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

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Proof that $\mathbb{E}(|y_n|) <\infty$?

Let $\{x_n,\mathcal{F}_n,n\ge 1\}$ be a martingale differences sequence. Put $x_n'=x_n \mathbf{1}_{\{ |x_n|\le a_n \}}$ with $a_n\ge 0$ and $$y_n=x_1'+\cdots+x_n'-\mathbb{E} (x_2'\mid\mathcal{F}_1) - \...
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1answer
28 views

Two notions of square-integrability

It seems to me there are two notions for random variables / processes which get labeled square-integrable: $EX^2_t<\infty \; \forall t$ $E \int^t_0 X_s^2 \; ds < \infty \; \forall t$ I ...
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14 views

Is expectation finite almost surely or not almost surely?

Let $\{w_n\}$ be a sequence of non-negative numbers and $\{X_n,\mathcal{F}_n, n\ge 1\}$ be a uniformly bounded martingale differences. Put $M_n= \sum_{k=1}^n w_k^2 <\infty$. I solved that \begin{...
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2answers
64 views

Trying to understand $\mathbb E[X\mid \mathcal F]$ and Martingale concept

Q1) Let $X$ a r.v. on a probability space $(\Omega ,\mathcal F,P)$ and $\mathcal G\subset \mathcal F$ a sub $\sigma -$algebra. What does $$\mathbb E[X\mid \mathcal G]\ \ ?$$ I understand what is $\...
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Why do they add addition $\mathbb{E} (s_1^-)<\infty$ when they have $\{ s_n, \mathcal{F}_n,n\ge 1\}$ be a submartingale?

By definition, if $\{ s_n, \mathcal{F}_n,n\ge 1\}$ is a submartingale then $\mathbb{E} (|s_n|)<\infty$ for all $n\in \mathbb{N}^*$. But according to a arcticle, in a theorem, author add addition: $\...
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1answer
21 views

Doob's submartingale theorem

According to Doob's theorem, If $\{X_n,\mathcal{F}_n, n\in \mathbb{N}^* \}$ is a submartingale and $L_1$ - bounded, it means $$\sup\limits_{n\ge 1} \mathbb{E} (|X_n|)<\infty,$$ then $\{X_n\}$ ...
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0answers
14 views

Rate of convergence for martingales, “merging of opinions” results

Let $(\Omega, \mathcal{F})$ be a measurable space, and let $P$ and $Q$ be probability measures on this space. Let $(\mathcal{F}_{n})_{n \in \mathbb{N}}$ be a filtration on $\Omega$. Assuming that the ...
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0answers
17 views

Markov process and Doob-Meyer decomposition

$(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq0}),\mathbb{P})$ - a filtered probability space supporting a 1-dimensional Brownian motion $B=(B_t)_{t\geq0}$, where \begin{equation} \mathcal{F}_t=\sigma(...
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15 views

How can be a stopped martingale martingale?

A martingale has constant expectation, but if we stop a martingale, then after the stopping the process becomes a constant. So how can it remain a martingale?
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1answer
19 views

Exponential martingale, Lévy-process and stopping times, definition quesiton.

I feel there is some ambiguity for the definition of the exponential martingale for a levy process which I do not understand. For a Lévy process it can be shown that $E[e^{iuX_t}]=e^{t\eta(u)}$, ...
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24 views

Martingale Convergence Theorem

I have a Question regarding MCT which I am stuck in, the question goes like this: Let $X_0 = 1$ and assume that $X_n$ is distributed uniformly on $(0,X_{n-1})$. and $Y_n = 2^nX_n$. the questions ...
4
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26 views

Comparison of stopped sub- and supermartingales when the future is discounted

Suppose we have a submartingale $X=\{X_t\}_t$ and a supermartingale $Y=\{Y_t\}_t$ which are adapted to the same filtration on a bounded set and have a common initial value $X_0=Y_0$. Suppose that $\...
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42 views

