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.
17
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 ...
0
votes
1answer
49 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 ...
1
vote
1answer
47 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
93 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 ...
4
votes
1answer
273 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 ...
3
votes
2answers
239 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
0answers
38 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. ...
1
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0answers
37 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 ...
2
votes
0answers
88 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
=\...
0
votes
1answer
624 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....
2
votes
0answers
239 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
118 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 ...
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?
-1
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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?
0
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0answers
51 views
$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 ...
4
votes
1answer
1k 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{ABRACADABRA}$...
0
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0answers
16 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$$
4
votes
1answer
928 views
Martingales: Why must expected posterior equal prior?
For a posterior distribution to be plausible in the Bayesian sense (Bayes' Plausible), it is said that:
$\mathbb{E}(\mu_{t+1} | \mu_t) = \mu_t$
where $\mu_t$ is the posterior distribution at time $...
5
votes
1answer
2k views
Semi-martingale vs. martingale. What is the difference?
Can someone please explain (preferably in layman's terms) what is the difference between semi-martingales and martingales?
I have found the following sentence on Wikipedia: In probability theory, a ...
1
vote
0answers
73 views
The best constant in a martingale moment inequality
Suppose we have a stochastic integral of the form $M_{t}=\int_{0}^{t}{H_{u}dW_{u}}$ and we know $M$ is a (true) martingale. It is known that for all $p\geq1$,
$$
\mathbb{E}[|M_{t}|^{p}]^{1/p} \leq C(p)...
2
votes
1answer
114 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
101 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
$$E(M(t)^{-1}\exp(\...
2
votes
1answer
3k views
Difference between random walk and martingale
I am trying to understand the diffrence between random walk and martingale. According to my understanding, a random walk without drift is
$$ y_{t} = y_{t-1} + u_{t} $$
where $u_{t}$ is $i.i.d.(0, \...
0
votes
1answer
69 views
martingales, stochastic processes
Suppose Xn, $n\geqslant0$ is a Markov chain on $\varphi =\left \{ 0,1,2,...,d \right \}$ and $P(x,y)=\frac{\binom{2x}{y}\binom{2d-2x}{d-y}}{\binom{2d}{d}} $. States 0 and d are absorbing states for ...
1
vote
0answers
53 views
Does the LHS of $E[X_n | \mathscr F_{n-1}]$ make sense even if $X_n$ is not integrable or adapted?
Let $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ be a filtered probability space. Then $X_n$ is a $(\{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)-$martingale if:
$X_n$'s are ...
3
votes
1answer
806 views
How to compute expectation of square of Riemann integral of a random variable?
How does one compute $E[(\int_0^T W_s ds)^2]$ where $(W_t)_{t \in [0,T]}$ is standard Brownian motion in $(\Omega, \mathscr F, \mathbb P)$?
Apparently proving
$$\int_0^T W_s ds = \int_0^T (T-s) dW_s ...
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 $(\{\mathscr F_n\}, \mathbb P)-$...
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
2answers
53 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
216 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, ...,...
2
votes
3answers
154 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 ...
4
votes
2answers
249 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, ...
6
votes
3answers
588 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 ...
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 ...
6
votes
4answers
28k views
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 ...
3
votes
1answer
161 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 $...
1
vote
1answer
445 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
142 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) ...
