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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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 ...
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 ...
6
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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 ...
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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:
$$X_t=X_{t-1}+a_t$$
$$E[X_{t+1}\mid X_1 + \dots+X_{t-1}]$$
$$=E[X_{t-1}+2a\mid X_1 + \dots+X_{t-1}]$$
$$=E[X_{t-1}\mid X_1 + \dots+X_{t-1}]+E[2a\mid X_1 + ...
6
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1answer
132 views
Martingale process
Let $\zeta(t)$ be a process with independent increments and $M(t)=E(\exp(\zeta(t))) < \infty $, show that $M(t)^{-1}\exp(\zeta(t))$ is a martingale.
So what I need to show is
$$E(M(t)^{-1}\exp(\...
6
votes
3answers
741 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 ...
5
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1answer
136 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 $\mathbb{E}[X]=X_0$...
5
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1answer
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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 ...
5
votes
3answers
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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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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 ...
5
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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 $...
4
votes
2answers
351 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
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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{...
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votes
1answer
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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)-$...
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2answers
874 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
69 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 ...
3
votes
1answer
210 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 $...
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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. ...
3
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0answers
153 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
2answers
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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 ...
2
votes
2answers
189 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 ...
2
votes
2answers
295 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
151 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
1answer
4k 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} $$
where $u_{t}$ is $i.i.d.(0, \...
2
votes
1answer
49 views
How to show that an m.d.s is not independent?
I have to prove that this Martingale Difference: $x_t = u_t u_{t-1}$ where $u_t \sim^{iid} (0, \sigma^2)$ is not serially independent, but am failing to do such thing.
I also have to prove that it's ...
2
votes
1answer
343 views
When is the posterior distribution equal to the prior?
So I have heard that if the prior distribution is in the subexponential class, applying Bayes rule does not change the belief. I have been trying to find an example of this but I am unable to do so. I ...
2
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3answers
347 views
How to show $M_n = X_n^2-n$ is a martingale?
Let $X_n, n = 0, 1, 2, . . .$ denote an unbiased Normal Random Walk.
$X_0 = 10$, and $X_{n+1} = X_n + Y_{n+1}$, with $\{Y_n\}$ are i.i.d. $N(0, 1)$.
Then how can I show that:
A) $M_n = X_n^2-n$ is a ...
2
votes
1answer
62 views
Prove Doob's using a certain Lemma
I am to prove Doob's (d) in the red box below:
What I tried:
Since $T < \infty$ a.s., we have
$$E[X_T] = E[\lim X_{T \wedge n}].$$
By Fatou's Lemma, we have
$$E\left[\lim X_{T \wedge n}\right] ...
2
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0answers
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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
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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\...
2
votes
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 ...
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0answers
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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
384 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 ...
2
votes
1answer
305 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
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1answer
32 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 ...
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vote
2answers
190 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
2answers
55 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 ...
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1answer
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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 ...
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vote
1answer
174 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
536 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 , ...
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vote
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)...
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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 ...
0
votes
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....
0
votes
1answer
228 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
73 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
19 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 ...
0
votes
0answers
30 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 ...
0
votes
0answers
17 views
Is unit root the same with martingale
I'm confused between these two concepts.
Could you give an explanation?
Many thanks.
0
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0answers
36 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?
...
0
votes
0answers
63 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 = \...
