Questions tagged [martingales]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
8 views

Does this stopped process converges to the original process ucp (uniformly on compacts in probability)?

Let $M$ be a continuous local martingale with null at zero. Let $T(n)=\inf(t:|M_t|>n)$ be a stopping time. Then, the stopped process $M^{T(n)}$ is a bounded martingale (Lemma 29.1, Rojas and ...
2 votes
0 answers
17 views

If $X$ is a continuous supermartingale, why is, for every $N$, $(X_s)_{s\leq N}$ uniformly integrable?

If $X$ is a continuous supermartingale, why is, for every $N$, $(X_s)_{s\leq N}$ uniformly integrable? This statement is used, without proof, in another statement. I haven't been able to prove it. Is ...
2 votes
1 answer
24 views

Derive the 0-1 law from a functional equation of a martingale limit

I am trying to understand a proof given by Jabbour-Hattab in the paper "Martingales and Large Deviations for Binary Search Trees". In this paper, we consider the limit of a non-negative ...
4 votes
1 answer
67 views

Uniform Integrability and Proving martingale of Poisson product process

So I have been stuck on the following question that I stumbled across in a textbook. It is trying to show the following product process is a martingale. $$M_n = n! \prod_{k=1}^{n} X_k \ \text{where } ...
1 vote
2 answers
43 views

Book recommendation for stochastic integral wrt local martingales

I am looking for a book (or a chapter of a book) for stochastic integral wrt local martingales. The book should contain a rigourous introduction to the definition. It should also contain proofs for ...
0 votes
1 answer
23 views

Question about the integrand space of stochastic integral wrt martinagles

I am reading the book "Introduction to Stochastic Integration" by Hui-Hsiung Kuo. In Chapter 5, he introduces the definition of stochastic integral wrt martingale: $$I(f) = \int_a^b f(t) dM(...
1 vote
1 answer
131 views

Choquet-Deny theorem

Problem (Choquet-Deny theorem): Let $\sigma$ be a probability measure on $\mathbb R$, and let $f : \mathbb R \to \mathbb R$ be a bounded (Borel) measurable function. We will prove the $f$ satisfies ...
1 vote
0 answers
24 views

Constant in the Banach valued Marcinkiewicz-Zygmund inequality

Let $(\xi_i)_{i=1}^n$ be a sequence of real-valued independent random variables with zero mean. The classical Marcinkiewicz-Zygmund inequality is the following \begin{align} (E(\sum_{i=1}^n\xi_i)^p)^{...
0 votes
1 answer
30 views

Exercise 4.3.11 of Durrett's Probability: Theory and Examples

I am trying to solve Exercise 4.3.11 from Durrett's book. The question is: Show that if $P(\lim Z_n/\mu^n = 0)<1$ then it is equal to $\rho = P(Z_n = 0 \text{ for some n}$) where $Z_n$ is the ...
5 votes
2 answers
5k 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 ...
-2 votes
1 answer
37 views

Creating a martingale given [closed]

Let $\{X_t\}_{t=1}^\tau$ be an iid sequence of bounded r.vs defined on $(\Omega, \mathcal{F}, \mathbb{P})$. Assume that $P(X_1=1)=\frac{1}{2}=P(X_1=-1).$ Given a filtration $\{ \mathcal{F} \}_{t=0}^\...
0 votes
1 answer
14 views

maximum between martingale and constant

I want to understand if $(M_t\vee K)_{t\geq0}$ is a martingale?. Where $M$ is a martingale and $K$ is a positive constant. It is easy to see that it is a submartingale, but I am wondering if it is ...
1 vote
2 answers
34 views

The almost sure event in the law of the iterated logarithm for the Brownian motion: what it looks like

I am trying to explicit the almost sure event in the statement of the law of the iterated logarithm. Let us focus on the result about the $\limsup$. Throughout, let $(\Omega, \mathcal{H}, \mathbb{P})$ ...
2 votes
2 answers
46 views

Is the stochastic integral process $\left(\int_{0}^{t}e^{-\lambda\left(t-s\right)}\mathrm{d}B_s\right)_{t\geq 0}$ a martingale?

