Questions tagged [martingales]

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

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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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1 vote
1 answer
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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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1 vote
1 answer
90 views

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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5 votes
2 answers
135 views

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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1 vote
0 answers
39 views

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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0 votes
1 answer
27 views

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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2 votes
0 answers
24 views

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 ...
1 vote
0 answers
19 views

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. ...
0 votes
1 answer
55 views

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 ...
4 votes
1 answer
72 views

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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0 votes
1 answer
27 views

$E[X_t-X_{t+1}|F_t]=[X_t-X_{t+1}|X_0,...,X_t]?$ [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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2 votes
0 answers
75 views

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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1 vote
1 answer
54 views

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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1 vote
0 answers
25 views

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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1 vote
1 answer
87 views

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 ...
0 votes
0 answers
71 views

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 ...
0 votes
0 answers
19 views

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^...
1 vote
0 answers
29 views

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}<...
2 votes
1 answer
50 views

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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0 votes
1 answer
35 views

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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1 vote
0 answers
21 views

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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0 votes
1 answer
67 views

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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0 votes
1 answer
25 views

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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2 votes
1 answer
68 views

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,...
1 vote
0 answers
24 views

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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1 vote
1 answer
44 views

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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5 votes
1 answer
87 views

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 $f(a,b) = b-a^2b$...
0 votes
1 answer
66 views

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 ...
  • 13.3k
1 vote
1 answer
81 views

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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6 votes
2 answers
168 views

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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2 votes
1 answer
48 views

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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0 votes
0 answers
39 views

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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2 votes
2 answers
79 views

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 ...
1 vote
1 answer
50 views

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\...
3 votes
1 answer
154 views

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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2 votes
1 answer
69 views

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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0 votes
1 answer
68 views

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 ...
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7 votes
1 answer
135 views

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 $\frac{n}{2n+2}$. ...
0 votes
0 answers
31 views

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 $Y_t$ ...
1 vote
0 answers
30 views

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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3 votes
1 answer
133 views

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 vote
1 answer
45 views

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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2 votes
0 answers
33 views

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 votes
2 answers
130 views

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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2 votes
1 answer
69 views

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 \...
1 vote
0 answers
34 views

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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2 votes
0 answers
61 views

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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3 votes
0 answers
91 views

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 votes
0 answers
19 views

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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3 votes
2 answers
82 views

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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