Questions tagged [martingales]

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

Filter by
Sorted by
Tagged with
-1 votes
0 answers
11 views

Martingales and stopping rules [closed]

\textbf{Question:} Let (X_1, X_2, \ldots, X_n) be a sequence of independent and identically distributed (iid) uniform variables on the interval ([-2, 2]), and define (S_n = X_1 + X_2 + \ldots + X_n). ...
kmil's user avatar
  • 1
0 votes
1 answer
22 views

Convergence of a specific martingale

Let $(S_n)_{n\geq 1}$ be a martingale s.t. $\mathbb{E}[{S_n}^2]<K<\infty$ for all $n$. Assume that $Var(S_n)\to 0$ as $n\to\infty$. Now I would like to prove that $S_n$ converges almost surely ...
Tob4U's user avatar
  • 43
1 vote
0 answers
23 views

$M$ is a continuous local martingale. Prove the consequences

$M$ is a continuous local martingale. Show that: a) If $M_0=0$ and $\mathbb{E} (< M >) _t <\infty$ for $t\ge 0$ then $M \in \mathcal M^{2,c}$ b) There is a sequence of stopping moments $\...
timofiej8384's user avatar
1 vote
1 answer
46 views

How can I check that the following process is a martingale?

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t), \Bbb{P})$ be a filtered probability space. Net $N$ be a poisson process with parameter $\lambda>0$. Let $h$ be a bounded measurable function and define $...
user1294729's user avatar
  • 2,008
0 votes
0 answers
38 views

Simple random walk, Martingales, stopping time

Suppose $S_n = X_1 + \dots + X_n $ is a simple random walk starting at 0. For any K, let $$T = \min \{n: | S_n| =K\}. $$ $\bullet$ Explain why for every j, $$ \mathbb{P}\{T \leq j +K | T > j\} \geq ...
Win_odd Dhamnekar's user avatar
1 vote
1 answer
42 views

P and Q martingales

Let $\mathbb{P},\mathbb{Q}$ be two equivalent probability measures. Could someone come up with an example of a $\mathbb{P}$-martingale, which is not a $\mathbb{Q}$-martingale and another example of a ...
Tob4U's user avatar
  • 43
-2 votes
0 answers
44 views

Is the following equation correct? [closed]

Let $N$ be a Poisson process and $t>s$, is the following equation correct? \begin{align*} &\mathbb E[N_s(N_t-N_s)|\mathcal F_s] = E[N_s|\mathcal F_s]E[(N_t-N_s)|\mathcal F_s] \end{align*}
Walter Venanzetti's user avatar
1 vote
2 answers
56 views

expected sum after rolling dice until getting 6 five times

We are rolling dice until we get 6 exactly five times. What is expected value of total sum? My approach: Lets denote $X_i$ - number rolled in $i$-th roll, and $S_n = \sum_{i=1}^{n}X_i$. So now i want ...
Kombajn's user avatar
  • 412
0 votes
0 answers
37 views

Wald's identity proof

Let $Y_1, Y_2, \ldots$ be iid. random variables in $\mathcal{L}^1$, $\mu:=\mathbb{E} Y_i$, and $\tau \geq 1$ a stopping time (w.r.t. the natural filtration), $\mathbb{E} \tau<\infty$. Let $S_n=\...
veganwithabeef's user avatar
0 votes
1 answer
34 views

Extending Doob’s optional stopping time

Let $\tau \geq 0$ be a stopping time, $\mathbb{E} \tau<\infty$. (b) Based on the identity $$ \left|X_{\tau \wedge n}-X_0\right|=\left|\sum_{k=1}^n\left(X_k-X_{k-1}\right) \cdot \mathbf{1}\{\tau \...
veganwithabeef's user avatar
0 votes
0 answers
21 views

Optional stopping time of a simple symmetric random walk on the square lattice

Let $S_n$ be a simple symmetric random walk on the square lattice $\mathbb{Z}^2$ with $S_0=(0,0)$. That is, the walker starts from the origin and at each step independently, she steps one unit to East,...
veganwithabeef's user avatar
2 votes
1 answer
30 views

Computing expected value of hitting time for a Feller process

We consider a Feller-Dynkin Markov process $X$ with generator $G$, which, when restricted to $C^2$ functions with compact support, is given by $Gf(x) = \frac{c(x)}{2}f''(x)$, where $c$ is a positive ...
IstEsOverFurMich's user avatar
0 votes
0 answers
20 views

book recommendations for martingale theory (studying measure theory)

Any recommendations for someone studying martingale theory. My course recommends Probability with martingales - David Williams and Probability-2 Albert N. Shiryaev but I've found both to be very dry, ...
veganwithabeef's user avatar
2 votes
1 answer
41 views

Is continuous, uniformly integrable martingale indistinguishable from BMO martingale?