Up to now, I was only presented with stochastic integral processes of the form $\left(\int_{0}^{t}\phi_s\mathrm{d}B_s\right)_{t\geq 0}$ and the general way to show that such a process is a martingale ...
1 vote
0 answers
29 views

Are processes derived from deterministic and square-integrable functions martingales?

Problem Let $f:\mathbb{R}_{\geq 0}\to\mathbb{R}$ be a deterministic and square-integrable function, i.e. $\int_0^\infty f(t)^2\mathrm{d}t < \infty$, and define the process $X=(X_t)_{t\geq 0}$ by $...
7 votes
1 answer
1k views

Skorokhod Embedding theorem proof

It occurs in Durrett's proof of Skorokhod embedding that he needs the following. Suppose that we have a Brownian motion $B_t$ in $1$-d that starts at $0$. (To be clear, it is not necessarily ...
2 votes
1 answer
43 views

$\mathcal{F}_n$ measurable

Let $(X_n)$ be a sequence of random variables and $(\mathcal{F}_n)$ is associated naturel filtration. We define for each $k,n$ $S_{n,k}:= S_{n,k}(X_0,X_1,...,X_n)$ a function that depend on $X_0,X_1,.....
2 votes
1 answer
66 views

Optional Stopping Theorem for martingales bounded except at the stopping time

One form of the Optional Stopping Theorem is as follows. Let $X$ be a submartingale, $T$ a stopping time, and $a$ a constant such that for all $n$ we have $|X_{n \wedge T}| \le a$ almost surely; then ...
2 votes
1 answer
25 views

Example of a process that yields a non-martingale

I'm attending a class on stochastic processes and as a “side quest” to understand a discrete martingale transform properly we were given a task: Provide/construct a not previsible adapted process $C$ ...
-1 votes
0 answers
27 views

Is there a relationship between Bayes Theorem and Martingale's law [closed]

I'm interested in gambling theory: Martingale's law, the law of large numbers. One question is bothering me. If I play a game of gambling (let's assume or 30% chance of winning). If I lose at one ...
1 vote
2 answers
44 views

Definition of Left-Closable Martingale

I am currently studying martingales with Resnick's book A Probability Path. He defines a martingale as closed on the right if there is an $X \in L_1$ such that $X_n = \mathbb{E}[X \mid \mathcal{B}_n]$ ...
4 votes
1 answer
122 views

Equivalent characterization of the martingale property

I saw in several books and articles that the following lemma has been used: Let $M_t$ be a stochastic process such that $\mathbb{E}[|M_t|] < \infty$ for all $t$. Is that true that $M_t$ is a ...
9 votes
3 answers
2k views

Gambler's Ruin - Probability of Losing in t Steps

I would be surprised if this hasn't been asked before, but I cannot find it anywhere. Suppose we're given an instance of the gambler's ruin problem where the gambler starts off with $i$ dollars and ...
0 votes
1 answer
31 views

Optional Stopping Theorem and Stopped $\sigma$-fields

This is a simple exercise needed to prove the Optional Stopping Theorem that I'm working on. Suppose $(X_n)$ is a supermartingale and we have stopping times $T, S$. Then we already know in general ...
0 votes
0 answers
31 views

Expected number of steps to get "ABRACADABRA" [duplicate]

Suppose you have a 26-sided die, each face is labelled from A-Z, what is the expected number of steps to observe the sequence "ABRACADABRA" for the first time? ANS = $26^{11} + 26^4 + 26$ A ...
1 vote
1 answer
1k views

Understanding proof of martingale transform being supermartingale

I am reading "Probability with Martingales" by David Williams. On page 97 the following theorem and proof is stated: Theorem: Let $C$ be a bounded non-negative previsible process so that, ...
0 votes
1 answer
28 views

Is $(e^{i \lambda B_t + \frac{1}{2}\lambda^2t})_{t\geq 0}$ a martingale?

