1
vote
2answers
16 views

Question on Doob's martingale convergence theorem

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and $(\mathcal F_k)_{k\in\mathbb N}$ a filtration of $\mathcal F$ such that $\mathcal F=\sigma(\mathcal F_k\mid k\in\mathbb N).$ Let ...
0
votes
0answers
40 views

Is the martingale propertey preserved by taking weak$^*$-limits?

Let $(\Omega,\mathcal F)$ be a measurable space, $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. Let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that ...
0
votes
0answers
14 views

Show $W_{\infty}$ is not tail measurable but {$W_{\infty}=0$} is

The random walk: ({$\omega_{n}$},$\mathbb{P}$) simple random walk on d-dimensional integer lattice $\mathbb{Z}^{d}$ and the random environment: $\eta$={$\eta(n,x):n\in\mathbb{N}, x\in \mathbb{Z}^{d}$} ...
2
votes
1answer
22 views

arbitrage free price in martingale measures

Consider a one-period market with $S^1_t,\cdots,S^n_t$, with $t=0,1$ the price process of $n$ assets, where $S_1$ is a risk-free asset: $S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the ...
0
votes
1answer
34 views

Question on Martingales and Brownian Motion

I've run into an issue and I am unsure of how to proceed. In the text I'm working through, the following is left "as an exercise to the reader." Normally proofs listed as such tend to be fairly simple ...
1
vote
1answer
66 views

Levy's Martingale Using Radon Nikodym

Let P and Q be two probability measures on the same space $(\Omega,\mathcal{F},\mathcal{P})$ and let $\mathcal{F_n}$ be filtration. Assume that $Q \ll P$. Let $X_n$ denote the Radon-Nikodym derivative ...
2
votes
1answer
35 views

question about martingale

In my lecture notes,I found the following problem: Let $X$ an $F_{t}$ adapted continuous process and $G_{t}\subset F_{t}$. show that $$E\left(\left. \int^{t}_{0}X_{s}ds ...
1
vote
1answer
23 views

Expectations of martingales

Consider a martingale $(M_n)_{n \geq 0}$ adapted to a filtration $(\mathcal F)_{n \geq 0}$ on a probability space $(\Omega, \mathcal F, P)$. Prove that, for each $k \leq n$; $$E(M_n M_k) = E(M_k^2)$$ ...
1
vote
0answers
50 views

Is Zn a Martingale with mean 1?

Consider a sequence of independent tosses of a coin, and let $P_h$ be the probability of a head on any toss. Let $A$ be the hypothesis that $P_h = a$, and let $B$ be the hypothesis that $P_h = b$. Let ...
-2
votes
1answer
38 views

Show that $E[Z_n^2]= \sum_{i=1}^n E[(Z_i-Z_{i-1})^2] $ for a martingale with $Z_0=0$

I was just wondering, if we let $(Z_n)_{n\geq 0}$be a martingale with $Z_0=0$, is it true then $$ E[Z_n^2]= \sum_{i=1}^n E[(Z_i-Z_{i-1})^2] $$ Please let me know and if it is true, can someone show ...
1
vote
1answer
19 views

A bound on the maximum of a submartingale

I'm trying to prove the following: If $\{(S_i, \mathcal{F}_i) \mid i \in [N]\}$ is a nonnegative submartingale with $\mathbb{E}S_i^p < \infty$, let $M = \max_i S_i$. Then $\|M\|_p \leq q ...
0
votes
0answers
38 views

Martingale and Stochastic equation

Using the Ito formula, I can show that the martingale $$ Z_{t}=\frac{1}{\sqrt{1-t}}\exp -\frac{B_t^2}{2(1-t)}\qquad 0\leq t<1 $$ admits the following differential $$ dZ_t=-\frac{B_t}{1-t}Z_tdB_t. ...
1
vote
2answers
52 views

$4^{Brownian(t)}$ martingale proof

Let $B(t)$ a Brownian motion. I like to prove that $4^{B(t)}$ = martingale I rewrote the expression into an exponential form (like $\exp(\ln(4) B)$), but then I don't know how to proceed.
4
votes
1answer
50 views

Transforming semimartingale to local martingale by change of measure

Consider a continuous $\mathbb{P}$ - semimartingale X which can be decomposed as M+A (M is local martingale and A is bounded variation process). Is it possible to change measure to $\mathbb{Q}$ s.t. ...
3
votes
2answers
77 views

Normal variables, uniform integrability and limit of a martingale

I'm stuck in the following problem! Would be great if you can help. Suppose you have $(Y_n)_{n\in \mathbb N}$, a sequence of independent and identically distributed random variables, $Y_1 \sim ...
1
vote
0answers
98 views

Upper bound for mean hitting time of two-dimensional random walk

Let $U_n=\sum_{i=1}^n X_i,V_n=\sum_{i=1}^n Y_i$, $n\geq 1$, be a two-dimensional random walk with iid increments $(X_n, Y_n)$, where $X_n, Y_n$ are discrete random variables with joint pmf ...
0
votes
1answer
33 views

