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Questions tagged [martingales]

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

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Markov versus Martingale

My text provides an illuminating example about the fact that Markov processes are not martingales in general and martingales are not Markov processes in general. First, the standard brownian motion $(...
Quasar's user avatar
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The sum of independent martingales

How to the following statements. If $X=(X_t)$ is a $\mathcal{F}^X$-adapted martingale and $Y=(Y_t)$ is a $\mathcal{F}^Y$-adapted martingale and $\mathcal{F}^X, \mathcal{F}^Y$ independent, then $(X_t + ...
Maxwell Zhan's user avatar
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One question about exponential martingale inequality at a paper of Ann. of Math.

I saw a strange inequality about martingales: If $\operatorname{M}\left(s,t\right)$ is an continuous $L^{2}$ martingale start at $s$, $\left[\operatorname{M}\right]\left(s,t\right)$ is its quadratic ...
shanlilinghuo's user avatar
6 votes
2 answers
71 views

Is is true that $\lim E[Z_n| F_n] = E[Z|F_{\infty}] $, where $Z_n \rightarrow Z$ a.s., $(Z_n)$ bounded and $F_\infty =\cap_{n\geq1}F_n$

Let $(Z_n)_{n \geq 1}$ be a sequence of bounded random variables converging a.s. to some $Z$. Let $(\mathcal F_n)_{n \geq 1 }$ be a sequence of decreasing filtration and let $\mathcal F_\infty = \...
Jeffrey Jao's user avatar
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Maximum of two semimartingales

I get curious after reading this post on maximum of two Brownian motion: Process properties of the maximum of two independent linear Brownian motions. It occurs that for two stochastic processes $A=(...
John He's user avatar
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176 views

Help needed findint the expected value of this stopping time

Let $\xi_i$ be iid random variables with $\mathbb{E}[\xi_i]=0$, and define: $$S_{(k)} = \sum_{i=1}^N \xi_{i+k}$$ Now, define: $$\tau = \min \left\{ k : S_{(k)} \notin (a,b) \right\}$$ How can I find $\...
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If $X_n$ is martingale, $N$ is a stopping time, is $X_{n+N}$ a martingale?

Is this true? If it is, can we change martingale to sub or super? My attempt (On submartingale): $\mathbb{E}[X_{n+N+1}\vert X_{n+N}]=\mathbb{E}[\mathbb{E}[X_{n+N+1}\vert N,X_{n+N}]\vert X_{n+N}]\ge \...
Ho-Oh's user avatar
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Questions in proving $\mathbb{P}\left(T_a<\infty\right)=1$ with $T_a:=\inf \{t>0: B_t \ge a\}$

Let $\left(B_t, t \geq 0\right)$ be a one-dimensional Brownian motion starting from the origin (i.e, $\left.B_0=0\right)$. Let $\mathcal{F}_t:=\sigma\left(B_s: s \leq t\right)$ be the filtration ...
Ho-Oh's user avatar
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Asymptotic behaviour of the difference process of the Polya urn

Assume we have a Polya urn with $(a,b)$ initial (time t=0) red and green balls, respectively. At each point in time we draw a ball at random and add it back together with further $S$ balls. This means ...
Irene's user avatar
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Show this truncated stochastic process is a martingale

Consider a filtered space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq 0}, P)$. Let $(M_t)$ be a uniformly integrable (UI) martinagle w.r.t. $\{\mathcal{F}_t\}_{t\geq 0}$. Let $\tau, \nu$ be two ...
Mingzhou Liu's user avatar
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65 views

Gambler ruin's: Probability of $k$ consecutive win before j consecutive loss

Assume that a stock has a probability of $p$ to win, a probability of $q$ to lose, and a probability of $(1-p-q)$ to remain every day. What is the probability of $k$ consecutive wins occur before $j$ ...
Zhihao Xu's user avatar
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Continuity of a Family of Supermartingales

Suppose $\{Z_t(\alpha)\}_{\alpha}$ is a family of supermartingales of class (D) that is known to be jointly continuous, almost surely, as a function of $(t, \alpha)$. Say $(t, \alpha) \in [0, T] \...
qp212223's user avatar
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Expected stopping time for biased random walk with increasing stepsize

Let $S_n$ be a stochastic process, with $$ S_{n+1} = S_n - 1 + \begin{cases} n^2 & \mathrm{with\;probability\;}\frac{1}{2}\\ -n^2 &\mathrm{else} \end{cases}. $$ Let now $T:=\inf\{n\geq0:S_n\...
lorenzw's user avatar
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Renewal reward process's reward tail probability

Suppose we are given a dice with $K$ faces, denoted by $k=1,\dots,K$, where the probability of realizing a face $k$ is $p_k\in[0,1]$ with $\sum_{k=1,\dots,K}p_k=1$. Now, we roll the dice repetitively. ...
hshlmh's user avatar
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Is there a closed form solution to the distribution of the running maximum for Gaussian processes?