Showing that $(e^{\lambda B_t - \frac{1}{2}\lambda^2t})_{t\geq 0}$ is a $\mathbb{R}$-valued martingale Let $B$ be a standard $\mathbb{R}$-valued Brownian motion and let $\lambda\in\mathbb{R}$. From $...
2 votes
1 answer
33 views

Calculate $\mathbb{E}(\exp (- \lambda T_x ))$ for $\lambda > 0$ where $X$ is a Brownian Motion with drift

In a course I am studying on Stochastic Processes, I encountered the following exercise: Let $X_t = B_t + ct$ for some $c \in \mathbb{R}$ and where $B$ is a standard Brownian Motion. Now define $T_x$ ...
2 votes
1 answer
64 views

Expected value of the square of a stopping time

Problem Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[S_a^2\...
5 votes
1 answer
74 views

Expected value of the exponential of a stopping time

Problem Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[e^{-\...
3 votes
1 answer
45 views

Expectation of the indicator function

Define: For $n \geq 0$, on note $X_n=(n+1) \mathbb{1}_{[n+1,+\infty}$, and $\mathcal{F}_n=\sigma(\{1\},\{2\}, \ldots,\{n\},[n+$ $1,+\infty[)$ and $\forall k \in \mathbb{N}^*, \mathbb{P}(\{k\})=\frac{1}...
2 votes
1 answer
59 views

Stirling approximation of the probability that the stopping time is finite

Let $\left(Y_n\right)_{n \geq 1}$ a sequence of i.i.d random variables such that $\mathbb{P}\left(Y_1=1\right)=\mathbb{P}\left(Y_1=\right.$ $-1)=1 / 2$. Define $S_0=0, \mathcal{F}_0=\{\Omega, \...
1 vote
1 answer
55 views

Proving martingale properties

I'm new to stochastic processes and have problems understanding martingales, conditional probability, $\sigma$-algebras etc. I have two proofs that I'm now sure how to handle. Problem 1. Prove that an ...
2 votes
1 answer
140 views

Convergence of Pólya urns: what does it mean "expanding real polynomials in a basis"?

I'm reading these notes, and in the proof of this Proposition 2, which states: Let $d\ge 2$ and $S\ge 1$ be integers. Let also $(\alpha_1,\dots,\alpha_d)\in\mathbb{N}^d \setminus\{ 0 \}$. Let $(P_n)_{...
2 votes
1 answer
31 views

$Cov[X_m, X_n] = \mathbb{E}[(X_m- \mathbb{E}(X_m)) (X_n- \mathbb{E}(X_n))]$ for $S_n := \sum_{i=1}^n X_i$ a martingale

Let $(X_n)_n$ be a sequence of square-integrable RVs and let $\mathcal{F}$ be given by $\mathcal{F}_n = \sigma (X_1, \ldots, X_n)$. Suppose that $S_n := \sum_{i=1}^n X_i$ is an $\mathcal{F}$-martingal....
0 votes
0 answers
35 views

Exponential martingale and random walks

Consider the assymtric random walk $(S_n)_n$ on $\mathbb{Z}$. Determine $(a_n)_{n \ge 0}$ such that $\exp(\lambda S_n-a_n)_{n \ge 0}$ is a martingale. (We assume that $\lambda \in \mathbb{R}$ such ...
1 vote
1 answer
51 views

Seeking clarity on the concept of equivalent martingale measures

Question Consider a one-step trinomial tree, where there are two traded assets, a bond with risk-free rate, $r$, a stock with initial price, $S_0$, and terminal price $$S_T = \begin{cases} S_0u,& ...
1 vote
1 answer
276 views

The conditional expectation of i.i.d. random variables

Let $Y_1,Y_2,\dots,Y_n$ be a sequence of i.i.d. random variables. Each of them is integrable. Let $X_1=(Y_1+Y_2+\cdots+Y_n)/n,X_2=(Y_1+Y_2+\cdots+Y_{n-1})/(n-1),\dots,X_{n-1}=(Y_1+Y_2)/2,X_n=Y_1$ Show ...
1 vote
0 answers
22 views

