# Questions tagged [martingales]

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

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### Martingales and stopping rules [closed]

\textbf{Question:} Let (X_1, X_2, \ldots, X_n) be a sequence of independent and identically distributed (iid) uniform variables on the interval ([-2, 2]), and define (S_n = X_1 + X_2 + \ldots + X_n). ...
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### Convergence of a specific martingale

Let $(S_n)_{n\geq 1}$ be a martingale s.t. $\mathbb{E}[{S_n}^2]<K<\infty$ for all $n$. Assume that $Var(S_n)\to 0$ as $n\to\infty$. Now I would like to prove that $S_n$ converges almost surely ...
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### Optional stopping time of a simple symmetric random walk on the square lattice

Let $S_n$ be a simple symmetric random walk on the square lattice $\mathbb{Z}^2$ with $S_0=(0,0)$. That is, the walker starts from the origin and at each step independently, she steps one unit to East,...
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### Computing expected value of hitting time for a Feller process

We consider a Feller-Dynkin Markov process $X$ with generator $G$, which, when restricted to $C^2$ functions with compact support, is given by $Gf(x) = \frac{c(x)}{2}f''(x)$, where $c$ is a positive ...
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### book recommendations for martingale theory (studying measure theory)

Any recommendations for someone studying martingale theory. My course recommends Probability with martingales - David Williams and Probability-2 Albert N. Shiryaev but I've found both to be very dry, ...
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### Under what conditions, if any, is a continuous function of a Martingale still a martingale?

Consider a martingale $M_s$ and a function $f(M_s) = \frac{1-e^{-kM_s}}{M_s}$. If $M_s$ is an Ito process, it is obvious from Ito's Lemma that $f(M_s)$ is not a Martingale. Does the restriction hold ...
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### Expectation and Variance of $Y_t = \int_0^t sdW_s$ + Martingale property

Denote the process $Y_t = \int_0^t sdW_s$. I want to answer the following question: Q) Calculate the expectation and variance of $Y_T$. Is $Y_T$ a martingale? A) I have read somewhere that it was ...
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### Relation between measurability of intensity, compensator, and martingale in Doob-Meyer decomposition

Consider a sub-martingale (e.g., a counting process) $N_t$ adapted to a filtration $\mathcal{F}_t$. According to the Doob-Meyer decomposition, we have: $$N_t = M_t + A_t,$$ where $M_t$ is a martingale ...
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### Is a martingale difference in terms of some random variables $X_t$ also a martingale difference in terms of $f(X_t)$ for a measurable function f?

The context is that in a version of central limit theorem for martingale difference (Billingsley 1961), the condition is for some random variable $u_n$ $$\mathbb{E}[u_n|u_1,...,u_{n-1}]=0\tag{1}$$ ...
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### Proof that the stochastic exponential is a local martingale

I'm struggling to understand the proof as to why the stochastic exponential is a continuous non-negative local martingale. My notes say the following: where 3.6 is $Z_t = 1 + \int^t_0 Z_s dX_s$. I ...
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### Martingale theorem representation using two martingales

Suposse we have two martingales $N$ and $M$, and the hypothesis of martingale representation theorem holds. Then \begin{equation} N_t=\int_0^t\psi_sdW_s, \,\, M_t=\int_0^t\phi_sdW_s \end{equation} So, ...
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### class of processes guaranteed to exceed any thresholds

Is it correct to say that any martingale (e.g., a GBM or anything similar), taking values in some real state space and not converging to a degenerate random variable, has the property that, given a ...
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### Polya's urn and SSRW.

I've got a question on a martingale that i've been stuck on for a good while. An urn contains $n$ white and $n$ black balls. We draw them one by one without replacement. We pay £1 for any black ball ...
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### Brownian motion is not of bounded variation

I would like to prove that Brownian motion, denoted $B_t$, is not of bounded variation using the fact that its quadratic variation is finite. Here is my attempt: Consider $[t,s]\subset[0,+\infty)$ and ...
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### Almost Sure Convergence (Durrett 2.4.1)

Suppose the $j^{th}$ light bulb burns for an amount of time $X_j$ and then remains burned out for time $Y_j$ before being replaced. Suppose the $X_j$,$Y_j$ are positive and independent with the $X_j$’...
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### Limit and Convergence of Doob Martingales

Let $(\Omega, \mathcal{F}, \mathcal{P})$ be a probability space for $X\in L^1(\mathcal{P})$ and $(\mathcal{F}_n)$ a filtration. Then $$Y_n = \mathbb{E}(X|\mathcal{F}_n)$$ is called Doob Martingale. ...
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### Almost sure convergence in density of infinite dimensional product measure of normal distributions unsing Kakutani's theorem

In measure theory Kakutani's Theorem is used to determine if two infinite product measures are equivalent, meaning they are both absolutely continuous with respect to each other. In one of my text ...
I'm analyzing a simple urn model, defined for $N>1$ an integer. The setup is as follows: we have an urn with $\frac{N}{2}$ white balls and $\frac{N}{2}$ black balls. At each step $k$, we sample $N$ ...
Prove that: A right-continuous process $X$ adapted to the filtration $\{\mathcal{F}_t\}$ is a martingale if and only if for any bounded stopping time $T$, $X_T \in L^1$ and $E[X_T] = E[X_0]$. I know ...