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

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$L^p$ Martingale convergence theorem

I am trying to prove the $L^p$ Martingale convergence theorem for martingale $X=(X_n)^{\infty}_{n=0}$ on $(\Omega,\mathcal{F},(\mathcal{F}_n)^\infty_{n=0},\mathbb{P})$ which is bounded in $L^p$ for ...
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44 views

prove this martingale inequality

The problem is like this: Let $Y_1,Y_2,\ldots$ be nonnegative i.i.d. random variables with $E(Y_m)=1$. Let $X_n=\prod_{m\leq n} Y_m$, show that $\lim_{n\rightarrow \infty}X_n=0$ if $P(Y_m=1)<1$. ...
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1answer
36 views

Uniformly integrable martingale in a finite time horizon

Let $\{ M (t) \mid t \in [0,T] \}$ be a martingale and $\{ \tau_n \mid n = 1, 2, \ldots\}$ be an increasing sequence of stopping times such that $\tau_n \rightarrow \infty$ as $n \rightarrow \infty$. ...
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0answers
19 views

Harmonic functions and null recurrence

Let $(X_n)_{n \geq 0}$ be a irreducible Markov chain defined on a countable state space $S$. It is known some ways to figure out if this chain is recurrent or not looking for superharmonic functions ...
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1answer
14 views

Positive component of a submartingale is a submartingale

I am trying to prove the Doob's Upcrossing Lemma and the first step requires to prove that: If $X$ is a submartingale, then $(X-a)_+$ is a submartingale. I found it intuitive but i failed to prove. ...
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1answer
35 views

A proof about martingales and variance

We consider a martingale $(S_n)$ with $\mathbb E(S_n^2)<K<\infty$. Suppose that $\mathrm{ Var}(S_n)\rightarrow0$. Prove that $S=\lim_{n\rightarrow \infty}S_n$ exists and is constant a.s. I ...
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1answer
33 views

Criterion of two measures are absolute continuous or singular

I was reading the Durrett's book: probability theory and example and stuck at some stages about the radon-nikodym derivatives related topic: Here is the setting: Let $\mu$ and $\nu$ be measures on ...
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109 views

Condition for a process to be a supermartingale

I am struggling in this question: Let $W$ denote a Brownian motion. Given that $ X_t = e^{- \lambda t} X_0 + \int_0^t \sigma e^{- \lambda (t-s)} \,dW_s$ solves the SDE \begin{equation} dX_t = - ...
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26 views

Prove the process is a martingale with respect to the natural filtration

Let $\{M_n\}_{n\ge 0}$ be a symmetric simple random walk. Fix a real $b$. Prove that the process $S_n = e^{bM_n} (\frac{2}{e^b + e^{-b}})^n$, $n = 0,1,2,....$, is a martingale w.r.t. the natural ...
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32 views

Showing martingale property of a series

I want to show that the following series is a martingale. $P(X_1=1)=P(X_2=-1)=0.5$ and $P(X_i=X_{i-1})=p$ and $ P(X_i=-X_{i-1})=1-p$ $S_n=X_1+...+X_{n-1}+\frac{1}{2(1-p)}X_n$ We need to show ...
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1answer
31 views

How to show that this is a martingale process?

$X_1,X_2,...,X_n.. \ $ are independent and $X_n\sim Pois (n)$. How can I show that $S_n=X_1+X_2+...+X_n-n(n+1)/2$ is a martingale with respect to the natural filtration? Thanks in advance!
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16 views

Interpreting a sequence and showing that it is a martingale.