$(X_n)$ transient Markov chain with transition probability matrix $P = \|P_{ij}\|$. Define $u(i) = \sum P_{i0}^{(n)}$. Then $u(X_n)$ is submartingale

Let $(X_n), n \geq 0$ be a transient Markov chain on the non-negative integers with transition probability matrix $P = \|P_{ij}\|$. Define $u(i) = \sum_{n=0}^{+\infty} P_{i0}^{(n)}$. Then $u(X_k)$ is ...
2
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1answer
45 views

a.s. convergence and conditional expectation

I have a stochastic process $X(t,\omega)$ which is a martingale. It is showed that there exists a r.v. $X(\infty)$ such that in $L^1(\Omega)$, $\lim_{t \rightarrow \infty}X(t) = X(\infty)$. In my ...
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14 views

Reference for *optimal* stopping theorem for supermartingales

Can anyone introduce a good reference about optimal (not optional!) stopping times for submartingales / supermartingales? I am looking for some theorem like the one mentioned in this question. I ...
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14 views

Question about Supermartingales

I came across the following problem: In my setting I have two sequences of non-negative integrable random variables (measurable with respect to some filtration $F_n$) which are called $X_n$ and $Y_n$. ...
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0answers
12 views

Drift analysis of an absorbing Markov chain

Consider a set $S$, and suppose we have a sequence of random subsets $$ \zeta_t = \{x_1, \dots, x_n\} $$ for $x_1, \dots, x_n \in S$. We do not know with which probability density the points of each $\...
3
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1answer
55 views

Law of large numbers for moving mean

Consider the following process: For $n = 1,\ldots$ $U_n \sim U[0, 1]$, that is, uniformly distributed on $[0, 1]$, $X_n = U_n 1_{U_n > q_n}$, where $q_n = \frac{1}{n-1} \sum_{i=1}^{n-1} X_i$, ...
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1answer
26 views

Martingale wrt natural filtration: $e^{ \sum_{i=1}^t X_i - t/2}$

Let all $X_i$ be standard normal and iid for $i \in [1,T]$, let $X_0 = 0$. Define for each $t \in [0,T]$ $S_t = e^{( \sum_{i=1}^t X_i) - t/2}$ Is this process a martingale wrt its natural filtration?...
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35 views

Time-changed Brownian Motion

Let $B_t$ be a standard Brownian motion and let $\tau_{-1}= \inf \{ t \geq 0: B_t(\omega) = -1\}$. By the Continuous Time Stopping Theorem we know that \begin{align} Z_t = B_{t \wedge \tau_{-1}} \...
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0answers
45 views

Concentration of a random variable in a random sequence

Given a sequence of random variables $X_0, X_1, \cdots$. Assume that $X_0=0$ and $X_{n+1}= X_n+U[0,1]$, where $U[0,1]$ denotes a uniform distribution over $[0,1]$. Define $N=\min\{n: X_n\geq 2\}$. Q:...
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1answer
32 views

Show that a cadlag adapted martingale is a local martingale (help using DCT to show uniform integrability)

EDIT 2: With the correct definition, I think I have a proof. Want to show $\lim_{M\to\infty} \sup_t E[|X_{t\wedge n}|; |X_{t\wedge n}|\ge M]=0$. Fix $n$. Note that $\sup_t E[|X_{t\wedge n}; |X_{t\...
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1answer
37 views

Doob Decomposition Theorem - submartingale iff increasing

Probability with Martingales To prove $b$ I tried: $$A_n \ge A_{n-1}$$ $$\iff E[X_{n} - X_{n-1} | \mathscr F_{n-1}] \ge 0$$ $$\iff E[X_{n} | \mathscr F_{n-1}] \ge X_{n-1}$$ That ...
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1answer
24 views

Doob Decomposition Theorem in Williams is working backward? Unique modulo indistinguishability?

