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: $\eta$={$\eta(n,x):n\in\mathbb{N}, x\in \mathbb{Z}^{d}$} ...
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 $k \leq n$; $$E(M_n M_k) = E(M_k^2)$$ ...
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. ...
1
vote
2answers
66 views

Martingale Proofs

I havent been able to find an analogous question and our textbook is lacking in good examples, so I could use a little help with this rather straight forward martingale problem: Let X=(Xn) be a ...
2
votes
1answer
176 views

Sequence of independent random variables: Convergence, martingales, uniform integrability

I am having some problems with the following exercise: Let $(Y_n ,n ≥ 1)$ be a sequence of independent random variables such that: $P(Y_n = e^n − 1) = e^{−n}$, $P(Y_n = −1) = 1 − e^{−n}$, $∀n ...
0
votes
1answer
59 views

How to prove that $S_n^2 − Var(S_n )$ is a martingale

I would be grateful for some help with the following exercise: Let $(X_n ,n≥1)$ be a sequence of independent random variables with $E[X_i]=0$, and $Var(X_i)=σ_i^2<\infty, ∀i ∈\mathbb{N}$. ...
0
votes
1answer
29 views

Martingales involving exponents

I'm trying to solve the following problem, and am having problems with the expectation operator: Let $(X_n)_{n\geq1}$ be independent such that $E(X_i)=m_i$, $var(X_i)=\sigma_i^2$, $i\geq1$. Let ...
1
vote
2answers
164 views

$(S_n^2-n)_{n\ge 0}$ martingale and bounded stopping times

Consider the random walk $$S_n=\sum_{k}^{n}X_{k}$$ Where $X_k$'s are iid, $$\mathbb P(X_1=1)=\mathbb P(X_1=-1)=\frac{1}{2}$$ and $\mathcal{F}_{n}=\sigma(X_i,0\leq i\leq n)$. How do I prove that ...
1
vote
0answers
107 views

Stuck on proof of a martingale equality (similar to doob's inequality)

The question is: Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. Prove that for every $x > 0$ $$P\{\sup_{t\geq 0} M_t > x \,| \,F_0\} = 1\wedge ...
4
votes
1answer
121 views

Gambling Game: Martingales

This is a multipart question; if there's a strong preference for breaking this into separate questions I'll do that. Imagine a game between a gambler and a croupier. Total capital in the game is ...
1
vote
1answer
95 views

Conditional Expectation With Respect to Filtration

I'm trying to solve this problem: Let $\left(X_n\right)_{n\geq 1}$ be independent such that $\mathbb E\left(X_i\right)=m_i$ and $\mathrm{Var}(X_i)=\sigma_{i}^{2}$ for $i\geq 1$. Let $\displaystyle ...
0
votes
2answers
138 views

Martingale Stopping Time

Let $(S_n)_{n \ge 0}$ be a $(\mathcal F_n)$-martingale and $\tau$ a stopping time with finite expectation. Assume that there is a $c > 0$ such that, $\forall n, \mathbb E (|S_{n+1} - S_n | | ...
1
vote
2answers
46 views

Finding sequences such that function of sum of r.v's is martingale

Let $\left(X_n\right)_{n\geq 1}$ be independent such that $\mathbb E\left(X_i\right)=m_i$ and $\mathrm{Var}(X_i)=\sigma_{i}^{2}$ for $i\geq 1$. Let $\displaystyle S_{n}=\sum_{i=1}^{n}X_i$ and ...
1
vote
1answer
144 views

Nonnegative Superharmonic Function is Constant for $d>2$?

I have to do the following: Let $\alpha>0$ be fixed, $(X_i)_{i\geq 1}$ be i.i.d., $\mathbb R^{d}$-valued random variables, uniformly distributed on the ball $B(0,a)$. Set $\displaystyle ...
1
vote
0answers
43 views

Supermartingale Lemma + related problems

Given the following Lemma: Let $A_{t}=\int_{0}^{t}a_{s}dB_{s}$ where $a$ is an adapted process satisfying $\mathbb{P}\Big(\int_{0}^{T}a^{2}_{u}du < \infty\Big) = 1$ and $B$ is a standard Brownian ...
1
vote
1answer
406 views

Checking for Martingales on Stochastic processes

I am confused about how to check whether a process is Martingale. I know, I have to check for clear drift but a bit confused about to approach this problems. I need to apply Ito's first i think. For ...
5
votes
0answers
101 views

Martingale inequality related to Kolmogorov's maximal inequality [duplicate]

The problem This is a homework question from Durrett's Probability: Theory and Examples. Hints would be appreciated. Let $\xi_i$ be independent with expected value zero and variances $\sigma^2_i ...
1
vote
1answer
230 views

d-dimensional Brownian motion and martingales

I was solving questions from the Martingales chapter in "Stochastic Processes" by Richard Bass. There was a question regarding d- dimensional Brownian motions(BM): Let $(W_t^1,...,W_t^d)$ be a d ...
0
votes
0answers
144 views

Further Clarification Needed: Martingale to prove f(x+s) = f(x)

Given $f$ is a bounded continuous function on $\mathbf{R}$ and $\mu$ is a probability measure such that for all $x \in \mathbf{R}$ $$ f(x) = \int_\mathbf{R}f(x+y)\mu(dy) $$ Please help to show that ...
0
votes
1answer
65 views

Convergence of a martingale

Let $X_n$ takes its values in [0,1] and $p$ is a fixed number in $[0,1]$ Now if $ X_{n+1} = 1-p+pX_n $ with probability $X_n$ and $X_{n+1} = pX_n$ with probability $1-X_n$ . I know that $X_n$ is a ...
1
vote
1answer
261 views

Optional Sampling Theorem Application on a Martingale

I'm struggling to figure out how to apply the optional sampling theorem to this problem. Here's the problem verbatim (it's problem 5.7 out of Lawler's Introduction to Stochastic Properties): ...
1
vote
0answers
96 views

Show that the assumption of right-continuity in the statement of the stopping theorem cannot be omitted

In our homework assignment, we were supposed to find an example showing that the assumption of right-continuity in the statement of the stopping theorem cannot be omitted in general (cf. ...
2
votes
1answer
196 views

Unbounded stopping times and optional stopping theorem

Given that ${X_n}$ is a submartingale with $\left|X_{k}-X_{k-1}\right|\leq M<\infty$, and defining stopping times $\tau_{1}<\tau_{2}$ with $E(\tau_{2})<\infty$, eventually I want to show ...
8
votes
1answer
375 views

A problem related to basic martingale theory

In our probability theory class, we are supposed to solve the following problem: Let $X_n$, $n \geq 1 $ be a sequence of independent random variables such that $ \mathbb{E}[X_n] = 0, ...
1
vote
1answer
647 views

Showing a function of random walk is a martingale

I would like a hint for the following problem: Consider a biased random walk on the integers with probability $p<1/2$ of moving to the right and probability $1-p$ of moving to the left. Let ...
5
votes
1answer
417 views

Exercise 7.7.1 in Grimmett & Stirzaker's 'Probability and Random Processes'

I'm having trouble solving exercise 7.7.1 in Grimmett & Stirzaker's Probability and Random Processes, which reads: Let $X_1,X_2,\ldots$ be random variables such that the partial sums ...
0
votes
2answers
583 views

Quadratic variation of continuous local martingales

Dear all, I hope you can help me with the proof of the following result: Fact If $X$ is a continuous local martingale, then $[X]_t < \infty $ a.s. for every $t \geq 0$, where $[X]$ denote the ...