Questions tagged [martingales]
For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.
3,344
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Calculate the expectation of $\tau$=inf{$n\in\mathbb{N}_0|\exists{k}\lt{n}:S_{k}\le-1, S_n=1$}
Let $(Z_k)_{k\in\mathbb{N}} \text{ be a sequence of i.i.d. random variables on } (\Omega,\mathscr{F},\mathbb{P}) \text{ with }$
$$\mathbb{P}(Z_1=-1)=\frac{1}{2}=\mathbb{P}(Z_i=1)$$
and symmetric ...
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Proving a claim that $Z_n = \max\{(X_n^{(1)})^2, X_n^{(2)}\}$ is a submartingale
Let $X_n^{(1)}, X_n^{(2)}, n \in \mathbb{N}$ be martingales. Let $\mathcal{F}_n$ be a filtration. I believe that $Z_n := \max\{X_n^{(1)}, X_n^{(2)}\}$ is a submartingale. I tried to show it by using ...
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If $E[Y|X]=X$ and $Z$ independent of $(X,Y)$, what is the conditional distribution of $E[Y|X,X+Y+Z]$ given $(X,Y)$?
Let $X,Y$ be random variables such that $E[Y|X]=X$ and $Z$ independent of $(X,Y)$. Let $W=X+Y+Z$, and $V=E[Y|X,W]$. $V$ is a random variable that depends on $X$ and $W$. What is the conditional ...
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Prove random variable is in a set a.s and determine its distribution
This was one of the questions in my exam that i did not figure out.
For $\epsilon \in ]0,1[$, $\operatorname{f}:[0,1]\to[0,\infty[$ is a continuous function with $$\operatorname{f}(x) \le \min(x,1-x) ,...
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What is the underlying probability space where the stopping time is defined? [closed]
The book which I am studying defines a stopping time as follows:
Definition
Given a filtration $(\mathcal{F}_{n})_{n\in\mathbb{N}}$ and a map $T:\Omega\to\mathbb{N}\cup\{\infty\}$, we say that $T$ is ...
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Show process is a martingale
Problem 12.2 4 from Grimmett (Probability and Random Process):
Let $(M,\mathcal{F})$ be a martingale with $M_0 = 0$ having differences $D_r = M_r - M_{r-1}$. Show that, $X_n = M_n^2 - Q_n$ is a ...
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Maximal inequality regarding the empirical distribution function
Let $F$ be a distribution function and $F_n$ the corresponding empirical distribution function. By the empirical process theory, we have the following stochastic equicontinuity condition
: for some ...
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Show{$(X_n^{'},B_n), n\ge 0$} is again a positive supermartingale.
Suppose {$(X_n,B_n), n\ge 0$} is a positive supermartingale and $v$ is a stopping time. Define $X_n^{'}:=E(X_{v∧n}|B_n), n\ge 0.$
Show{$(X_n^{'},B_n), n\ge 0$} is again a positive supermartingale.
...
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1
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Dice Problem, optimal stopping
In a dice game, if you roll 1-5, you get money for the number of points you've rolled, and if you roll 6, you lose all the money you've accumulated, so you can roll any number of dice, so how much is ...
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Martingale that is not a Markov process: one technical issue in existing examples.
My question is technically not new. However, I am a bit confused by certain setups. A thread appeared here:
Martingale that is not a Markov process
The spirit of the answer provided by Did is clear. ...
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$E[X_t-X_{t+1}|F_t]=[X_t-X_{t+1}|X_0,...,X_t]?$ [closed]
Recently I have learned about filtration in probability theory, But I don't have a good grasp about it. What I want to confirm is the $F_t$ has nothing different with the history of the stochastic ...
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David Williams' Exercise 4.5 $\mathbb{P}(\text{limsup}(\frac{X_n}{\sqrt{2\log{n}}})\leq1)=1$
I am attempting Exercise 4.5 from David Williams' Probability with Martingales, which is about Borel-Cantelli lemma. The question states the follows.