Let $X=(X_t)_{t\geq0}$ be a zero mean Gaussian process with variance $\sigma(t)=\mathbb{E}X_tX_t^\top$ and define $$ S_T=\sup_{t\leq T}X_t\quad T\in[0,\infty) $$ the running maximum of $X$. Question: ...
Daan's user avatar
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On the Optimal Transport Distance in Equivalent Martingale Measures

I've been working on this idea for a while and could use some guidance. Right now I'm wondering if the formulas I am writing out make sense. Any help is very much appreciated. Problem Statement Idea ...
jeffery_the_wind's user avatar
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Equivalent characterizations of a Poisson process

Consider a counting process $N(t)$, and the following three conditions that I believe are equivalent: $$ \begin{aligned} (A) \quad& N(t) \text{ is a Poisson process of intensity }λ \\ (B) \quad&...
Julius Plenz's user avatar
1 vote
1 answer
113 views

Expected stopping time of summation of ranking supermartingale is finite

Given a stochastic process (that takes on real numbers) $X_n$, which is a ranking supermartingale, which we defined as $\mathbb{E}(X_{n+1}-X_n|\mathcal{F_n})\leq \epsilon\ < 0$. Let now $Y_n$ be ...
lorenzw's user avatar
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Extending Law of Large Number for Square integrable Martingale

Suppose we have a square integrable martingale $\{M_n\}$ with $\lim_{n\to\infty}M_t=\langle M\rangle_n=\infty$ a.s. Now, if we have a non-decreasing function $g:[0,\infty)\to[0,\infty)$, and also $$\...
Arbitor Lunae's user avatar
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$X\in L^{\infty,p},Y\in L^{2,q},prove \int_{0}^{t} XY_{s}dB_{s}$ is martingale

$L^{\infty,p}$ is X is progressively measurable and $E[\max_{0\le t\le T} \left | X_{t} \right |^{p} ]<\infty$ $L^{2,q}$ is Y is progressively measurable and $E[(\int_{0}^{T}\left | Y_{t} \right |^{...
Yu GongLian's user avatar
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1 answer
27 views

Showing that $M^T(N-N^T)$ is a continuous local martingale for all stopping times $T$ and continuous local martingales $M,N$

Given is that a cadlag adapted process $X=(X_t)_{t\geq 0}$ is a martingale if and only if $\mathbb{E}X_T=\mathbb{E}X_0$ and $X_T\in L^1$ for every bounded stopping time $T$. Now let $M,N$ be two ...
Daan's user avatar
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20 views

Optional sampling theorem for a continuous martingale with two bounded stopping time

My question is related to this one, in which it claims that if $X$ is a martingale,$\tau\ge\sigma$ are two bounded stopping times, then $$\mathbb E[X_\tau|\mathcal F_\sigma]=X_\sigma$$ I know one ...
PPP's user avatar
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A question on the proof of the ratio limit theorem for Brownian motions.

I am trying the prove the following ratio limit theorem, and I think I got it but I am not entirely sure that the mathematical logic in the last part of the proof is correct, and so I am asking if ...
Daan's user avatar
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2 answers
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Alternative form of Kolmogorov extension theorem on filtered space

In an thesis HERE the author proved (categorically) an alternative form of Kolmogorov extension theorem as follows (p. 129~p. 130): let $I$ be an nonempty poset, $(\Omega,(\mathcal{F}_i)_{i\in I}, \...
Westlifer's user avatar
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2 votes
1 answer
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Can we always find a martingale part in a càdlàg supermartingale?

Abstract: By Doob-Meyer decomposition theorem, for any càdlàg supermartingale $Z$, there exists a unique predictable increasing process $A$ starts from $A_0=0$ such that $Z+A$ is a local martingale ...
Hirofumi Shiba's user avatar
3 votes
1 answer
59 views

When does a local supermartingale become a proper supermartingale?