Approximation of quadratic variation of martingales

Let $\{X_t\}$ be a square integrable martingale, $X^n_t\rightarrow X_t$ in $L^2(\Omega,\mathcal{F},P)$ for each $t$ (i.e., in the sense of mean square). Do we have $[X^n,X^n]_t$ converges to $[X,X]_t$ ...
3 votes
1 answer
676 views

Variance of the stopping time. Matching Problem

I have a following problem: $N$ guests leave their hats in a heap and collect them in random order. Those who by chance get back their own hats happily go home. The remaining ones yet again throw ...
8 votes
4 answers
2k views

Show rigorously that Pólya urn describes a martingale

We work with the famous Pólya urn problem. At the beginning one has $r$ red balls and $b$ blue ball in the urn. After each draw we add $t$ balls of the same color in the urn. $(X_n)_{n \in \mathbb N}...
2 votes
1 answer
35 views

A martingale $(X_t)_{t\geqslant 0}$ which is bounded in $L^{1}$ but not uniformly integrable.

Is there any examples of the martingale $X(t)$ which is bounded in $L^{1}$ but not uniformly integrable? I've just known that $$M_t=\mathrm{e}^{aW_t-a^2t/2}$$ with $a$ not $0$ and $W$ a Brownian ...
5 votes
1 answer
651 views

Quadratic variation of the compensated Poisson process $N_t - \lambda t$

Consider a Poisson process $\{N_t, F_t\}_{0 \le t < \infty}$. Then the martingale $M_t = N_t - \lambda t$ has quadratic variation $\lambda t$. The quadratic variation of $M$ is defined here as ...
2 votes
0 answers
35 views

Martingale defined by an integral

Consider a probability space $(\Omega,\mathcal{F},P).$ Let $f \in C^{\infty}_{c}(\mathbb{R}^d,\mathbb{R}),p \geq 2.$ $(X_r^{y})_{(r,y) \in \mathbb{R}_+ \times \mathbb{R}^d}$ is a stochastic process ...
0 votes
1 answer
80 views

On the proof of convergence of Pòlya urns

I'm reading the proof of Proposition $2$ of the Appendix here. The proposition states Let $d\ge 2$ and $S\ge 1$ be integers. Let also $(\alpha_1,\dots,\alpha_d)\in\mathbb{N}^d \setminus\{ 0 \}$. Let $...
4 votes
3 answers
213 views

Biased random walk with unequal step size

Consider an asymmetric random walk $(X_n)$ in which the initial point is one ($X_0 = 1$). It increases by $a$ with a probability of 0.3, remains the same with a probability of 0.4, and decreases by $b$...
1 vote
1 answer
59 views

Why do these three conditions imply divergence in probability?

This is a follow up to A martingale converging in distribution but not a.s. or in probability. Suppose we have a sequence of integer-valued RVs that satisfies (1) $P(X_n=a~i.o.)=1$ for each $a=-1,0,1$ ...
0 votes
1 answer
38 views

Comparing unbiased random walks with unequal step sizes and single boundary

I understand that the probability of absorbing eventually is one in a simple random walk with a single absorbing boundary. However, if we depart from this simple random and consider biased random ...
4 votes
1 answer
346 views

A martingale converging in distribution but not a.s. or in probability

I am now working on R. Durrett's Probability Theory and Examples. In his book, I am asked to construct a martingale $(X_n)$ satisfying the following three conditions. (1) $P(X_n=a$ i.o.$)=1, a=-1,0,...
2 votes
0 answers
26 views

Jacod & Protter's proof of Doob's Upcrossing Inequality

Let $(X_n)_{n\geq0}$ be a submartingale, $$T_0=0\text{,}\quad S_{n+1}=\inf\{k>T_n:X_k\leq a\}\text{,}\quad T_{n+1}=\inf\{k>S_{n+1}:X_k\geq b\}\text{,}$$ and $U_n=\sup\{k:T_k\leq n\}$ the number ...

1
2 3 4 5
72