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

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28 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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1answer
16 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} X_k = ...
5
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0answers
30 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 ...
10
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1answer
75 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
30 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
28 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 ...
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1answer
34 views

To test whether a process is a Martingale (Stochastic calculus)?

If $W_t$ is a standard Brownian motion, I was trying to prove $Y_t = \exp (\int_{0}^{t} s\cdot dW_s)$ is a martingale ! First I started finding $dY_t$ using Ito formula. But I am confused how to ...
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0answers
31 views

Prove that ($Y_n$,$F_n$) is martingale. What does ($Y_n$,$F_n$) mean in this problem?

$X_1$, .... , $X_n$ are independent variables, $P(X_i=1)=p$, $P(X_i=-1)=q$, where $0<p<1$. $F_n=\sigma(X_1,.....,X_n)$ and $Y_n=(\frac{q}{p})^{X_1+....+X_n}$. The task is to prove that ...
3
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0answers
67 views

stochastic exponential uniformly integrable martingale

$N$ is a continuous local martingale and $T_c:=\inf\{t>0:[N]_t>c\}$, $c>0$ . I need to show that the stochastic exponential $\mathcal{E}(-N)$ is a uniformly integrable martingale if and only ...
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1answer
21 views

definitions of martingals

Let X - discrete time stochastic process A discrete-time martingale: is $E[|X_{n}|] < \infty$ $$ E[X_{n+1}|X_{1}, X_{2}, ..., X_{n}] = X_{n}. $$ The definition of martingale for filtration: $$ ...
2
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1answer
31 views

Buyer's price in terms of risk-neutral measures

Let us consider a finite arbitrage-free market model $(B,S)$, where $B$ is a bank account and $S$ is a share. Let $X$ be a claim. We define a buyer's price of $X$ as follows:$$\Pi^b_0(X)=\sup \lbrace ...
1
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0answers
31 views

$n$ times integrated Brownian motion martingale process

According to this post, we found that a $n$ times integrated Brownian motion could be expressed as, \begin{align} V_n(t) = \int_0^t V_{n-1}(s)\ ds = \frac{1}{n!} \int_0^t (t-s)^n\ dB_s, \end{align} ...
1
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1answer
88 views
+50

Stochastic Integral with respect to Compensated Poisson Process

Proposition: Let $N_t$ be an $\mathcal{F}$-Poisson process and $M_t=N_t-\lambda t$ its compensated process. Then for any $\mathcal{F}$-predictable bounded process $H_t$, the stochastic integral ...
5
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0answers
48 views

Brownian Motion Third Power Martingale using Ito Integral

Let $(B_t)_{t \geq 0}$ be a standard Brownian motion and $M_t = B_t^2 - t$. According to this and this posts we know that \begin{align} [M] = [B^2] = 2 \int_0^t B_s^2\ ds. \end{align} Now, without ...
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0answers
11 views

Gaussian blur over (or random walk in) a surface mesh

Let $V$ be the set of mesh vertices, connected by edges $E$, forming a mesh that represents a surface embedded in $\mathbb{R}^3$. On this mesh a function $f:V\rightarrow\mathbb{R}$ is defined. For ...
0
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1answer
42 views

Quadratic Variation Brownian motion martingale (2)

Let $B_t$ be a standard Brownian motion and $M_t = B_t^2 -t$. From here we are aware of the identity \begin{align} [M]=[B^2]. \end{align} Now, I want to apply Itô's formula to $B_t^2$ and from that ...
0
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1answer
32 views

Radon-Nikodym derivative as a Martingale

Let $(\Omega,\mathscr{F}, P)$ be a probability space, let $\nu$ be a finite measure on $\mathscr{F}$, and let $\mathscr{F}_{1}$, $\mathscr{F}_{2}$,... be a non-decreasing sequence of $\sigma$-fields ...
1
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1answer
19 views

Quadratic Martingale Problem

Let $S_n=\sum_{i=1}^n X_i$, where the $X_i$ are identically and independently distributed with $P(X_i=1)=p$ and $P(X_i=-1)=1-p$. Show that there does not exist such $c(n)$ that makes ...
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2answers
48 views

If $M_n$ is a Martingale, How to Show that $\log M_n$ is a Supermartingale?

