# 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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### Random walks and martingales

In class, our professor explained that the martingale process is the in between case of random walk type I (innovations are i.i.d.) and random walk type II (innovations are serially uncorrelated). ...
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### Projection operator: why squared norm of the sum of them is equal (or smaller) than the sum of the squared norms?

I am working through the proof of Lemma 2 in this paper (page 25, need it for my own research) and I am stuck at the very first step. Here, I will formulate a bit simplified version of this step. ...
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### Additive hazard model: Estimating a martingale residual process

Consider an additive hazard model with a single categorical covariate $$\alpha(t | x_i) = \beta_0(t) + \beta_1(t) x_i,$$ where e.g., $x_i \in \{0, 1\}$. To assess the goodness-of-fit of this model, ...
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### Pulling out constants out conditional expectation

Given $p_t = p_{t-1} + \varepsilon_t$ and $\varepsilon ~$i.i.d. with$~E(\varepsilon_t) = 0~\forall~t$ I wanna derive the conditional expectation for $p_t$ and I know it is $p_{t-1}$ since it is a ...
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### 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 ...
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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 ...
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### 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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### 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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### 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 ...
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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 $\mathbb{E}[X]=X_0$...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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\...
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### What is the meaning of $X \in \mathcal {F}$ in probability space $(\Omega ,{\mathcal {F}},P)$?

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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### 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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### 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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### 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....
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### 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 ...
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### 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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### 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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### 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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### $M_t$ 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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### 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 ...
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 $p\geq1$, $$\mathbb{E}[|M_{t}|^{p}]^{1/p} \leq C(p)... 1answer 163 views ### How to Simplify the Representation of Local Martingales? This is a follow-up to my previous question on MathOverflow. Is there a way to combine the Dambis-Dubins-Schwarz theorem and the Martingale Representation Theorem to get the following result? Let ... 1answer 139 views ### Martingale process Let \zeta(t) be a process with independent increments and M(t)=E(\exp(\zeta(t))) < \infty , show that M(t)^{-1}\exp(\zeta(t)) is a martingale. So what I need to show is$$E(M(t)^{-1}\exp(\...
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, \...