Abstract: When a local supermartingale is bounded from below, is it a proper supermartingale? Question: In remark 4.2 (p.16) of the lecture notes by Martin Hairer, the conditions when $$ [0,\infty)\ni ...
Hirofumi Shiba's user avatar
2 votes
0 answers
30 views

Empirical concentration of sum of Bernoulli variables around unknown expectation

Setup Let $S_n = \frac{1}{n}\sum_{i=1}^n X_i$ where $X_i\in \{ 0,1 \}$ are i.i.d. Bernoulli's with some unknown probability $p\in[0,1]$. We want to provide concentration for $S_n$ around its mean $p$ (...
DivergentSeries's user avatar
2 votes
1 answer
74 views

Quadratic covariation of martingale transforms to simple processes.

Let $X$ and $Y$ be simple processes, that is $X_t=\sum_{n=0}^\infty\xi_n{1}_{(t_{n},t_{n+1}]}(t)$ for a uniformly bounded sequence $(\xi_n)_{n\in\mathbb{N}}$ of random variables so that $\xi_n$ is $\...
Daan's user avatar
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2 votes
1 answer
49 views

Sum of hitting times and hitting times of sum of Brownian Motion

I have came across a question that asks to prove that for a brownian motion $B$, the first hitting time of $T_a=\inf\{t≥0 : B_t=a\}$ has a $1/2-$stable distribution, in that if we have $n$ independent ...
R.V.N.'s user avatar
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1 vote
1 answer
64 views

How can I show that $\mathbb{E}(T)=\frac{a^2}{\sigma^2}$?

Let $(X_i)_{i=1,2,...}$ be a sequence of iid random variables such that $\mathbb{P}(X_i=\pm 1)=\frac{1}{2}$ and with $\operatorname{Var}(X_i)=\sigma^2>0$. Set $S_t=\sum_{k=1}^t X_i$ for $t=1,2,...$ ...
user123234's user avatar
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2 votes
1 answer
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Prove that for symmetric random walk $S_n$ it holds that $S_{min\{n,\tau\}}^2 - \min\{n,\tau\}$ is a uniform integrable martingale

I want to show that for a stopped symmetric random walk $S_n$ we have that $S_{min\{n,\tau\}}^2 - \min\{n,\tau\}$ is a uniform integrable martingale. Where $\tau = \inf\{n \ge 0 : S_n \in \{-a,b\}\}$. ...
user007's user avatar
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1 vote
0 answers
42 views

Ville-like inequality for double-indexed martingales

Given a non-negative (super)-martingale $M_t$ (with respect to a filtration $(F_t)$), such that $M_0\equiv 1$, Ville's inequality states that $$\mathbb P(\exists t>0: M_t\geq 1/\delta)\leq \delta\,,...
ECL's user avatar
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4 votes
1 answer
80 views

An example such that $(X_n)$ and $(Y_n)$ are martingales with respect to their natural filtration and $(X_n +Y_n)$ is no martingale [closed]

I know that if $(X_n)$ and $(Y_n)$ are martingales with respect to their natural filtration then $(X_n +Y_n)$ does not have to be a martingale with respect to its natural filtration. However, I did ...
user007's user avatar
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1 vote
1 answer
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Show $M^2$ is of class D if $M$ is a martingale bounded in $L^2$

Let $M$ be a martingale null at $0$ and bounded in $L^2$. Then $M^2$ is a submartingale (just apply Jensen's inequality), and by Doob's $L^2$ inequality, we have $$E\left[\sup_t M_t^2\right]\leq 4E\...
Mingzhou Liu's user avatar
3 votes
1 answer
29 views

Counterexample that $(X_n)_n$ is not a martingale

Suppose we have a sequence $(X_n)$ of integrable random variables such that $X_{n+1} - X_n$ and $X_n - X_{n-1}$ have the same distribution for all $n \in \mathbb{N}$ and $X_0 = 0$. I know that ...
MathMaestro's user avatar
2 votes
0 answers
78 views

Intuition: Why is quadratic variation finite for martingales

(Disclaimer: I'm not well-read in this topic, so might be getting some details wrong. Hopefully not wrong enough to make my question for intuition moot) For any martingale $(X_t)_{t \geq 0}$ with $X_0=...
Bananach's user avatar
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1 answer
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Analyzing Expected Profit in a Symmetric Random Walk with Trading Actions

Problem Formalization: I am examining a problem where a stock price $X_t$ follows a symmetric random walk starting at 10, and increments or decrements by 1 unit at each step with equal likelihood. The ...
XiaoBanni's user avatar
2 votes
1 answer
41 views

Is stopping time for martingale with continuous trajectories finite almost surely?