I think that the problem should use Jensen's inequality. This is because my textbook states that with Jensen's inequality, if $\phi$ is a convex function and $M_n$ a martingale, then $\phi(M_n)$ is a ...
2
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1answer
43 views

show that the process is a martingale [closed]

maybe you will have an idea how to show that : the process $(exp(X_t-\frac{1}{2}Y_t))$ is a martingale? Where $h \in L^2([0,T])$, $T< \infty$, $X_t=\int_0^th(s)dW_s$ and $Y_t=\int_0^th^2(s)ds$ for ...
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2answers
25 views

Understanding the $\sigma$-algebra of a sum of random variables

I've been studying discrete martingale theory and I have been wondering about the relationship between $\sigma\{X+Y\}$, and $\sigma\{X\}$ and $\sigma\{Y\}$ for two random variables X and Y. Is it ...
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0answers
26 views

Quadratic Variation of Brownian motion martingale

Let $B_t$ be a standard Brownian motion and let $M_t = B_t^2 -t$. I want to show that $[M]=[B^2]$. Therefore I want to use the linearity of quadratic variations \begin{align} [\alpha X + \beta Y, Z] ...
4
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0answers
117 views

Convergence of an implicitly defined sequence of random variables

Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...
4
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2answers
53 views

process with integral is martingale

How to show that the process $X_t=tW_t - \int_0^t W_s ds $ is a martingale? I guess I have to use the definition of martingale and properties of Wiener process, but I stack with this integral. ...
0
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0answers
12 views

Partial Integration for Semimartingales

Let $X,Y$ be 2 continuous semimartingales. It could be shown that for every $t>0$, \begin{align} X_tY_t = X_0Y_0 + \int_0^t X_s dY_s + \int_0^t Y_s dX_s + \langle X, Y \rangle _t. \end{align} Let ...
0
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1answer
44 views

non negative super martingale

Let $(X_n)_{n\geq0}$ be a non-negative supermatingale and $T = \inf\{n \geq 0 : X_n = 0\}$. Show that on the event $\{T < \infty\}$, $X_{T+n} = 0$ for all $n \geq 0$ a.s. My approach: $0 \leq ...
3
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1answer
67 views

Using the martingale central limit theorem

Suppose a box has $2n$ tickets half of which are labelled $+1$ and half $-1$. Labeling the draws without replacement by $X_1, ...$, define $S_m = X_1 + ... + X_m$. For any $t \in (0,1)$ ...
2
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0answers
26 views

Is every discrete martingale a time-changed simple random walk?

While going through the book by Revuz and Yor titled 'Continuous Martingales and Brownian Motion', I came accross the notion of time change. In a nutshell, if X is a stochastic process and C is an ...
2
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2answers
51 views

Convergence of $\sum_{i \leq n} X_i/n$

I have a question like this: Let $(X_n)$ be an i.i.d sequence of random variables with values in $\{-1,1\}$, and define $Y_n:= \sum_{i \leq n} X_i/n$. Show that $(Y_n)$ converges almost surely and in ...
2
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1answer
32 views

$T$ can be $\infty$ with positive probability

From Williams' Probability with Martingales How exactly do we know $T$ can be $\infty$ with positive probability or $$P(T = \infty) > 0 \text{ ?}$$ I'm guessing that that means there ...
4
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1answer
45 views

Show local martingale

I have $\exp(\lambda X_t-\frac{\lambda ^2}{2}t)$ is a local martingale, now i have to know if $X_t$ is also a local martingale. Can anybody help me how i can show this correctly?
0
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1answer
42 views

Prove $X_{\infty} < \infty$

From Williams' Probability with Martingales: How exactly do we prove $X_{\infty} < \infty$ a.s.? $$E[|X_{\infty}|] = E[|\lim X_n|] = E[|\liminf X_n|] = E[\liminf |X_n|]$$ $$ \le \liminf ...
2
votes
1answer
33 views

$P($partial sums of $\sum X_k$ are bounded$)>0 \to \sum_{k=1}^{\infty} X_k < \infty\ \text{a.s.}$

From Williams' Probability with Martingales How is the remark deduced from the proof of $b$? I really don't see it.
0
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1answer
49 views

If $M_n \to M_{\infty}$ in $\mathscr L^{2}$, then inequality holds with equality

From Williams' Probability with Martingales I tried rewriting the RHS to: $$\sum_{k=n+1}^{\infty} E[(M_k - M_{k-1})^2] = \sum_{k=n+1}^{n+r} E[(M_k - M_{k-1})^2] + \sum_{k=n+r+1}^{\infty} ...
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1answer
27 views

Prove $A_{\infty} < \infty$?