Squared Poisson Martingale

I know that $M_t=N_t-\lambda t$ is a martingale for $N_t$ a rate $\lambda$ poisson process and that for a brownian motion, $B_t^2-t$ is a martingale. I'm wondering, is there something similar for ...
2
votes
1answer
60 views

Probabilistic Proof That An Absolutely Continuous Function is Differentiable Almost Everywhere

Consider the probability space $([0,1), \mathcal{B}, \lambda)$ where $\mathcal{B}$ is the Borel $\sigma$-algebra and $\lambda$ is the uniform measure. Let $A_{i,n} = [(i-1)2^{-n}, i2^{-n})$ for $i \in ...
6
votes
1answer
90 views

Monkey typing ABRACADABRA and gamblers

Problem: A monkey is sitting at a typewriter, typing a letter (A-Z) independently and with uniform distribution each minute. What is the expected amount of time that passes before ABRACADABRA is ...
2
votes
2answers
47 views

Conditional expectation as Borel function

Let $X,Y$ be random variables with $E|X|< \infty$. Prove that there is a Borel function $h:\mathbb{R}\rightarrow \mathbb{R}$ such that $E[X|\sigma(Y)]=h(Y)$ almost surely. (Here $\sigma(Y)$ is ...
1
vote
1answer
54 views

Find a function f(t) such that Y is a martingale

Let $(X_t)$ be a process with independent increments such that $X_0=0$ and $E(X_t)=0$ Let $F_t$ be a natural filtration of $X_t$ Let $a$ and $b$ be arbitrary real numbers and let $(Y_t)$ be a random ...
4
votes
0answers
74 views

Submartingality of generalized stochastic exponential of a BMO martingale

I attended a talk today on BMO martingales. It was my first encounter with the subject, and this may explain my inability to solve this myself. We take a continuous local martingale $L$, and say ...
3
votes
0answers
74 views

Levy's extension of the Borel-Cantelli Lemmas

Following is the statement and proof of Levy's extension of the Borel-Cantelli Lemmas, as given in Williams' "Probability with Martingales" (1991), in section 12.15 on page 124. I understand most of ...
1
vote
1answer
21 views

Infinite oscillation of random signs

Suppose that $\left(a_n\right)$ is a sequence of real numbers and that $\left(\varepsilon_n\right)$ is a sequence of IID RVs with $$P\left(\varepsilon_n = \pm 1\right) = \frac{1}{2}$$ According to ...
0
votes
0answers
27 views

square integrable martingale gaussian conditionally on its quadratic variation

First of all, I recall that if $M = (M_t)_{t \in [0,T]}$ is a cadlag square-integrable martingale, its quadratic variation process $[M]$ is defined as the unique cad increasing process such that ...
2
votes
1answer
49 views

Showing that a nonnegative integer-valued random variable is NOT a stopping time

Suppose that $\left(A_n\right)$ is an adapted process, and that $B\in\mathcal{B}$. Let $L = \sup\left\{n:n\leq10;A_n\in B\right\}$, $\sup\left(\emptyset\right)=0$. Convince yourself that $L$ is NOT ...
2
votes
1answer
29 views

Martingale and mean squared error

In preparation for a course I am doing later in the semester I have been trying to brush up on my knowledge about martingales. But I am struggling with the following problem: Let ...
3
votes
1answer
53 views

The branching process $\mu^{-n}Z_n = \mu^{-n}\sum_{k=1}^{Z_n{-1}}X_{n,k}$ is a martingale

Let $\{X_{n,k} : n,k \geq 1\}$ be a collection of i.i.d. $\mathbb{Z}_+$-random variables with finite variance $\sigma^2 > 0$ and mean $\mu > 0$. Define $(Z_n)_{n\geq 0}$ recursively by ...
2
votes
1answer
56 views

Determine a sequence of random variables is a martingale

I'm trying to solve a problem from an old exam. This is an easy but a bit lengthy exercise, divided into subproblems. Since they are based on each other and probably are quite short, I was hoping that ...
1
vote
0answers
49 views

An application of the Dambis-Dubins-Schwarz theorem. Is my argument correct?

I attended a lecture today, in which the professor went through an example with a lot of tedious calculations to show something which I'd think would follow directly from the Dambis-Dubins-Schwarz ...
0
votes
1answer
32 views

Bilinearity of quadratic variation

Fix a filtered probability space satisfying the usual conditions. Let $\mathcal{M}^2_0$ be the vector space of cadlag martingales null at $0$ bounded in $L^2$. We state without proof the following ...
0
votes
1answer
33 views

product of martingales bounded in $L^2$

Let $(M_t)_t$ and $(M_t)_t$ be two càdlàg martingales on the same filtered probability space. We know that $M_{\infty}$ and $N_{\infty}$ are orthogonal in $L^2$. Is it true that $(M_t N_t)_t$ is a ...
0
votes
1answer
38 views

Extended stochastic exponential

I have encountered a problem, which is hopefully rather easily solvable. I just can't get my head around it at the moment. First we extend a well known notion. We call a stochastic process $M$ a ...
4
votes
1answer
86 views

A weak-type submartingale inequality

Let $(X_n)_{n\in\mathbb N_0}$ be a submartingale or a supermartingale. Show that, for all $n\in\mathbb N$ and $\lambda>0$, $$\lambda P[|X|^*_n\ge \lambda ]\le 12 E[|X_0|]+9E[|X_n|].$$ ...
2
votes
1answer
104 views

What are some good books about martingales?