Let $(M_t,F_t)_t$ be martingale with continuous trajectories such that $M_t - t^{3/2}$ is also a martingale, and $M_0=3$. Let $T=\inf\{t : \lvert M_t \rvert = 8\}$. Show that $T$ is finite almost ...
romperextremeabuser's user avatar
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Markov and Supermartingale with stopping time $\tau$

Let $X_n$, $n\geq 0$, be a non-negative integer valued integrable markov chain with transition probabilities $$ p(j|i) = \begin{cases} p_n & j=(i-1)\vee 0 \\ 1-p_n & j=i+1 \\ 0 &\text{...
Kumar's user avatar
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4 votes
0 answers
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Question regarding a value in a one-period model

There is a script at my university (can't post it for copyright reasons) for a course on discrete time financial mathematics. I decided to give it a try and found this problem: Consider a financial ...
ryen.xain's user avatar
2 votes
1 answer
32 views

Martingale inequality with Convex Function

I want to know how to prove following equation appearing in the book 'Stochastic approximation and recursive algorithms and applications' without proof. Let $\{Mn,Fn\}$ be a martingale, $q(\cdot)$ be ...
出木杉英才's user avatar
0 votes
1 answer
21 views

Is any (integrable) centered, adapted stochastic process with independent increments a martingale?

I came across the following question: Assume $(X(t), t\geq 0)$ is an integrable, adapted stochastic process on a filtered probability space $(\Omega, \mathcal{F},(\mathcal{F}_t, t\geq 0), \mathbb{P})$ ...
Frank's user avatar
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10 votes
2 answers
720 views

How to apply Ito's Formula to show that this is a martingale?

In the book Brownian Motion, Martingales and Stochastic Calculus by J.F. Le Gall, in order to give an alternatice derivation of the distribution of $L_{U_{a}}^{0}(B)$ where $L^{0}_{t}(B)$ is the Local-...
Dovahkiin's user avatar
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4 votes
1 answer
109 views

Showing a basic market admits no arbitrage

Setting We work in $\left(\Omega, \mathcal{F},\left(\mathcal{F}_t\right)_{t=0}^1, \mathbb{P}\right)$. Let $d=1, T=1$ and assume the discounted price equals the non-discounted price. Take $S_0^1 \in \...
portero's user avatar
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1 vote
0 answers
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How to prove this formula about expectation ?

The continuous-time Markov chain has an infinitesimal generating element Q. For all $f \gt$ $0$,define $$Z(t)=f(X_t)\exp\left(-\int_0^t\left(\frac{Qf}{f}\right)(X_s)ds\right).$$ Define $\tau_n$ as the ...
Jie's user avatar
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4 votes
1 answer
50 views

If $X$ is a continuous supermartingale, why is, for every $N$, $(X_s)_{s\leq N}$ uniformly integrable?

If $X$ is a continuous supermartingale, why is, for every $N$, $(X_s)_{s\leq N}$ uniformly integrable? This statement is used, without proof, in another statement. I haven't been able to prove it. Is ...
math_undergrad_questions's user avatar
3 votes
1 answer
44 views

Derive the 0-1 law from a functional equation of a martingale limit

I am trying to understand a proof given by Jabbour-Hattab in the paper "Martingales and Large Deviations for Binary Search Trees". In this paper, we consider the limit of a non-negative ...
CampFire's user avatar
  • 178
1 vote
2 answers
51 views

Book recommendation for stochastic integral wrt local martingales

I am looking for a book (or a chapter of a book) for stochastic integral wrt local martingales. The book should contain a rigourous introduction to the definition. It should also contain proofs for ...
Mingzhou Liu's user avatar
0 votes
1 answer
26 views

Question about the integrand space of stochastic integral wrt martinagles

I am reading the book "Introduction to Stochastic Integration" by Hui-Hsiung Kuo. In Chapter 5, he introduces the definition of stochastic integral wrt martingale: $$I(f) = \int_a^b f(t) dM(...
Mingzhou Liu's user avatar
4 votes
1 answer
101 views

Uniform Integrability and Proving martingale of Poisson product process

So I have been stuck on the following question that I stumbled across in a textbook. It is trying to show the following product process is a martingale. $$M_n = n! \prod_{k=1}^{n} X_k \ \text{where } ...
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