Probability with Martingales This is my understanding of what is going on in the proof above: We first assume $X$ has such Doob Decomposition in order to figure out what $A$ to use in ...
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0answers
43 views

Help with change of measure and martingales

Consider two three stochastic processes $X$, $Y$ and $Z$ in probability space $(\Omega, (\mathcal F_t)_{t \geq0},\mathbb P)$ such that $$ X_t = \exp\left(\int_0^t f_s ds\right), $$ $$ Y_t = \exp\...
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1answer
27 views

Prove $M_{S(k) \wedge n}$ is bounded in $\mathscr L^2$

Probability with Martingales: To prove $$\sup E[M_{S(k) \wedge n}^2] < \infty,$$ how can we use 12.12c? There aren't any stopping times there.
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45 views

Angle bracket (quadratic variation) process for martingales

Probability with Martingales: What is the relation between $\langle M_{S(k) \wedge n}\rangle \ = A_{S(k) \wedge n}$ and $\{N_n\}, \{ N_{ S(k) \wedge n } \}$ being martingales? It seems that $$\...
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0answers
68 views

Prove the stopped process $A^{S(k)}$ is previsible

Probability with Martingales: I have a different attempt in mind, but I'm guessing it's wrong because if it were right, the book would've used it. It seems that we must show that $$A_{S_k \...
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1answer
30 views

Prove $S_k$ is a stopping based on $A$ being previsible

Probability with Martingales: It looks like we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ ...
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1answer
61 views

showing a process martingle from ito's lemma.. [closed]

**by ito formula, Ft: filtratoin $M(t) = (aB(t) - t) \exp( 2B(t) - 2t ) $ find constant a for $M(t)$ to be a martingale plz help!**
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1answer
26 views

Why is $M$ bounded in $\mathscr L^2$ iff $E[\lim A_n] < \infty$?

Probability with Martingales: About $(c)$ My understanding is that $(c)$ is equivalent to: $$\sup E[A_n] < \infty \iff E[\lim A_n] < \infty$$ Why is that so?
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18 views

Preservation of ui condition [closed]

I have a stopping time $\tau_n$ with $\mathbb{P}(\tau_n=\infty)\rightarrow 1$ for $n \to \infty $. With this stoppingtime $M^{\tau_n}$ is a uniformly integrable martingale. I deduced that $M$ is a ...
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0answers
22 views

Example of a semimartingale with special properties

I have to find an example of a semimartingale X such that $\lim_{t \rightarrow \infty} X_t$ exists a.s. and $X$ is not a semimartingale up to infinity. I think it could be a deterministic function ...
2
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0answers
43 views

martingale square integrable

Let $X_t=\int_0^te^{W_s}dW_s$ and $Y_t=\int_0^tW_sdX_s$. How to show that $X$ and $Y$ are martingale square integrable? ($W_t$ - Wiener) It it enough to show that $\mathbb{E}X_t^2<\infty$, $\...
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Showing that Polya's Urn is a martingale - is this correct definition of Polya's urn?

Possibly related: Show rigorously that Pólya urn describes a martingale Suppose we had $b$ and $r$ blue/red balls in urn at time $0$, and at each $n\ge 1$ we draw a ball randomly and then put it ...
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0answers
18 views

Proving mean of random variables does not converge to zero almost surely. [duplicate]

We have a sequence of i.i.d random variables $(X_n)$ such that $$ P(X_n=n)=P(X_n=-n)=\frac{1}{2(n+1)\log(n+1)},$$ and $$P(X_n=0)=1-\frac{1}{(n+1)\log(n+1)}$$ I want to show that $\dfrac{S_n}{n}$ (...
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2answers
24 views

Natural filtration of martingales

I don't quite understand what the natural filtration really is. Imagine e.g. a sequence of independent and identically distributed random $N(0,1)$ variables. What is their natural filtration, and how ...
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1answer
45 views

Stochastic differential equation substitution reasoning?