From Williams' Probability with Martingales How do we know that $A_{\infty} < \infty$? If $T = \infty$, then $$E[A_{T \wedge n}] \le (K+c)^2$$ $$\to E[A_{n}] \le (K+c)^2$$ $$\to ...
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0answers
21 views

Prove $|M_{T \wedge n}| \le c + K$

From Williams' Probability with Martingales Is $\sigma_k^2$ random (and not constant)? How can that be? As far as I know unconditional variance and unconditional expectations are supposed ...
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0answers
49 views

$X_n$ doesn't converge to a limit in $[-\infty, \infty] \to$ Is this supposed to be a stronger version of $\lim X_n$ doesn't exist?

From Williams' Probability with Martingales: What's the difference between saying that '$X_n(\omega)$ does not converge to a limit in $[-\infty,\infty]$' and '$\lim X_n$ does not exist' ? ...
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0answers
30 views

Brownian Motion with Levy's Characterization 2

Let W be a $\mathbb{R}$-valued Brownian motion. To prove that $(B_t)_{t\geq 0}$, where: $B_t:=W_t-\int_0^t\frac{W_u}{u}du$, is a Brownian Motion with respect to $\mathcal{F}^B$, I showed $[B]_t=t$ and ...
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0answers
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Friedman's urn is a supermartingale or a submartingale?

Here is the urn model: At time zero there are $r$ red and $g$ green balls in an urn. At each time-step, we draw out a ball at random and replace it along with $c$ of the same color and $d$ of the ...
2
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0answers
54 views

Brownian motion with Lévy’s Characterization

I want to show that: if for all $\lambda \in \mathbb{R}$ the process $(exp(\lambda X_t-\frac{\lambda ^2}{2}t))_{t\geq0}$ is a $\mathcal{F}^X$ local martingale, then the $\mathbb{R}$-valued process X ...
0
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1answer
62 views

Gambling strategy for coin-flip with 51% chance to win.

So I'm curious if there is a decent gambling strategy for this site I play on. Essentially it's a coin flip where you can have anywhere from 49% to 51% chance to win against the other player. ...
3
votes
1answer
37 views

Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale?

Let $(x_n,\mathcal{F}_n, n\ge 1)$ be a martingale diference. Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale and why?? $a_n$ is a constant.
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0answers
20 views

showing a processes is martingale using ito's lemma

Let $Y(t) = t^2W_t - 2 \int_0^t sW_s \ ds$ where $W_t$ is brownian motion. I am trying to show it is a martingale by showing it is driftless. I set $Z(t,W_t) = t^2W_t$ and ito's gives $dZ = 2tW_t \ dt ...
1
vote
1answer
17 views

Conditional expectation: when does $X_t=E[X_t\mid \mathcal{F}_s]$ for $s<t$

I came across a calculation (1$^\circ$ calculation, 2$^{nd}$ step) that stated, for $s<t$ $$E[B_s(B_t^2-t)]=E[B_sE[(B_t^2-t)\mid\mathcal{F}_s]]$$ I know the expectation here is zero, however, I ...
0
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1answer
41 views

Martingale and local martingales

I have to show that $e^{B_t^1}\cos(B_t^2)$ is a martingale ($B=(B^1,B^2)$ is a two-dimensional Brownian Motion). I used Ito's formula and got $e^{B_t^1}\cos(B_t^2)=1+\int_0^t ...
0
votes
0answers
47 views

Martingal-property of stochastic Integral w.r.t. Brownian Motion

To Show that $(e^{B_t^1}cos(B_t^2))_{t \in \mathbb{R_+}}$ (where: $B=(B_s^1,B_s^2)$ is a 2-dimensional Brownian Motion) is a Martingal I used Ito's Lemma and showed that this is equal to: $ 1+ ...
4
votes
1answer
34 views

How to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson Process?

I am trying to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson process with rate $\lambda$. So far, what I have done is: \begin{align*} E\left((X(t)-\lambda t)^2 ...
1
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0answers
61 views

Probability of the martingale staying non-negative.

Here is a question on martingales (given after third graduate lecture on the subject). Let $X_n$ a martingale with respect to the natural filtration and such that $X_0 = 0$, assume that $\frac{1}{2} ...
2
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1answer
18 views

Bounding expectation of a supremum process

This is exercise 3.9(c) on page 15 of Karatzas and Shreve's Brownian Motion and Stochastic Calculus. Let $N_t$ be a Poisson process with intensity $\lambda$. In particular, if $t$ is fixed, $N_t$ is ...
0
votes
2answers
42 views

Show $(X_n+a)^2$ is a submartingale

Let $(X_n)$ be a martingale, and let $EX_n^2 < \infty$ - then I am told to show $E(X_n+a)^2 $ is a sub martingale. I wrote $$(X_n+a)^2 = ((X_{n-1} + a) + (X_n - X_{n-1}))^2 $$ then $$E((X_n+a)^2 | ...