I'm looking for suggestions for well written books dealing with martingale theory, not necessarily exclusively. I'm also looking for a nice compilation of problems, preferably with answers, on this ...
3
votes
1answer
65 views

Textbook suggestion for studying martingales

I've been studying probability from Davar Khoshnevisan's graduate textbook since the beginning of the semester. A month ago, I came to the chapter of "Martingales". Since it's my first encounter with ...
2
votes
0answers
103 views

Local martingale is locally uniformly integrable martingale?

Is a local martingale locally uniformly integrable martingale ? Here I define a local martingale to be the process with a localizing sequence $\tau_n$ such that the stopped process is martingale. ...
0
votes
1answer
48 views

Show that a process is no semimartingale

_Hello everyone! I got a little question about how to show that the process $X_t:=|B_t|^{\frac{1}{3}}$ is NOT a semimartingale. So far I tried to apply Ito. Since if $X_t$ was a semimartingale so is ...
1
vote
1answer
62 views

Convergence of a conditional expectation

Consider a filtered probability space ($\Omega,\mathcal{F}_t,\mathcal{F},\mathbb{P}$) and a random variable $X$ defined on $\Omega$ with values in a set $E$. We consider the process $X_t = ...
3
votes
0answers
46 views

Absolute continuity of quadratic variation of continuous local martingales

I am interested to know if there are any simple sufficient conditions on continuous local martingale to have absolutely continuous quadratic variation. In general , we know only that quadratic ...
2
votes
1answer
44 views

Hoeffding-type bound covering all partial sums? (better than naive union bound)

I think this must have been done before, possibly with martingales, but I can't find anything online! Given $X_1,\dots,X_n$ independent, each $X_i \in [a_i,b_i]$, letting $S_i = \sum_{j=1}^i X_j$, do ...
1
vote
1answer
50 views

Fake Brownian Motion

Does there exist a martingale which has Marginal distributions same as Brownian Motion marginals but the process itself not being Brownian motion? Any references are highly appreciated. Thanks.
4
votes
1answer
144 views

Can you make money on coin tosses when the odds is against you?

The strategy Given an initial investment $n$ dollars and a "bet buffer" $b$. Calculate the bet size $x=\lfloor\frac{n}{2^b-1}\rfloor$ dollars. Wager $x$ dollars on random variable $C$ that $C=1$ ...
4
votes
0answers
90 views

Time scaling of Brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion and $A_t$ be an increasing continuous process adapted to the filtration generated by the Brownian Motion and $A_0 = 0$. I am trying to prove ...
4
votes
1answer
96 views

Karatzas and Shreve Problem 3.3.38

Let $X$ be a continuous process and $A$ a continuous, increasing process with $X_0 = A_0 = 0$, a.s. Suppose that for every $\theta \in \mathbb{R}$, the process $$Z_t^{\theta} = ...
2
votes
1answer
52 views

Prove $Y_n:=\sup|\hat{F}_n(x)-F(x)|$ is a reverse submartingale

Suppose $\{X_j,j \ge 1\}$ are iid with common distribution $F$ and let $\hat{F}_n$ be the empirical distribution based on $X_1,\dots,X_n$. Show $$Y_n:=\sup|\hat{F}_n(x)-F(x)|$$ is a reverse ...
0
votes
1answer
44 views

Example of a sequence of r.v.'s with constant stopping time that is not a Martingale

Could anybody give me a simple example of a sequence of random variables $(X_{n})_{n \geq 0}$ that has constant expectation, but is not a martingale?
1
vote
1answer
38 views

How $\Pi_{i=0}^n\xi_i$ converges a.s. to $0$ provided $\xi_n>0 $, iid and $E(\xi_n)=1$

suppose $\{\xi_n,n \ge 0\}$ are iid and positive random variables. $E(\xi_0)=1$. show $\Pi_{i=0}^n\xi_i$ is a positive martingale converging to $0$ provided $P[\xi_0=1]\not=1$ It's easy to prove ...
1
vote
1answer
95 views

Variance With Martingales Problem - Answered; Ignore the Bounty

Let $(X_{j})_{j \geq 1}$ be random variables such that $X_{j}$ is $\mathcal{F}$-measurable for each $j$, where $(F_{j})_{j\geq 1}$ is an increasing sequence of $\sigma$-algebras. Assume ...
1
vote
1answer
81 views

a problem about martingale, to prove $X_n \le E(X_\infty|B_n)$

Suppose $\{(X_n, B_n), n\ge0\}$ is an $L_1$-bounded martingale. If there exists an integrable random variable Y such that $X_n \le E(Y|B_n)$ then $X_n \le E(X_\infty|B_n)$ for all $n \ge0$ where ...