I need help to explain reasoning behind why we choose certain substitutions to solve SDE. After choosing the correct substitution the solution of the SDE are often quite easy. However I have trouble ...
4
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1answer
148 views

Limit theorem for changed time

This post seems long, but its almost everything proofed in this post. Only one step seems to be left, for the desired proof. I would be very gratefull for any help. The setup Given a Levy-Process $U_{...
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1answer
32 views

From brownian bridge to brownian motion proof

Let $B_t$ be a brownian motion. and let $\{W_t=B_t-tB_1:0\le t\le 1\}$ be a brownian bridge. Now let $Y_t=(1+t)W_{t\over 1+t}$. Proof that $Y_t$ is a brownian motion in $[0, \infty)$ My attempt: 1) $...
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2answers
56 views

The Second Hearts Problem

According to the last part of these lecture notes, if we have a standard deck of playing cards and turn cards until the first heart appears, the probability that the next card is a heart is $\color{...
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0answers
39 views

Show that $W^2 _t - t$ is a $\mathbb{P}$-martingale.

Claim: $V_t = W^2 _t - t$ is a $\mathbb{P}$-martingale. I have shown via Ito's formula, that $dV_t = 2 W_t \, dW_t$. For reference, I will list this "Proposition": If $X$ is a stochastic process ...
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1answer
34 views

Find constants such that transformed simple symmetric random walk is martingale

Let $$S_0 :=0, \quad S_n = X_1 + ... + X_n \quad \forall n \in \mathbb{N}$$ be the simple symmetric random walk on $\mathbb{Z}$, i.e. the $X_i$ are i.i.d. with $$P[X_i = +1] = P[X_i = -1] = 1/2.$$ ...
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34 views

A stochastic process $X$ is a martingale $\iff$ $X$ is driftless.

A Collector's Guide to Martingales: If $X$ is a stochastic process with volatility $\sigma _t$ (that is, $dX_t = \sigma _t dW_t + \mu _t dt$) which satisfies $\mathbb{E}[(\int_{0}^{T} \sigma^2 _s \, ...
0
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1answer
33 views

Expected time until pattern (1,0,0,1)

Let $(X_n)_{n\geq 0}$ be i.i.d. with $\mathbb P(X_n = 0 ) = \mathbb P(X_n = 1) = \frac{1}{2}$. Let $\tau_a$ be the stopping times defined as $$\tau_a = \inf\{n: (X_{n-3}, ... , X_n) = (1,0,0,1)\}$$ I ...
2
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2answers
29 views

Divergence of asymmetric not-simple random walk

Consider a (not simple) random walk $S_n = \sum_{k=0}^n X_k$ where X_k are i.i.d and the mean $\overline{X}<0$. Is there is simple proof or a reference showing $P( \lim \limits_{k \to \infty} S_k = ...
5
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1answer
43 views

Functions of a random walk and martingales

Let $\xi_1,\xi_2,\ldots$ be a sequence of iid random variables, such that $$\mathbb{P}(\xi_i=1)=p\ne \frac{1}{2},\,\mathbb{P}(\xi_i=-1)=q=1-p.$$ Consider the corresponding random walk $X_n=\xi_1+\xi_2+...
10
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1answer
124 views

Hard Question in Stochastic processes - variance Martingales

I got some hard challenge to solve and I am looking for a small clue/help. My question goes like this: 10 Englishmen are trying to leave a pub in a rainy weather. They do it in the following ...
3
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0answers
34 views

$2D$ random walk stopping time

A $2D$ random walk starts at $(X_0, Y_0) = (k, k)$ where $k>0$ is an integer. At each step $(X_{n+1}, Y_{n+1}) = (X_{n}-1, Y_{n})$ or $(X_{n+1}, Y_{n+1}) = (X_{n}, Y_{n}-1)$ with the same ...
3
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0answers
31 views

Two Similar Measures on a Probability Space

Let $(\Omega,\mathscr{F}, P)$ be a probability space and let $Q$ be another probability measure on $\mathscr{F}$, and let $\mathscr{F}_n=\sigma(Y_1,\ldots,Y_n)$ be a non-decreasing sequence of $\...