Definition: BMO: Let $M$ be a martingale in $\mathcal{H}^2$. $M$ is said to be in BMO if there exists a constant c such that for any stopping time T we have $$ E\{(M_\infty-M_{T_-})^2 \mid\mathcal{F}...
KinoOrange's user avatar
1 vote
0 answers
59 views

Show that $Z(t)=\exp{\sigma X(t)-(\sigma \mu + \frac{1}{2} \sigma^2)t}$ is a martingale

I'm trying to show that $Z(t)=\exp{(\sigma X(t)-(\sigma \mu + \frac{1}{2} \sigma^2)t)}$ is a martingale. Attempt: I want to show that $E[Z(t)|\mathcal{F}(s)] = Z(s)$ $E[Z(t)|\mathcal{F}(s)] = E[Z(t)/ ...
George's user avatar
  • 63
0 votes
0 answers
22 views

Conditional Expectation of random walk square $E^{F_n}X_{n+1}^2$

$X_n$ is simple random walk It seems like $E^{F_n}[X_{n+1}^2]=X_n^2+1$ And $E^{F_n}[X_{n+1}^4]=X_n^4+6X_n^2+1$ I think the first expression is due to $X_n^2$ being positve But how does it reach the ...
dodo's user avatar
  • 718
1 vote
0 answers
33 views

Minimum submartingale inequality

I have a problem that goes like this: Let $(X_n)_{n\geq 0}$ be a submartingale and the constant $\lambda>0$. Show that $\lambda\mathbb{P}(\min_{0\leq k\leq n}X_k<-\lambda)\leq\mathbb{E}[X_n]-\...
Tob4U's user avatar
  • 43
2 votes
1 answer
45 views

Expected number of coin flips to get THTHTT

Suppose I am flipping a fair coin. What's the expected number of coin flips to get THTHTT? I know how to do this using a Markov chain approach. However, I end up with a system of six linear equations. ...
user1691278's user avatar
  • 1,341
1 vote
1 answer
72 views

A positive supermartingale is closed on the right

I would like some help with proving the following statement: Let $(X_t)_{t \in I}$ be a positive $(F_t)_{t \in I}$-supermartingale, that is, $E[X_t |F_s] \leq X_s$ for all $s \leq t$. Then $(X_t)_{t \...
somethingsomething69's user avatar
1 vote
1 answer
46 views

Conditional expectation of functions of a random variable

The conditional expectation $\mathbb{E}(X|Y)$ of a random variable $X$ given a random variable $Y$ on a probability space $\Omega$ is understood heuristically as our best guess of $X$ given the ...
xyz's user avatar
  • 878
5 votes
0 answers
93 views

High-probability tail bounds for a quantitative supermartingale inequality.

Let $(Y_t)_{t \in \mathbb{N}}$ be a non-negative stochastic process and $c \in (0,1)$. Suppose that $Y_1 = 1$ and that for all $t \in \mathbb{N}$ it holds that $$\mathbb{E}[Y_{t+1} \mid Y_1,\dots, Y_t]...
Bob's user avatar
  • 5,603
2 votes
1 answer
101 views

Under what conditions, if any, is a continuous function of a Martingale still a martingale?

Consider a martingale $M_s$ and a function $f(M_s) = \frac{1-e^{-kM_s}}{M_s}$. If $M_s$ is an Ito process, it is obvious from Ito's Lemma that $f(M_s)$ is not a Martingale. Does the restriction hold ...
Giorgi Chavchanidze's user avatar
1 vote
1 answer
50 views

Expectation and Variance of $Y_t = \int_0^t sdW_s$ + Martingale property

Denote the process $Y_t = \int_0^t sdW_s$. I want to answer the following question: Q) Calculate the expectation and variance of $Y_T$. Is $Y_T$ a martingale? A) I have read somewhere that it was ...
Itrytobehelpful's user avatar
0 votes
1 answer
50 views

Relation between measurability of intensity, compensator, and martingale in Doob-Meyer decomposition

Consider a sub-martingale (e.g., a counting process) $N_t$ adapted to a filtration $\mathcal{F}_t$. According to the Doob-Meyer decomposition, we have: $$N_t = M_t + A_t,$$ where $M_t$ is a martingale ...
Mingzhou Liu's user avatar
1 vote
0 answers
34 views

Is a martingale difference in terms of some random variables $X_t$ also a martingale difference in terms of $f(X_t)$ for a measurable function f?

