# Questions tagged [martingale]

In probability theory, a martingale is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings and only the current event matters.

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### Are these Bernoulli variables independent?

I was reading a paper in which it was assumed that $\varepsilon_1,\cdots,\varepsilon_n$ conditional on $X$ possess serial (non-linear) dependence, such that \begin{equation} P[\varepsilon_t\geq0\mid\...
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### What is the meaning of $X \in \mathcal {F}$ in probability space $(\Omega ,{\mathcal {F}},P)$?

I'm studying Martingales. In the text I keep finding $X \in \mathcal {F}$ with random variable $X:\Omega \rightarrow \mathbb{R}$. And $\mathcal {F}$ a $\sigma$-algebra of $\Omega$. What does it ...
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### weird outlier in a cox regression model

I'm using normal deviate residual to identify outliers, and I'm confused that my plot seems to suggest that there are unreasonably a lot of outliers...? Has anyone seen something like this? ...
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### Conditional expectation of random variables defined off of each other

First of all, when we say that $X_n \sim \text{Unif}(0,X_{n-1})$, what does that mean, rigorously? Does it mean that for every $\omega \in \Omega$, $X_n(\omega)\sim \text{Unif}(0,X_{n-1}(\omega))$? ...
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### Martingale Difference & Conditional Heteroskedasticity

I was reading this passage in a book and I am confused about the last line. As far as I understand, the white noise process does not allow for conditional heteroskedasticity due to its i.i.d. property....
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### Is a random walk necessarily a martingale?

I read in the following notes (http://www2.econ.iastate.edu/classes/econ672/Falk/_notes/lecture_4_martingales.pdf, p.2) that "a random walk is a martingale." Although it seems logical with the used ...
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### Upper bound for randomly weighted sum of independent random variables

I have a sequence of independent random variables {$\epsilon_j$} with mean 0. I also have another sequence of Bernoulli random variables $\delta_1, \delta_2,\dots$ which are dependent on the previous ...
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### 1_event set notation

I do not understand the notation as given here: The source for the image is page 17 in this paper after equation 18. I understand the first subset notation but what does 1_\e_t mean?
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### Why white noise process and IID process are considered martingale [closed]

Can anyone explain to me why A white noise process (εt) and An IID process (εt) are martingales?
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### $M_t$ Local Martingale if and only if satisfies $\frac{\partial f}{\partial t}+\frac{1}{2}\frac{\partial^2 f}{\partial x^2}=0$ proof?

I was reading the wikipedia article on local martingales and it states the following theorem Let $W_t$ be standard Brownian motion and let $f:[0,\infty)\times\mathbb{R}\rightarrow\mathbb{R}$ such ...
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### Semi-martingale vs. martingale. What is the difference?

Can someone please explain (preferably in layman's terms) what is the difference between semi-martingales and martingales? I have found the following sentence on Wikipedia: In probability theory, a ...
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I am trying to understand the diffrence between random walk and martingale. According to my understanding, a random walk without drift is $$y_{t} = y_{t-1} + u_{t}$$ where $u_{t}$ is $i.i.d.(0, \... 1answer 72 views ### martingales, stochastic processes Suppose Xn,$n\geqslant0$is a Markov chain on$\varphi =\left \{ 0,1,2,...,d \right \}$and$P(x,y)=\frac{\binom{2x}{y}\binom{2d-2x}{d-y}}{\binom{2d}{d}} $. States 0 and d are absorbing states for ... 0answers 57 views ### Does the LHS of$E[X_n | \mathscr F_{n-1}]$make sense even if$X_n$is not integrable or adapted? Let$(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$be a filtered probability space. Then$X_n$is a$(\{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)-$martingale if:$X_n$'s are ... 1answer 1k views ### How to compute expectation of square of Riemann integral of a random variable? How does one compute$E[(\int_0^T W_s ds)^2]$where$(W_t)_{t \in [0,T]}$is standard Brownian motion in$(\Omega, \mathscr F, \mathbb P)$? Apparently proving $$\int_0^T W_s ds = \int_0^T (T-s) dW_s ... 1answer 99 views ### Prove Z_n = X_n1_{n \le T} + Y_n1_{n+1\ge T} is a martingale Given a filtered probability space (\Omega, \mathscr F, \{\mathscr F_n\}, \mathbb P), let X = (X_n)_{n \in \mathbb N} and Y = (Y_n)_{n \in \mathbb N} be (\{\mathscr F_n\}, \mathbb P)-... 1answer 57 views ### Prove Doob's using a certain Lemma I am to prove Doob's (d) in the red box below: What I tried: Since T < \infty a.s., we have$$E[X_T] = E[\lim X_{T \wedge n}].$$By Fatou's Lemma, we have$$E\left[\lim X_{T \wedge n}\right] ... 2answers 54 views ### Probability of being$\$5$ up after $25$ plays of a game of Heads and Tails (fair coin) [duplicate]

In a game of heads and tails with a fair coin - you win $\$1$if heads; lose$\$1$ if ...