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.
45
questions
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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 ...
6
votes
4answers
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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 ...
6
votes
1answer
2k views
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 ...
6
votes
1answer
107 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
$$E(M(t)^{-1}\exp(\...
6
votes
3answers
647 views
Power martingales for change detection: M goes to zero?
I'm trying to apply the power martingale framework by [Vovk et al., 2003] to change detection in unlabeled data streams, just like in [Ho and Wechsler, 2007]. The basic idea involves using a power ...
4
votes
2answers
295 views
Finding $b$ such that $e^{5B_t - bt}$ is a martingale
I have $X_t = e^{5B_t}$ and
Where $B_t$ is brownian motion at time $t$.
$M_t = X_t \cdot e^{-bt}$
I need to find a value for $b$ such that $M_t$ is a martingale.
I am encountering difficulty, ...
4
votes
1answer
1k views
How to compute expectation of square of Riemann integral of a random variable?
How does one compute $E[(\int_0^T W_s ds)^2]$ where $(W_t)_{t \in [0,T]}$ is standard Brownian motion in $(\Omega, \mathscr F, \mathbb P)$?
Apparently proving
$$\int_0^T W_s ds = \int_0^T (T-s) dW_s ...
4
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1answer
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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 $\small \text{ABRACADABRA}$...
4
votes
1answer
602 views
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 ...
4
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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 $...
3
votes
2answers
435 views
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 ...
3
votes
1answer
97 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 $(\{\mathscr F_n\}, \mathbb P)-$...
3
votes
1answer
179 views
Basic question about the filtration in the martingale formed by sum of iid random variables
It is known that if $X_i$ are iid, $E(|X_i|) < \infty$ and $E(X_i) = 0$ then $S_n = \sum_1^n X_i$ is a martingale. Suppose all $X_i$ are defined wrt sample space $\Omega$.
I don't understand why $...
3
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0answers
49 views
How to approach Basketball “Beat the Pro” drill with Markov Chain
Suppose, similarly to Gambler's ruin problem a Basketball player is doing "Beat the Pro" drill. That is, for every shot made, he scores one point, and for those missed, two points are to be deducted. ...
3
votes
0answers
127 views
Upper bound for randomly weighted sum of independent random variables
I have a sequence of independent random variables {$\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 ...
2
votes
1answer
3k 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, \...
2
votes
1answer
147 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 ...
2
votes
1answer
127 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 $...
2
votes
3answers
172 views
How to show $M_n = X_n^2-n$ is a martingale?
Let $X_n, n = 0, 1, 2, . . .$ denote an unbiased Normal Random Walk.
$X_0 = 10$, and $X_{n+1} = X_n + Y_{n+1}$, with $\{Y_n\}$ are i.i.d. $N(0, 1)$.
Then how can I show that:
A) $M_n = X_n^2-n$ is a ...
2
votes
0answers
31 views
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 ...
2
votes
0answers
93 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
$$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz
=\...
2
votes
0answers
270 views
Is a random walk necessarily a martingale?
I read in the following notes (http://www2.econ.iastate.edu/classes/econ672/Falk/_notes/lecture_4_martingales.pdf, p.2) that "a random walk is a martingale."
Although it seems logical with the used ...
1
vote
2answers
54 views
Probability of being $\$5$ up after $25$ plays of a game of Heads and Tails (fair coin) [duplicate]
In a game of heads and tails with a fair coin - you win $\$1$ if heads; lose $\$1$ if ...
1
vote
1answer
1k views
How to assess the functional form of covariates in the Cox model with martingale residuals in R?
I want to find if the functional forms of covariates in my Cox model are linear. I understand the way to do this is to plot the Martingale residuals against the covariate of interest.
I have found ...
1
vote
1answer
150 views
Limiting behavior of a martingale
This is a homework question:
Suppose that $X_0=1$ and that for $n\geq 1$ $$X_n\sim \left\{
\begin{array}{l l}
U(0,X_{n-1}) & \quad \text{with probability $1-X_{n-1}/2$}\\
U(X_{n-1},1) ...
1
vote
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
$$E\left[\lim X_{T \wedge n}\right] ...
1
vote
1answer
470 views
Martingale and deterministic functions
Suppose:
$u_t \sim N(0,1) \ iid.$, $X_t = g(X_{t-1}) \cdot u_t$ whereas $g(X)$ can be any deterministic function.
Is this sufficient to define a martingale?
So does it hold: $E(X_t|X_{t-1}, \ldots , ...
1
vote
1answer
79 views
log transform fixed PH in Cox model - how?
I have survival data to which I am fitting a Cox model with a continuous predictor. The cumulative martingale residual method (supremum test) of Lin, Wei and Ying
suggested that both proportional ...
1
vote
0answers
40 views
Azuma's inequality Use
For binary vectors, x = (x1,x2,...,xn) and y=(y1,y2,...,yn), the Hammond distance between them is defined by
d(x,y) = |x1 - y1| + |x2 - y2| + ... + |xn - yn|.
Let A be a finite set of such vectors ...
1
vote
0answers
84 views
The best constant in a martingale moment inequality
Suppose we have a stochastic integral of the form $M_{t}=\int_{0}^{t}{H_{u}dW_{u}}$ and we know $M$ is a (true) martingale. It is known that for all $p\geq1$,
$$
\mathbb{E}[|M_{t}|^{p}]^{1/p} \leq C(p)...
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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 ...
1
vote
1answer
244 views
Proving Y is a martingale using indicator functions
Prove the stochastic process $Y = (Y_n)_{n \geq 0}$ is a martingale w/rt the filtration $\mathscr{F} = (\mathscr{F_n})_{n \geq 0}$, where $\mathscr{F_n} = \mathscr{F_n}^{Y} \doteq \sigma(Y_0, Y_1, ...,...
0
votes
1answer
862 views
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....
0
votes
1answer
45 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?...
0
votes
1answer
79 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 ...
0
votes
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 ...
0
votes
0answers
23 views
Does a martingale difference sequence (mds) imply strong mixing?
I read this from an econometrics paper (Dufour and Pelletier, 2014)
"
The typical hypothesis which is imposed in the time series literature is that the $u_t$'s are either independent and identically ...
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0answers
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Is this weird pattern in my Martingale Residuals plot normal?
I'm running a cox model:
coxtest<-coxph(Surv(time=Age_Sample,time2=Age_event,event=LungCancer)~Cigarettes_Lifetime,data=mydata)
Then I do
...
0
votes
0answers
31 views
What is the Interpretation of the Alternative Hypothesis of the Conditional Predictive Ability Test [Giacomini and White (2006)]?
I am reading "Tests of Conditional Predictive Ability" by Giacomini and White (2006).
The Null Hypothesis of equal Conditional Predictive Ability is formulated as
$H_0: E[\Delta L_{m,t+\tau}|G_t]=0 \...
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votes
0answers
29 views
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 = \...
0
votes
0answers
16 views
Estimating conditional expectation using SVR
I have an estimation problem: $X_t=f(S_t)=E(X_T\mid S_t)$ given the observations $(X^i_{T},S^i_{t})$ for all $i$'s. I used support vector regression (SVR) to run a regression of $X_T$ on $S_t$. Here $...
0
votes
0answers
24 views
1_event set notation
I do not understand the notation as given here:
The source for the image is page 17 in this paper after equation 18.
I understand the first subset notation but what does 1_\e_t mean?
0
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0answers
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$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 ...
0
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
$$E^p|X_n|\leq E |X_n|^p$$
-1
votes
2answers
2k views
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?