The context is that in a version of central limit theorem for martingale difference (Billingsley 1961), the condition is for some random variable $u_n$ $$\mathbb{E}[u_n|u_1,...,u_{n-1}]=0\tag{1}$$ ...
toki's user avatar
  • 95
3 votes
1 answer
71 views

Proof that the stochastic exponential is a local martingale

I'm struggling to understand the proof as to why the stochastic exponential is a continuous non-negative local martingale. My notes say the following: where 3.6 is $Z_t = 1 + \int^t_0 Z_s dX_s$. I ...
Jamal's user avatar
  • 431
0 votes
0 answers
20 views

Martingale theorem representation using two martingales

Suposse we have two martingales $N$ and $M$, and the hypothesis of martingale representation theorem holds. Then \begin{equation} N_t=\int_0^t\psi_sdW_s, \,\, M_t=\int_0^t\phi_sdW_s \end{equation} So, ...
Don P.'s user avatar
  • 181
0 votes
0 answers
18 views

The conditions for an independent but not IID exponential random variable to converge in distribution to 0

Would anyone be able to check if I've missed anything for my answer to the following question. The requirements I've noted so far is that: For every bounded and continuous $f: \mathbb{R} \rightarrow \...
veganwithabeef's user avatar
0 votes
1 answer
38 views

How to solve these equations? (from 10.12.c, D. Williams, Probability with Martingales)

In Sec. 10.12 (hitting times of simple random works), eq. (c) (page 103) of Probability with Martingales (D. Williams 1991), the author said: For (any) $\theta\in \mathbb{R}$, let $\alpha:=\mathrm{...
Mingzhou Liu's user avatar
2 votes
1 answer
54 views

class of processes guaranteed to exceed any thresholds

Is it correct to say that any martingale (e.g., a GBM or anything similar), taking values in some real state space and not converging to a degenerate random variable, has the property that, given a ...
Jada's user avatar
  • 115
0 votes
0 answers
29 views

Polya's urn and SSRW.

I've got a question on a martingale that i've been stuck on for a good while. An urn contains $n$ white and $n$ black balls. We draw them one by one without replacement. We pay £1 for any black ball ...
carsck's user avatar
  • 1
2 votes
0 answers
54 views

Maximal inequality for martingales

Let $X_i$ be measurable for the $\sigma$-field $F_i$. Suppose that for some constants $a_i, c_i \in \mathbb{R}$, $$ \mathbb{E}\left(X_i-X_{i-1} \mid F_{i-1}\right)<a_i \quad \text { and } \quad\...
香结丁's user avatar
  • 397
0 votes
1 answer
117 views

Prove that gambling strategy with stopping time on seeing consecutive sequence of coins is a martingale.

recently I am trying to solve the following question: Let $\xi_{1}, \xi_{2}, ... $ be Bernoulli Random variable with $\mathbb{P}(\xi = 1) = \dfrac{1}{2}$ and $\mathbb{P}(\xi = 0) = \dfrac{1}{2}$. ...
엄익훈's user avatar
2 votes
1 answer
223 views

Prove that $L^2$ martingales with bounded increments that converge almost surely to a finite limit has converging quadratic variation

If $X_n$ is a sequence of $L^2$ martingales with bounded increments (i.e. $|X_n-X_{n-1}|<K$ for some $K>0$) such that $X_n$ converge almost surely to a finite limit, prove that the quadratic ...
YuiTo Cheng's user avatar
  • 4,695
0 votes
0 answers
20 views

Wiener Process Textbook or Reference Specifically Containing Ramp Intersection (or Exit Time) Analysis

I am looking for a reference (preferably a textbook so that additional preparation material is handy) that calculates the exit time of a Wiener process from a region bounded by sloped lines. Thank you,...
Gary's user avatar
  • 1
2 votes
0 answers
85 views

Martingale in a modified gambler's ruin problem

Let $\{Y_t\}_{t \in \mathbb{N}}$ be a random walk on $\mathbb{Z}$ defined as follows: $\mathbb{P}(x,x+1) = \begin{cases} p & \text{ if } x \geq 1\\ 1-p & \text{ if } x \leq 0 \end{cases}$ $\...
Daileon108's user avatar
0 votes
1 answer
59 views

How do I go from $E[Z_{n+1}\mid {\mathcal {F}}_{n}] = Z_n,\; n \in \mathbb N$ to $E[Z_{n}|{\mathcal {F}}_{k}]=Z_k,\;k<n$ using the tower rule? [closed]

How do I go from $E[Z_{n+1}\mid {\mathcal {F}}_{n}] = Z_n,\; n \in \mathbb N$ to $E[Z_{n}|{\mathcal {F}}_{k}]=Z_k,\;k<n$ using the tower rule?
kahan's user avatar
  • 11
1 vote
1 answer
65 views

Showing that a specific stochastic process is a martingale.