If $G$ is a random variable with the normal N(0,1) ...
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Distribution and Martingale in Polya's urn
This is exercise 10.1 from Probability with Martingale by David Williams:
At time $0,$ an urn contains $1$ black ball and $1$ white ball. At each time $1,2,3,...,$ a ball is chosen at random from the ...
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martingale proposition-more lated to conditional expectation-looking for details operations
Let $(\Omega,\Sigma,\mathbb{P})=([0,1),\mathscr{B}([0,1)),\lambda)$, where $\lambda$ is the lebesgue measure. For $k\in\mathbb{N}$, define $$\mathcal{D}_k= \Bigg\{\Bigg[\frac{i}{2^k},\frac{i+1}{2^k}\...
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Expected number of steps before 1-d biased random walk hits a number, given it's before another number [duplicate]
We have a biased 1-dimensional random walk on the number line. Each timestep, with probability $p$ such that $0<p<1$, it increments by $1$, otherwise it decrements by $1$. Conditioning on the ...
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71
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Probability that a biased 1-d random walk hits a number before another
We have a biased 1-dimensional random walk on the number line. Each timestep, with probability $p$ such that $0<p<1$, it increments by $1$, otherwise it decrements by $1$. Given positive ...
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Convergence of total quadratic variation to predictable quadratic variation for continuous martingales: proof clarification.
Suppose that $X\in\mathcal{M}^2_c$ is a continuous square-integrable martingale. By the Doob-Meyer decomposition there exists an increasing predictable process $\left<X\right>_t$ such that $X_t^...
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Inequality for expected value of sum of increments of a martingale (from Karatzas and Shreve)
The following computations are based on Lemma 5.9 of Karatzas and Shreve.
Suppose $X$ is a continuous-time martingale and that $\left|X_t\right|\leq K$ $\mathbb{P}$-a.s. . Let $0=t_{0,n}<t_{1,n}<...
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some details questions about doob's maximal inequality
Q1
I was so confused about this maximal sequence, I thought it was a constant random variable sequence at first, but then I realised it should be an increasing sequence (not really sure). Then I have ...
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1
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Downcrossing inequality of submartingales
On page 14 of "Brownian motion and stochastic calculus" by Karatzas and Shreve, (iii) of theorem 3.8, the book has the downcrossing inequalities for a submartingale
$$ED_{[\sigma,\tau]}(\...
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Is a nonincreasing expectation of a function value sequence a supermartingale sequence?
Let $f: \mathbb{R}^n \times \omega \to \mathbb{R}$ where $\omega \sim D$. Assume $(f(x_k, \omega_k))$ be a random sequence such that
$$
\mathbb{E}_{\omega^{k+1}} [f(x_{k+1}, \omega_{k+1})] \leq f(x_{k}...
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How to prove the Martingale's property if we have a special stopping time?
$(X_n)$ is a sequence of $(F_n)$-adapted integrable random variables, where $(F_n)$ is a Filtration and $X_0=0$.
I have to prove that
1)$X_n$ is a martingale with a respect to $F_n$
iff
2)for any ...
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Convergence of the bounded below supermartingale process [closed]
I am studying this book where Theorem 4.2.12. on page 222 is the following:
If $X_n \geq 0$ is a supermartingale then as $n \to \infty$, $X_n \to X$ almost surely and $\mathbb{E}[X] \leq \mathbb{E}[...
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Williams 'Probability with martingales' E9.2
I am unsure about my (supposed) solution to exrecise 9.2 from the book "Probability with martingales" by David Williams.
The problem is as follows:
Suppose that $X,Y\in \mathcal{L}^1(\Omega,...
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0
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Utilize the discrete-time martingale central limit theorem to prove the continuous-time case
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$.
We know that if $(M_n)_{n\in\mathbb N_0}$ is a square-integrable ...
