approximation of the probability that the stopping time is finite Let$\left(Y_n\right)_{n \geq 1}$a sequence of i.i.d random variables such that$\mathbb{P}\left(Y_1=1\right)=\mathbb{P}\left(Y_1=\right.-1)=1 / 2$. Define$S_0=0, \mathcal{F}_0=\{\Omega, \...
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I'm new to stochastic processes and have problems understanding martingales, conditional probability, $\sigma$-algebras etc. I have two proofs that I'm now sure how to handle. Problem 1. Prove that an ...