I saw that one guy already asked this question, but he did not get an answer and I wasn't able to comment his thread. So, hopefully this is allowed. I am wondering about the following problem: The ...
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1answer
64 views

conditional expectation of some solution of SDE

Let $(M_t)$ be a nonnegative martingale in a probability space $(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, \mathbb{P} )$ given by \begin{equation} dM_t = M_t \sigma_t dW_t \end{equation} for some ...
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1answer
23 views

deterministic expression of stochastic integral

Let $(M_t)$ be a non-negative martingale on a probability space $(\Omega, \mathcal{F}, \{ \mathcal{F}_n \} , \mathbb{P})$. Let $dM_t = M_t dW_t$. How can we write the following \begin{equation} ...
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1answer
22 views

Quadratic covariation of Martingales

I was succesful at showing that the quadratic covariation $\langle\cdot ,\cdot \rangle_t$ is a positiv semidefinit, symmetric and bilinear form for each $t$ on the set of local martigales. So the ...
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1answer
25 views

Interpret the following sequence $X_n^{(k)} = \underset{1 \leq i_1 < \dots <i_k \leq n }{\sum} \; \xi_{i_1} \dots \xi_{i_k}$

I'm working on a problem in which I have the following set up. Let $\xi_1, \xi_2, \dots$ be independent random variables with $E[\xi_i] =0$ for all $i$. Then they define the following sequence $ ...
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1answer
41 views

Where does this product of random variables converge to?

Consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ wich are independently normal distributed $N(0,\sigma^2)$. Set $M_0$=1 and $$ M_n =\exp \left( \sum_{i=1}^n X_i - ...
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1answer
52 views

Martingale $X_n \to \infty$ a.s.

Construct a martingale $X_n$ such that $X_n \to \infty $ a.s. I have trouble coming up with such an example and prove it. Can someone provide an example?
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1answer
21 views

local martingale bounded below by a DL process

Let a continuous adapted process $Z= (Z_t)_{t \geq 0}$ be of class DL if \begin{equation} \{ Z_{\tau \wedge t} : \, \tau \text{ is a stopping time } \} \end{equation} is uniformly integrable, for each ...
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0answers
11 views

would the following game be beatable with martingale

I want to mix 2 games with weight on game one 51.5% and 48.5% on game two player will be presented with 3 coins and he will be asked to click on 1 of the 3 coins and then all 3 coins are turned ...
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2answers
35 views

Lévy's upward theorem and $\mathcal{L}^p$ convergence.

Lévy's upward theorem: Let $Y \in \mathcal{L}^1(\Omega, \mathcal{F}, P)$, $(\mathcal{F}_n)_{n=1}^{\infty}$ a filtration of $\mathcal{F}$ and $\mathcal{F}_{\infty} = \sigma( \bigcup_{n=1}^{\infty} ...
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1answer
25 views

show that the solution is a local martingale iff it has zero drift

Most financial maths textbook state the following: Given an $n$-dimensional Ito-process defined by \begin{equation} X_t = X_0 + \int_0^{t} \alpha_s \,d W_s + \int_0^{t} \beta_s \,d s, \end{equation} ...
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1answer
18 views

Asymmetric simple random walk?

It comes from the book Probability: Theory and Example. I don't understand the part marked with red line. Why it cannot converge to an interior point of $(a,b)$? Can anyone help? Thanks so much!
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1answer
32 views

Property of submartingale and supermartingle?

Is it true that for a submartingale, $$E(X_n) \le E(X_m)$$ for $n \le m$. And for a supermartingale, $$E(X_n) \ge E(X_m)$$ for $n \le m$. If it is true, then why? I feel confused because the ...
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37 views

Where is the assumption of right continuity used in the following proof?

Lemma:If $X$ be a right-continuous positive local martingale then , $X$ is a generalized super martingale Proof: $\forall s<t$ $$E[X_t\mid F_s]=E[\lim_{n\to\infty} X_{t \wedge\tau_n}\mid F_s] \leq ...
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3answers
55 views

Problem similar to Kolmogorov's inequality using martingale.

Suppose that $X_k$ is a sequence of independent random variables with mean zero and variance $1$. Let $S_k=X_1+\cdots+X_k$ and let $$ h(\lambda)=\limsup_{n \rightarrow \infty}P\left(\max_{1\leq k\leq ...
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fractional Brownian motion is not a semimartingale. How to apply Ergodic theorem in the proof of this theorem?