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proving stochastic processes are martingale(Ito Integral)
I need to show that two random processes $\{X_t\}_{t\geq0}$ are martingale.
(1) $X_t=\int^t_af(u)dB_u$ ($f$ is continuous)
(2) $dX_t=b(t,X_t)dB_t$ ($b$ is continuous)
My attempt: To prove the second ...
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Can you factor out a known Brownian Motion during integration?
Say I have a problem which is of the form, where $W(t)$ is a Brownian Motion and $f(a,b)$ is just some function where we will plug in terms $u, W(u)$. For this example, let's say that $f(a,b) = b-a^2b$...
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proof of the martingale central limit theorem
Let $(M_n)_{n\in\mathbb N_0}$ be a square-integrable martingale on a filtered probability spacce $(\Omega,\mathcal A,(\mathcal F_n)_{n\in\mathbb N_0},\operatorname P)$ with $M_0=0$ almost surely such ...
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central limit theorem for continuous-time martingales
I'm searching for a proof of the following result: If $(M_t)_{t\ge0}$ is a square-integrable martingale with statinary increments such that $$\frac1t[M]_t\xrightarrow{t\to\infty}\sigma^2$$ in $L^1$ ...
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What is the probability that a 2D symmetric random walk will reach a point $(x,y)$ before returning to the origin?
Let $p_n(\vec{x})$ be the probability that a symmetric random walk (SRW) on $\mathbb{Z}^n$ starting at the origin reaches the point $\vec{x}\in\mathbb{Z}^n-\{0\}$ before returning to the origin. What ...
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Two questions about conditional expectations
When $Y$ is $\mathcal{F}$-measurable, what are the minimal conditions for $E[XY|\mathcal{F}]=YE[X|\mathcal{F}]$ to hold? Is it enough to know that both $X$ and $XY$ are $L^1$ or do we need some ...
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Grisanov theorem application
I have the following Ito process: $X_t=e^{B_t-\frac{t}{2}}$ with $\{B_t\}_{t\geq 0}$ is a $\mathbb{P}$-Brownian motion and I want to find under which probability measure $\mathbb{Q}$ the process $X_t^...
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A gambler's fortune converges 0 or 1? (Martingale convergence theorem)
Let $ X_0 = 1 , X_n = \prod_{i=1}^n Y_i, \space (Y_n : nonnegative \space independent, \space E[Y_i]=1) \space \Rightarrow \space then, \space \{X_n\} \space is \space martingale. $
Knowing this, Now ...
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Optional sampling theorem with $L^2$ bounded martingale with stopping time
In this reference, theorem 1.3.3 in page 16, it claims that if:
$$
E[|M_{n \wedge T}|^2] \leq C < \infty \implies E[M_T] = E[M_0]
$$
Halfway through the proof I was stuck by this claim:
$$
E[|M_n|1\...
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1
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Gambler's Ruin: Expected time of ruin using martingale and stopping time
Setup:
Let $X_n$ be a Gambler Ruin's game such that
$$X_n = x + \sum_{i =1 }^n Y_k$$
with $x \in \left\lbrace1,2,.., 99 \right\rbrace$ the initial value and $(Y_k)_{k \in \mathbb{N}}$ i.i.d with ...
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Martingale processes
I've been finding some difficulties in solving this exercise;
Let $\{B_t\}_{t\geq 0}$ a standard Brownian motion with respect to the natural filtration $\{\mathcal{F}_t\}_{t\geq 0}$ and define the ...
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Conditional Expectation of Integral over square of Brownian Motion - PDE Approach
I am looking to compute the following using Ito's formula.
$$u(t,\beta_t) = \mathbb{E}(\int_t^T\beta_s^2ds|\beta_t)$$
Knowing the properties of brownian motion, it is rather easy to show that the ...
user848541
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A stopping time problem for a random walk with transition probabilities dependent on states
The Problem: In a one-dimensional random walk, at position $n > 0$, the probability of moving to $(n-1)$ is $\frac{n+2}{2n+2}$, and the probability of moving to $(n+1)$ is $\frac{n}{2n+2}$. ...
