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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questions
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Are there good examples of martingale processes that are not simple random walks?
Are there non-trivial examples of martingale processes that aren't simple random walks?
I'm trying to better understand the difference between martingales and simple random walks. They look pretty ...
3
votes
1answer
52 views
Martingale property & limiting distribution for frequency of last names
Suppose that children always inherit their last names from their father (which implies that no new last names are ever created).
Pick a last name of interest (e.g. Smith), and let $X_n \in \left[0, 1\...
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1answer
26 views
Martingale in Cox Model
Can someone help me to show that
$$
\hat{A}(t, \beta_0) = \sum_{i=1}^{n} \int_0^t\frac{1}{\sum_{j}^n Y_j(s) e^{X_i^T \beta_0}} dN_i(s)
$$
is a martingale. The setup is the Cox proportional hazard ...
9
votes
2answers
221 views
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 ...
0
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0answers
78 views
Reverse-time martingale for non-polynomial approximating functions
Let $f(\lambda):(0,1)$ → $(0,1)$ be a continuous function. Given a coin with unknown probability of heads of $\lambda$, sample the probability $f(\lambda)$. One way to do so is to build randomized ...
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0answers
13 views
Markov process of order one a Martingale?
I have two questions, and I am very confused about the concepts
Can a Markov process of order one also be a Martingale?
Is any Markov process of order one also a Martingale?
can anyone help me solve ...
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0answers
20 views
How do I determine the set of all replicable claims?
Consider the one-periode model $(\Omega,F,\mathbb{P},S_0,S_1)$ with $\Omega=\lbrace \omega_1,\omega_2,\omega_3,\omega_4\rbrace ,\ F=$P$(\Omega),\ \mathbb{P}(\lbrace\omega_i\rbrace)>0$. There are ...
6
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1answer
158 views
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 $\mathbb{E}[X]=X_0$...
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0answers
36 views
How can I mathematically prove this time series when $e_t$ has i.i.d distribution?
Not really sure on how to simplify the $y_t$ because there is $y_{t-1}^2$
In order for a time series to be Martingale difference sequence the expected value given all the past value should be 0 and ...
2
votes
1answer
36 views
How can I test if a time-series data satisfies Markov's property and it is a martingale?
My question is about investigating some properties of time-series.
How can I test if my time-series data satisfies Markov's property?
How can I test if my time-series data is a martingale?
I wonder ...
2
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85 views
Carré du champ operator is a quadratic variation
Let $X_t$ be a real valued Markov process (starting at $x$) with generator $L$. Let $\Gamma(f)$ denote Carré du champ operator i.e. $L(f^2) - 2f \cdot L (f)$. As far as I know under suitable ...
2
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1answer
63 views
How to show that an m.d.s is not independent?
I have to prove that this Martingale Difference: $x_t = u_t u_{t-1}$ where $u_t \sim^{iid} (0, \sigma^2)$ is not serially independent, but am failing to do such thing.
I also have to prove that it's ...
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0answers
18 views
Is unit root the same with martingale
I'm confused between these two concepts.
Could you give an explanation?
Many thanks.
2
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1answer
45 views
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\...
2
votes
2answers
101 views
What is the meaning of $X \in \mathcal {F}$ in probability space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$?
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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0answers
43 views
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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3answers
576 views
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 $\omega \in \Omega$, $X_n(\omega)\sim \text{Unif}(0,X_{n-1}(\omega))$? ...
6
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2answers
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Prove that a simple random walk is a martingale
Note that $a$ has a mean of 0.
My approach:
$$X_t=X_{t-1}+a_t$$
$$E[X_{t+1}\mid X_1 + \dots+X_{t-1}]$$
$$=E[X_{t-1}+2a\mid X_1 + \dots+X_{t-1}]$$
$$=E[X_{t-1}\mid X_1 + \dots+X_{t-1}]+E[2a\mid X_1 + ...
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votes
2answers
290 views
Does a martingale difference sequence (mds) imply strong mixing?
I read this from an econometrics paper
"
The typical hypothesis which is imposed in the time series literature is that the $u_t$'s are either independent and identically distributed (i.i.d.) or a ...
3
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1answer
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Help to understand martingale example from Billingsley
I am studying Billingsley's section 35 about martingales and I have some difficulties understanding one of the examples.
This is the example 35.4.
Suppose we have a measurable space $(\Omega,\mathcal ...
0
votes
1answer
257 views
Martingale Difference Sequence
I saw that in the link on page 3 it is said $Y_t = e_t\cdot e_{t-1}$ is martingale difference sequence and dependent where $e_t$ is i.i.d with $N(0,\sigma^2)$ Could you provide me with the proof of it?...
