Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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.

A martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values.

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The magic money tree problem

I thought of this problem in the shower, it was inspired by investment strategies. Let's say there was a magic money tree. Every day, you can offer an amount of money to the money tree and it will either triple it, or destroy it with 50/50…
ElectronicToothpick
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Reference Request: Book on Unit Root Theory

In trying to do time series analysis, I almost regularly stumble upon unit root and cointegration tests. The design of most these tests is based on a null of unit root (for both linear and non-linear models) and the statistic's distribution is…
Dayne
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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 real valued process X is called a semimartingale if…
abu
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Distribution of $\frac{1}{1+X}$ if $X$ is Lognormal

Suppose $Z \sim \mathcal{N}(0,1)$. Suppose $X$ is a lognormally distributed random variable, defined as $X:=X_0exp^{(-0.5\sigma^2+\sigma Z)}$, in other words, $X$ is log-normal with $\mathbb{E}[X]=X_0$. Suppose we are interested in the variable of…
Jan Stuller
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Prove that a simple random walk is a martingale

Note that $a$ has a mean of 0. My approach: $$X_t=X_{t-1}+a_t$$ $$E[X_{t+1}\mid X_1 + \dots+X_{t-1}]$$ $$=E[X_{t-1}+2a\mid X_1 + \dots+X_{t-1}]$$ $$=E[X_{t-1}\mid X_1 + \dots+X_{t-1}]+E[2a\mid X_1 + \dots+X_{t-1}]$$ $$=E[X_{t-1}\mid X_1 +…
GarlicSTAT
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Cox PH linearity assumption: reading martingal residual plots

According to a lot of ressources about Cox PH model, continuous numeric variables should be tested for linearity assymption by plotting the Martingale residuals. In R, you can use survminer::ggcoxfunctional() to easily plot these residuals for the…
Dan Chaltiel
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Martingale process

Let $\zeta(t)$ be a process with independent increments and $M(t)=E(\exp(\zeta(t))) < \infty $, show that $M(t)^{-1}\exp(\zeta(t))$ is a martingale. So what I need to show is $$E(M(t)^{-1}\exp(\zeta(t))|F_s)= M(s)^{-1}\exp(\zeta(s))$$ What I've…
Parinn
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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 $$\int_0^T W_s ds = \int_0^T (T-s) dW_s \tag{*}$$ might need to assume that $E[(\int_0^T…
BCLC
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Power martingales for change detection: M goes to zero?

I'm trying to apply the power martingale framework by [Vovk et al., 2003] to change detection in unlabeled data streams, just like in [Ho and Wechsler, 2007]. The basic idea involves using a power martingale of the form $$M_n^{(\epsilon)} :=…
snikolenko
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Is it a valid algorithm to win at the casino roulette?

I would like to try the following algorithm in order to win in the roulette: Be an observer until there are 3 same parity numbers in a row ($0$ has no defined parity in this context) Once there were achieved 3 numbers with the same parity in a row:…
0x90
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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 $\omega \in \Omega$, $X_n(\omega)\sim \text{Unif}(0,X_{n-1}(\omega))$? This doesn't make much sense as $X_n(\omega)$ is a…
D.R
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ABRACADABRA Problem

As a complement to this answer for those not familiar with martingales. What is the expected number of keystrokes (or "time") it would take a monkey to type the string $\small \text{ABRACADABRA}$? Step-by-step intuitive solution light on math…
Antoni Parellada
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Martingales: Why must expected posterior equal prior?

For a posterior distribution to be plausible in the Bayesian sense (Bayes' Plausible), it is said that: $\mathbb{E}(\mu_{t+1} | \mu_t) = \mu_t$ where $\mu_t$ is the posterior distribution at time $t$, and consequently the prior at $t+1$ and…
Fatsho
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Finding $b$ such that $e^{5B_t - bt}$ is a martingale

I have $X_t = e^{5B_t}$ and Where $B_t$ is brownian motion at time $t$. $M_t = X_t \cdot e^{-bt}$ I need to find a value for $b$ such that $M_t$ is a martingale. I am encountering difficulty, however. $$\mathbb{E}[ e^{5B_t}e^{-bt} | \mathcal{F}_s]…
elbarto
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Help to understand martingale example from Billingsley

I am studying Billingsley's section 35 about martingales and I have some difficulties understanding one of the examples. This is the example 35.4. Suppose we have a measurable space $(\Omega,\mathcal F)$ and let $Q$ and $P$ be probability measures…
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