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Questions tagged [martingales]

For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.

In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Expressing the following mathematically:

A sequence of random variables $X_0, X_1, \dots$ with finite means such that the conditional expectation of $X_{n+1}$ given $X_0, X_1, X_2, \dots, X_n$ is equal to $X_n$, i.e., $$\mathbb{E}[X_{n+1}\mid X_0, X_1, \dots, X_n] = X_n.$$

A one-dimensional random walk with steps equally likely in either direction $(p=q=\frac12)$ is an example of a martingale.

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Martingale convergence theorem for $L^2$

Let $(\Omega, F, P)$ be probability space with probability measure $P$. Theorem Let $X\in L^1(P)$, let $F_k$ be an increasing family of sigma algebras, $F_k \subset F$ and $F=\cup_{k=1}^{\infty} \sigma(F_k)$. Then, $$E[X|F_k] \to E[X|F] \mbox{ as…
Seongqjini
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Martingales and optional stopping in biased games

As a martingale is a model of a fair game, and using optional stopping theorem assumes that you are dealing with a martingale, is it even possible to use optional stopping to find stopping time in an unfair situation? And how? (In conncetion to the…
User123456789
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How can you model the quicksort algorithm as a martingale sequence?

A sequence of random variables $X_0 , X_1,....,X_n$ is called a martingale sequence if: $E[X_i | X_0,X_1,....,X$i-1$]$ = $X$i-1 The text that I am reading from says the following: Let us analyse quicksort on a fixed n input array and track the…
Aditya Naidu
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Martingale linearity property

Let $X_1, X_2,\ldots$ be i.i.d r.v's with mean $\mu$. Let $$S = \sum_{i=1}^n X_i$$ Let $F_n$ denote the information contained in $X_1,\ldots, X_n$ Show that $$E[S_n \mid \mathcal{F}_m] = S_m + (n-m)\mu$$ $$m
JKnecht
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Confused regarding this notation for scalar Xt?

In the above, we are told that both |Xt$\preceq$| $\preceq$ |Xt| and |Xt$\succ$| $\preceq$ |Xt|. Would appreciate clarification on 1) why this is the case, and 2) what the superscripts for |Xt$\preceq$| and |Xt$\succ$| are actually representing.
user277225
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Non-martingale with respected to the natural filtration, and satisfies $E[M_{n+1} | M_n]=M_n$

I am thinking about the exercise: Exercise 5. Give an example of a random sequence ($M_n$) such that $E[ M_{n+1} | M_n ] = M_n$ for all $n\ge0$, but which is not a martingale w.r.t. the filtration $F_n = \sigma(M_0, \dots , M_n)$. From Exercise 5 of…
Shangjie Yang
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Questions on proving a stochastic process to be a martingale

I need to prove that a stochastic process $M_{t} $to be a martingale, is it necessary and sufficient to prove that $E[M_{t}]=M_{0}$ and if so, can it be proved rigorously? Thank you!
oamc
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Is $Z$ a martingale?

$M$ is a continuous, strictly positive martingale. $Z$ is defined by: \begin{equation*} Z(0) = 1,~dZ = \frac{dM}{M} \end{equation*} Clearly $Z$ is a strictly positive local martingale. Is it a true martingale?
PJH
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martingale: Prouve that $(S_n=\frac{R_n}{n+2})$ is a martingale refer to $(R_n)$

A box has red balls and green balls. To each step, we take a ball and we put it back in the box with an other ball of the same color. At the beginning, the box has exactly one ball red and one ball red. Let $R_n$ the number of red balls after $n$…
idm
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Martingale: why $\mathbb E[S_{n+1}\mid R_0,...,R_n]=\frac{1}{m-1}\sum_{i=1}^{m-1}\mathbb E[X_i\mid Z_m,X_{m+1},...,X_N]$

Let $(X_k)$ a sequence i.i.d. of random variables such that $\mathbb E[|X_1|]<\infty $ and let fix $N\in\mathbb N$. We set, \begin{cases}Z_n=X_1+...+X_n\\ Y_n=\frac{1}{n}Z_n\\ R_n=Z_{N-n}\\ S_n=Y_{N-n}\end{cases} with $n\in\{0,...,N-1\}$. Show that…
idm
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Interpreting a sequence and showing that it is a martingale.

I saw that one guy already asked this question, but he did not get an answer and I wasn't able to comment his thread. So, hopefully this is allowed. I am wondering about the following problem: The problem How does one interpret that seq. and show…
Alumninou
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measurability in backwards martingales

$X$ is a backwards martingale with $X_0\in L^1 $ According to the convergence theorem:$X_{-n}\to X_{-\infty} $ a.s. But how to get the conclusion that $X_{-\infty}$ is $\mathcal F_{-\infty} $ measurable? Unlike the forwards…
Lookout
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Martingales and Integrals

Could someone explain why the following is a Martingale please? \begin{align} M_s = \int_0^s(1+u^2)dW_u \end{align} (where $W_t$ is standard Brownian motion). I'm used to determining martingales using the expectation operator. But I don't believe…
John Smith
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book recommendations for martingale theory (studying measure theory)

Any recommendations for someone studying martingale theory. My course recommends Probability with martingales - David Williams and Probability-2 Albert N. Shiryaev but I've found both to be very dry, with little to no examples or questions. I'm also…
veganwithabeef
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Polya's urn and SSRW.

I've got a question on a martingale that i've been stuck on for a good while. An urn contains $n$ white and $n$ black balls. We draw them one by one without replacement. We pay £1 for any black ball drawn but receive £1 for any white one. Denote by…
carsck
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