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

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How Solve these expectations

Let $T$ be the exit time of from the interval $[-b,a]$ of a standard Brownian Motion $X_t$, then how would we go about calculating the following two expectations: $E[T^2]$ (and) $E[\int_0^T X_tds]$? ...
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13 views

How to Prove Unboundedness

Suppose I have a submartingale $X_k$, what results/theorems can be useful if I want to show that $X_k$ is unbounded in the limit. There are results (basically bounding $\mathbb{E}X_k$) for convergence ...
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1answer
18 views

Supremal distribution of positive continuous martingale, which converges to zero a.s.

So the question is as follows: Let $M$ be a positive continous martingale, converging a.s. to zero as $t \rightarrow \infty$. Prove that for every $x>0$: $\mathbb{P}\{\sup_{\{t \geq 0 \}} M_t > ...
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1answer
11 views

Local martingale being true martingale

I am doing a question Let $X$ be a continuous local martingale and suppose $\mathbb{E}\left[\sup\limits_{0\leq s\leq t} |X_s|\right]<\infty$ for each $t\geq 0$. Then $X$ is a true martingale. In ...
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11 views

Continuous time Uniform Integrability and convergence in L1 for (super)martingale

I need to prove the following theorem by reasoning as in discrete-time, since we have proven this theorem in discrete time. Theorem Let $M$ be a right-continuous supermartingale that is bounded in ...
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1answer
19 views

Can this attempt to prove Ito Isometry for Elementary Processes be fixed?

So I have found this link which I will try after writing this post, but I would like to see if my original attempt (which is his/her attempt there) can be made to work. The reason I want this to work ...
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1answer
48 views

Justifying a step in proving $M_{S\wedge T} = \mathbb{E}[M_T | \mathcal{F}_S ]$

$S,T$ are stopping times and $M$ is a (right) continuous martingale. My lecturer set this as an exercise and I am given a solution(essentially split $M_T = M_T \mathbf{1}_{S≤T} + M_T ...
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1answer
25 views

Martingales and stopping times question

Let $X_n$ be iid r.v.s such that $P(X_n=1)=P(X_n=-1)=1/2$, and $S_n=\sum_{k=0}^{n}X_k$. Define $S_0=0$ a.s. . Prove that for all $k,n \in \mathbb{N}$, $\mathbb{E}[S^2_{n \wedge T_k}]=\mathbb{E}[{n ...
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1answer
37 views

Show that a certain functional of Brownian motion is a martingale

Question: Show that $(W^2_{t}-t)^2 - 4 \int_{0}^{t} W^2_{u} du$ is a martingale. I understand how to show that $(W^2_{t}-t)$ is a martingale, and I know that $4 \int_{0}^{t} W^2_{u} du$ is the ...
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43 views

2d random walk on the nonnegative quadrant using martingale techniques

I know the basics of (discrete time) martingales, and I'd appreciate any help and suggestions on how to prove the following using martingale techniques. Let $Z_n$, $n\ge 0$ be a random walk on the ...
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The square of an Ito integral is not a martingale

I just had a lecture on martingales and my teacher said something which I thought was interesting but he said wasn't important to the course. I was wondering if you guys could help me on this. We ...
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1answer
29 views

Not all martingales $Y$ can be represented $Y = H\bullet X$ for a given $X$

This is given as a counterexample that not all martingales $Y$ with $Y_0 = 0$ can be represented as $H\bullet X$ (= "discrete stochastic integral" ... wherever this term comes from??) for a given ...
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1answer
40 views

Exercise about martingale convergence

Let $p \in [0, 1]$, consider a stochastic process $(X_n)_{n\in\mathbb{N}_0}$ with $X_0 = x_0 \in [0, 1]$ and the following dynamics: For $n\in \Bbb{N}_0$, conditional on $X_0, X_1, \ldots, X_n$, we ...
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0answers
32 views

Convergence theorem for uniformly integrable martingales

This is a theorem in my textbook: Why "for all $n\in\mathbb{N}$" and not "for all $n\in\mathbb{N}_0$"? What's wrong with setting $n=0$, e.g. $$ X_0 =\mathbf{E}[X_\infty| \mathcal{F}_0] \; ?$$
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29 views

Riesz decomposition of a nonnegative supermartingale

$X=(X_n,\mathcal{F_n})$ is a nonnegative supermartingale, and moreover $EX_n\to 0$, i.e., it is a potential. If $X_n=M_n-A_n$ is the Doob decomposition, then $$EX_n=EM_n-EA_n=EX_0-EA_n,$$ so by the ...
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34 views

A characterization of quadratic variation for $L^2$ martingales

I am trying to prove the following statement but I am totally at a loss. Let $(A_t)$, $t \in \mathbb{R}^+$ be an adapted (with respect to the filtration $(\mathcal{F}_t)$) continuous integrable ...
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23 views