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A semimartingale differential/integration problem
Say i have a Markov chain $Y_t$ that is defined such that it has a semimartingale representation
$Y_{t,T}=Y_t+\int_t^T{\Lambda Y_s ds}+N_T$
where $\Lambda$ is the transition intensity matrix for $Y_t$ ...
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0
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Convergence of likelihood function
Given a probability space $(\Omega,\mathcal{F},(\mathcal{F}_n)_n)$ where $(\mathcal{F}_n)_n$ is a filtration of $\mathcal{F}$. Consider the sequence of random variables/vectors $(X_n)_n$ and $(\tilde{...
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Optimal strategy for doubling money in an unfair game
Assume we have $10\$$ and we want to double this money (so to end up with $20\$$ before going bankrupt) in an unfair game of coin tossing, where the probability of winning a single round is $p$ and of ...
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1
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45
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Choosing the stake before flipping a coin
I have a following problem:
"We play a game with a symmetric coin. Before $n$-th round, possibly relying on the results of previous games, we set the stake in the $n$-th game: we choose $b_n$, $1 ...
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Show that $W(Y_{\tau \land k})$ is a martingale where Y is a Markov chain and that it converges almost surley
My questions refers to this paper by T. Lyons. I'm quite new to stochastic processes.
Let $(Y_n)_{n \in \mathbb{N}}$ be a time-homogenous, recurrent, reversible Markov chain with statespace $X$ and ...
3
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Doob's Martingale Decomposition -- Proving that the Martingale component is indeed a Martingale
The Martingale-Part of the Doob decomposition for a stochastic process $(X_n)_n$ and filtration $(\mathcal F_n)_n$ is
$$M_n=X_0+\sum_{k=1}^n\bigl(X_k-\mathbb{E}[X_k\,|\,\mathcal{F}_{k-1}]\bigr)$$ (see ...
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Questions of the Proof of Kolmogorov's Strong Law of Large Numbers
From the Book of Jacod-Protter (Probability Essentials).
Theorem: Let $(X_j)$ be an independent and identically distributed sequence with $E\{|X_j|\} < \infty$. Then
\begin{equation*}
\lim_{n \...
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Determine $\mu \in \mathbb{R}$ so that an adapted process is a martingale and give the Doob decomposition of that adapted process
For all $n \in \mathbb{R}$ there is $(x_{n,i})_{i \in \mathbb{n}}$ a sequence of independently and identically distributed variables with $P(X_{1,1} \in \mathbb{N}) = 1$ and $\mu := \mathbb{E}(X_{1,1})...
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Martingale and $L^4$-integrability
Let $(\mathcal{F}_n)_{n \in \mathbb{N}}$ be a filtration and $(\mathcal{H}_n)_{n \in \mathbb{N}}$ be a reversed filtration. We consider a sequence $(X_n)_{n \in \mathbb{N}}$ of square integrable ...
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Martingale property and conditional expectation
We consider a cadlag stochastic process $(X_r)_r$ with non-decreasing sample paths and $X_0=0.$ Let $Y_r:=X_r-r.$ We suppose that $(Y_r)_r,(Y^2_r-r)_r,$ and $(Y_r^3-3rY_r-r)_r$ are martingales ...
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0
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Jump process: Integrand Phi(s) is left-continuous and adapted
Below is the statement on a theorem left continuous process in a jump process (Poissoon):
Source book: Stochastic Calculus Bk 2 Continuous Time model
Theorem: Assume that the jump process X(s) (...
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2
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Couldn't understand decomposition inside derivation of option valuation using Martingale method
In the derivation of option valuation using Martingale method in continuous time framework of my book named "Brownian Motion Calculus by Ubbo F Wiersema" I have faced some issue. I add the ...
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