Let $ X = (X_n, n \in \Bbb N_0)$ be a stochastic process with state space $\Bbb N_0$, with unitary mean value for each $n \geqslant 1$, with independent increments and such that $P(X_0 = 0) = 1.$ I ...
xyz's user avatar
  • 1,709
4 votes
0 answers
40 views

Convergence theorem for bounded continuous martingale in $L^2$

I would like to be sure to understand the proof this theorem. Consider the set $\mathcal{M}^2$ of continuous martingale bounded in $L^2$. If $(X_t)_{t\geq0}\in\mathcal{M}^2$, there exists $X_{\infty}\...
coboy's user avatar
  • 1,116
3 votes
1 answer
67 views

From maximal inequality in finite time to continuous time

In class we have seen a maximal inequality about discrete sub martingale. The setting was the following : Consider $T=\{0,1,..,N\}$ and $(X_t)_{t\in T}$ a sub martingale. Then we have for all $\lambda&...
coboy's user avatar
  • 1,116
1 vote
1 answer
52 views

Show that $\mathbb E[X\mid Z_0,Z_1,\ldots,Z_t]$ is a martingale, where $\mathbb E[X]<\infty$ and $Z_t$ is a martingale.

Let $X$ be a random variable such that $E[|X| < \infty$, and let $\{Z_t: :t = 0,1,\ldots\}$ be a random sequence. We define the random sequence $\{X_t: t = 0,1,\ldots\}$ by $X_t = E[X\mid Z_0, Z_1, ...
Nicrotte's user avatar
1 vote
1 answer
61 views

Gambler's Ruin and the expectation of a stopping time in the case the conditions of the Optional Stopping Theorem do not hold

In these lecture slides (pages 13-21) on optional stopping theorem (OST) and martingale I have found a great example of the gambler's ruin problem and what happens when the criterions of the OST are ...
Sussyphus's user avatar
3 votes
1 answer
155 views

Polya's urn, should I use martingales or LLN

I am trying to prove the following question, but I am finding it a bit tricky to determine the distribution of $X_i$ (the number of red balls drawn in the $i$-th round) and thus I do not know which ...
idlatva's user avatar
  • 143
0 votes
0 answers
17 views

Proof of a maximal inequality in a finite interval

I consider the set $ I = \left\{ 0,…, N\right\}$ and $X$ a sub martingale on this set. I would like to prove the following inequality for $\lambda>0$ $$ \lambda\mathbb{P}(max(X_0,… X_N)\geq\lambda)\...
coboy's user avatar
  • 1,116
1 vote
1 answer
98 views

Brownian motion is not of bounded variation

I would like to prove that Brownian motion, denoted $B_t$, is not of bounded variation using the fact that its quadratic variation is finite. Here is my attempt: Consider $[t,s]\subset[0,+\infty)$ and ...
coboy's user avatar
  • 1,116
0 votes
0 answers
38 views

Almost Sure Convergence (Durrett 2.4.1)

Suppose the $j^{th}$ light bulb burns for an amount of time $X_j$ and then remains burned out for time $Y_j$ before being replaced. Suppose the $X_j$,$Y_j$ are positive and independent with the $X_j$’...
28ADY0901's user avatar
  • 686
0 votes
1 answer
46 views

Limit and Convergence of Doob Martingales

Let $(\Omega, \mathcal{F}, \mathcal{P})$ be a probability space for $X\in L^1(\mathcal{P})$ and $(\mathcal{F}_n)$ a filtration. Then $$Y_n = \mathbb{E}(X|\mathcal{F}_n)$$ is called Doob Martingale. ...
Sussyphus's user avatar
4 votes
2 answers
71 views

Almost sure convergence in density of infinite dimensional product measure of normal distributions unsing Kakutani's theorem

In measure theory Kakutani's Theorem is used to determine if two infinite product measures are equivalent, meaning they are both absolutely continuous with respect to each other. In one of my text ...
Sussyphus's user avatar
2 votes
0 answers
39 views

Convergence of urn model with repeated binomial draws

I'm analyzing a simple urn model, defined for $N>1$ an integer. The setup is as follows: we have an urn with $\frac{N}{2}$ white balls and $\frac{N}{2}$ black balls. At each step $k$, we sample $N$ ...
ntrstd11's user avatar
  • 239
0 votes
0 answers
64 views

About Martingale Stopping Theorem

Prove that: A right-continuous process $X$ adapted to the filtration $\{\mathcal{F}_t\}$ is a martingale if and only if for any bounded stopping time $T$, $X_T \in L^1$ and $E[X_T] = E[X_0]$. I know ...
Jimmy Gao's user avatar

1
2 3 4 5
71