Conditional Borel-Cantelli lemma

Let $A_1, A_2, \ldots$ be events with $A_n\in\mathcal{F}_n$. Show that $$\biggl\{\sum_{n=1}^\infty \mathbf{P}[A_n|\mathcal{F}_{n-1}]=\infty\biggr\} = \limsup_{n\rightarrow\infty} A_n \text{ a. ...
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22 views

Martingale with bounded increments converges or diverges to $\pm \infty$ [duplicate]

Let $(M_n)$ be a martingale with $|M_n - M_{n-1}| \leq c$ for some fixed $c < \infty$. Check that the two disjoint events $$C:=\{M_n \text{ converges to a finite limit}\}, \; F:=\{\limsup M_n ...
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30 views

Proof of the optional sampling theorem

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}$ be a Filtration on $(\Omega,\mathcal{A})$ $X=(X_n)_{n\in\mathbb{N}_0}$ be a nonnegative $\mathbb{F}$-supermartingale ...
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14 views

Does Doob's optional sampling theorem hold for *backward* (sub)martingales?

Here is my setup: $X_1,X_2,\ldots\overset{\mbox{iid}}{\sim}\mathcal{N}(0,1)$. $N_k=\min\{n:|\bar{X}_n|\geq k/\sqrt{n}\}$. I want to show that $k_2>k_1$ implies ...
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1answer
50 views

$\mathcal{L}^2$-martingale with finite limit but infinite square variation

Find an $\mathcal{L}^2$-martingale $(M_n)$ with $M_n \rightarrow M_\infty$ almost surely for a finite real valued $M_\infty$ but $\langle M \rangle_n \rightarrow \infty$ almost surely. $\langle M ...
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27 views

How does expectation of martingales relate to “fair games”?

A martingale is defined as an $(\mathcal{F}_n)_{n\in\mathbb{N}}$-adapted stochastic process with $\mathbf{E}[|X_t|] < \infty$ for all times $t\in I\subset \Bbb{R}$ and ...
3
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1answer
55 views

Variance of the square variation process

Let $X = (X_n)_{n\in\mathbb{N}_0}$ a square integrable $(\mathcal{F_n})_{n\in\mathbb{N}_0}$-martingale. The predictable process $\langle X \rangle_n = \sum_{i=1}^n \Bigl(\mathbf{E}\bigl[X_i^2\vert ...
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1answer
52 views

Relation between Hermite polynomials and Brownian motion (on martingale property) [duplicate]

Let us define Hermite polynomials as $H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n} e^{-x^2/2}.$ One can prove that $e^{\theta x-\frac{1}{2} \theta^2}=\sum_{n=0}^{\infty} \frac{1}{n!}H_n(x)\theta^n \quad ...
3
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1answer
41 views

Computation of a stochastic integral with respect to a local martingale

I am trying to compute the stochastic integral $$\int_{(0,t]}\mathbb{1}_{[a,b)}(s)dM_s$$ where $0 < a < b< \infty$ are constant and $M$ is a continuous local $L^2$-martingale. I am guessing ...
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1answer
46 views

Proof of stopping theorem for bounded stopping times

Let $\tau$ be a bounded stopping time and $X=X_n$ a martingale. Then $X_\tau$ is integrable and $E(X_\tau)=E(X_0)$. I need help with the proof at discrete time, at one step I am not sure I ...
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1answer
74 views

Quadratic Variation of a square-integrable Lévy process

I am having a problem with the following question. I have tried using the definition of square integrable martingales and quadratic variation, but just can't seem to get anywhere. Can anybody offer me ...
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1answer
22 views

Upper bound for martingale at a stopping time

This seems like a simple question, but I cannot figure out the following. Let $\{M_i\}_{i\geq 0}$ be a martingale adapted to a filtration $\mathcal{F}_i$, with the following conditions: ...
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1answer
33 views

Azuma's/Hoeffding's inequality for geometric series

Let $X_1,X_2,\dotsc$ be a sequence of a.s. bounded, zero-mean random variables. For $\alpha \in (0,1)$ define $Z_t$ as the geometric series with $Z_t = \sum_{i=1}^t\alpha^{t-i}X_i$ and $\mathcal{F}_k ...
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1answer
45 views

Is the geometric series of a set of $n$ RVs a martingale?

Let $X_1,\dotsc,X_n$ be independent, zero mean random variables and define $Y_k = \alpha^{n-k}X_k$. Is $\{Z_k\}$ with $Z_k = \sum_{i=1}^k Y_i = \sum_{i=1}^k \alpha^{n-i}X_i$ a martingale? I suppose ...
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0answers
24 views

How to show that stochastic exponent is integrable?

I need to prove that if $u: [0,T]\rightarrow \mathbb{R}$ is a deterministic square integrable function then stochastic exponential process defined : $M_{t} = exp(-\int_0^t \! u(s) \, \mathrm{d}W_{s} ...
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1answer
24 views

(Bt)^2 it is a martingale?