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0answers
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Martingale Difference Sequence CLT
Could you provide me with the proof of the following:
$$n^{-1/2} \cdot\sum_t a_{t-j}e_t$$ converges to normal distribution as $n$ goes infinity by martingale difference sequence CLT where $a_t = \...
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votes
4answers
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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 ...
1
vote
2answers
211 views
How to check these sequences generated by i.i.d random variables are martingales?
Let $\{Y_n\}_{n\geq 1}$ be a sequence of independent, identically distributed random variables.
$P(Y_i=1)=P(Y_i=-1)=\frac12$
Set $S_0=0$ and $S_n=Y_1+...+Y_n$ if $n\geq 1$
I want to check if the ...
2
votes
2answers
417 views
log transform fixed PH in Cox model - how?
I have survival data to which I am fitting a Cox model with a continuous predictor. The cumulative martingale residual method (supremum test) of Lin, Wei and Ying
suggested that both proportional ...
2
votes
1answer
409 views
When is the posterior distribution equal to the prior?
So I have heard that if the prior distribution is in the subexponential class, applying Bayes rule does not change the belief. I have been trying to find an example of this but I am unable to do so. I ...
5
votes
1answer
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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 ...
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votes
2answers
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Cox PH model: managing continuous variables and linearity assumption
In an epidemiological study, I'm using martingale plot to assess the linearity of continuous variables.
Here are the Martingale Residuals (from Null Model) using R's ...
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0answers
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How to approach Basketball “Beat the Pro” drill with Markov Chain
Suppose, similarly to Gambler's ruin problem a Basketball player is doing "Beat the Pro" drill. That is, for every shot made, he scores one point, and for those missed, two points are to be deducted. ...
2
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0answers
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Azuma's inequality Use
For binary vectors, x = (x1,x2,...,xn) and y=(y1,y2,...,yn), the Hammond distance between them is defined by
d(x,y) = |x1 - y1| + |x2 - y2| + ... + |xn - yn|.
Let A be a finite set of such vectors ...
2
votes
0answers
104 views
Write expectation of brownian motion conditional on filtration as an integral?
Let $W_t$ be a Brownian motion, so $W_t=z_t \sqrt{t}$ where $z_t \in N(0,1)$ and the pdf of $z$ is
$f(z)=\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}$. So
$$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz
=\...
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1answer
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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....
2
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0answers
408 views
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 ...
3
votes
0answers
160 views
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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0answers
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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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2answers
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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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0answers
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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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1answer
3k views
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{...
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votes
0answers
18 views
An inequality about expectation [duplicate]
If $X_n$ is a martingales with $sup E|X_n|^p<\infty$ where $p>1$, How can show that
$$E^p|X_n|\leq E |X_n|^p$$
5
votes
1answer
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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 $...
9
votes
1answer
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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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0answers
86 views
The best constant in a martingale moment inequality
Suppose we have a stochastic integral of the form $M_{t}=\int_{0}^{t}{H_{u}dW_{u}}$ and we know $M$ is a (true) martingale. It is known that for all $p\geq1$,
$$
\mathbb{E}[|M_{t}|^{p}]^{1/p} \leq C(p)...
2
votes
1answer
153 views
How to Simplify the Representation of Local Martingales?
This is a follow-up to my previous question on MathOverflow.
Is there a way to combine the Dambis-Dubins-Schwarz theorem and the Martingale Representation Theorem to get the following result?
Let $...
6
votes
1answer
133 views
Martingale process
Let $\zeta(t)$ be a process with independent increments and $M(t)=E(\exp(\zeta(t))) < \infty $, show that $M(t)^{-1}\exp(\zeta(t))$ is a martingale.
So what I need to show is
$$E(M(t)^{-1}\exp(\...
2
votes
1answer
5k views
Difference between random walk and martingale
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, \...
0
votes
1answer
75 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 ...
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0answers
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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 ...
5
votes
1answer
2k 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
$$\int_0^T W_s ds = \int_0^T (T-s) dW_s ...
4
votes
1answer
111 views
Prove $Z_n = X_n1_{n \le T} + Y_n1_{n+1\ge T}$ is a martingale
Given a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}, \mathbb P)$, let $X = (X_n)_{n \in \mathbb N}$ and $Y = (Y_n)_{n \in \mathbb N}$ be $(\{\mathscr F_n\}, \mathbb P)-$...
2
votes
1answer
64 views
Prove Doob's using a certain Lemma
I am to prove Doob's (d) in the red box below:
What I tried:
Since $T < \infty$ a.s., we have
$$E[X_T] = E[\lim X_{T \wedge n}].$$
By Fatou's Lemma, we have
$$E\left[\lim X_{T \wedge n}\right] ...