Well i think no, because the expected value is t E[(Bt)^2)=t, so it´s not constant, change with the time. Am I right? Propz
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1answer
37 views

Let $X_n$ be the $n$-th partial sum of i.i.d. centralized rv and $\mathcal{F}_m:=\sigma(X_n,n\le m)$, then $\text{E}[X_n\mid\mathcal{F}_m]=X_m$

Let $(\Omega,\mathcal{F},\text{P})$ be a probability space $\left(Y_i\right)_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variables ...
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1answer
46 views

Prove $\sum E((X_n-X_{n-1})^2)$ is finite iff $X_n$ converges to $X$ in $L^2$

Let $(X_n)_{n \in \mathbb{N}}$ be a martingale. Prove $\sum_n E((X_n-X_{n-1})^2)$ is finite iff $X_n$ converge to $X$ in $L^2$. It is not hard at first glance, but I cannot figure it out after many ...
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1answer
35 views

How to prove this Brownian motion convergence?

Let $W_t$ be a Brownian motion. How do I show the following? $$ \alpha > \frac{1}{2} \Rightarrow \lim_{t\rightarrow\infty} \frac{W_t}{t^{\alpha}} = 0 \text{ a.s.} $$ Showing convergence of this ...
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1answer
41 views

How do I prove that a martingale has a constant expected value?

I can´t prove that a martingale has constant expected value. $$ \mathbf{E}[M_t]=\mathbf{E}[M_0] $$ Thanks people.
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1answer
26 views

How to show the sum is a martingale

In the hypothesis of the martingale central limit theorem, my book says that given a sequence of random variables $X_n$ with the condition that $E(X_n \mid \mathcal F_{n-1}) = 0$, then $S_n = ...
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1answer
32 views

Show that a function of a symmetric random walk is a martingale

Suppose $S_n = (X_n,Y_n)$ is a symmetric random walk on $\mathbb{Z}^2$. Show that $G_n = X_n^2 + Y_n^2 - n$ is a martingale. What is true about $E_{(x_0,y_0)}[|S_n|]$? Find an upper bound for ...
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1answer
31 views

Supermartingale of product of random variables

Let $Y_1, Y_2, \ldots$ be independent and identically distributed non-negative random variables. Define $X_n := \prod_{i=1}^n Y_i$ for $n\geq 1$. I have to show that if $(X_n)_n$ is a supermartingale ...
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24 views

Reference book on Backward Martingale theory

I am looking for an introduction to the topic of BACKWARD MARTINGALES possibly with good intuition (it can be either notes or a book) AND a reference book on the topic.
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121 views

Example: satisfying $E(X_{n+1}\mid X_n)=X_n$ but not a martingale

I am wondering if there is such a sequence of random variables $(X_n)_{n=0}^\infty$ such that $\mathbb{E}(X_{n+1}\mid X_n)=X_n$ for all $n\geq0$ but which is not a martingale with respect to the ...
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0answers
27 views

Supermartingales and optimal strategies for a game

Your winnings per unit stake on game $n$ are given by independent random variables $\epsilon_n$ such that $P(\epsilon_n=1)=p$, $P(\epsilon_n=-1)=q$ with $1/2<p=1-q<1$. Let $C_n$ be your stake on ...
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76 views

Is this stochastic process a martingale?

I have the following process: $X_t=tB_t-\int^{t}_{0}B_s \ ds$ where $B_t $ is a Brownian motion. Is this a Gauß-process and/or a martingale? Can someone help me with this? And how can I calculate ...
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22 views

Good reference on stopping times and continuous time change

I've been trying to look at stopping times and continuous time change in martingales but have trouble understanding without some concrete examples. Anyone knows of any good references that might be ...
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20 views

Question on exponential martingale

I was reading the first proof here on exponential martingale, https://fabricebaudoin.wordpress.com/2012/09/27/lecture-23-time-changed-martingales-and-planar-brownian-motion/ It says that "Let $ ...
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1answer
70 views

Does this stopping time always have infinite first moment?

Let $X_1, X_2, X_3, \ldots$ be i.i.d. random variables with zero mean and let $S_n := X_1 + \ldots + X_n$. Does $T := \inf\{n: S_n > 0\}$ always have infinite first moment? In the trivial case, ...
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1answer
61 views

Counterexample in optional stopping martingale

Problem: Give an example of submartingale $\{X_n\}$ with $\sup_nE |X_{n-1}-X_n|<\infty$ and stopping time $N$ with $E[N]<\infty$ such that $\{X_{n\wedge N}\}$ is not uniformly integrable. ...
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28 views

Size-Biased Galton Watson Tree.

First of all, i am not sure whether this question belongs here or to stack overflow. Let me write here, first, i will give a definition of size-biased distribution. then i will give definition of ...
2
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0answers
22 views

Martingale energy inequality

I am reading a book on BMO martingales which uses a so-called energy inequality. I have not been able to find a solid reference for this. Can someone please give a reference to these inequalities. ...
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1answer
34 views

Positive submartingales

Let $\{X_n\}$, $n>0$ be a positive submartingale with $X_{0} = 0.$ Let $V_n$ be random variables such that $V_n \in\mathcal F_{n−1}$ for all $n \geq 1$. $B > V_1 > V_2 > \dots > 0$ ...