Here is the proof of the theorem. I couldn't understand how to apply Ergodic theorem in this proof. Let $X=(X_t)_{t\geq0}$ be a fractional Brownian motion with self-similar parameter $H\in(0,1)$. We ...
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1answer
32 views

Martingales and variance

For a martingale $(Z_n)_{n\in \mathbb N}$ define $X_i=Z_i-Z_{i-1}$ with $Z_0=0$ Show: $$Var(Z_n)=\sum_{i=1}^nVar(X_i)$$ My attempt: We can write $Z_n=\sum_{i=1}^nX_i$, so we actually just have to ...
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32 views

martingale convergence proof

This is out of Durrett 5.5.7. Let $X_n \in [0,1]$ be adapted to $\mathcal{F_n}$. Let $\alpha,\beta > 0$ such that $\alpha + \beta = 1$. Suppose that $$ P(X_{n+1} = \alpha + \beta X_n | ...
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Doob's decomposition of $X_n=\sum _{m=1}^n1_{B_m}$, where $B_m \in \mathcal{F}_m$.

$X_n=\sum _{m=1}^n1_{B_m}$, where $B_m \in \mathcal{F}_m$. I want to find the Doob's decomposition. I think $X_n=Y_n+Z_n$, where $Y_n$ is a martingale, $Z_n$ is a predicable process. Then ...
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1answer
38 views

How to show that this process is a martingale

Consider the probability space $([0, 1), \mathcal{B}[0, 1), \lambda)$, where $B[0, 1)$ are the Borel sets on $[0, 1)$ and $\lambda$ is the Lebesgue measure. Let \begin{align*} I_k^n = \left[ ...
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1answer
29 views

A question on proving the existence of a martingle which has a deterministic square bracket

Let $g:\mathbb{R^+} \to \mathbb{R^+}$ be a non decreasing and continuous function . Show that there exists a continuous martingale M such that its square bracket $<M>_t=g(t)-g(0)?)$ I have ...
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1answer
17 views

Estimate of the expectation value

Consider a sequence $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. random variables on a probability space $(\Omega,\mathcal{F}, \mathbb{P})$ with expectation $\mathbb{E}[X_1]=0$ and ...
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0answers
32 views

Exercise on quadratic variation

I am faced with the following exercise: Let $X_{1},X_{2},...$ be independent random variables satisfying $\mathbb{E}(X_{n}^{2})<\infty$ and $\mathbb{E}(X_{n})=0$ for all $n\in\mathbb{N}_{0}$. ...
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27 views

Prove that $\text{Var} \tau = \frac{1 − (p − q)^2}{(p-q)^3} $ where $\tau$-stopping time

Let $S_n = \xi_1 + \dots + \xi_n$ be asimetric random walk such that $P(\xi_i = 1) = p > \frac{1}{2}$ and $P(\xi_i = -1) = q $. Let $\sigma^2 =1-(p-q)^2$ and let $X_n=(S_n-n-(p-q)n)^2 - \sigma^2n $ ...
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0answers
35 views

Prove that $X_n = e^{(\theta S_n-n \ln \phi(\theta))}$ is martingale.

Let $\xi_1 \neq const.$ be a random variable with moment-generating function $\phi(\theta) = Ee^{(\theta \xi_1)}$. Let $S_n = \xi_1 + \dots \xi_n$. Prove that $X_n = e^{(\theta S_n-n \ln ...
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The definition of terms in Doob's decomposition theorem for submartingales

The definition of $Z_n$ in the Doob's Decomposition Theorem, I think it is a predictable submartingale starting at $0$. Is that right? Thanks for your help!:)
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Doob's decomposition Thm_ Got stuck applying induction in proving $Z_{n+1}$ is $F_n$ measurable?

Already know $Z_0=0$, and $$Z_{n+1}=E(X_{n+1}|F_n)-X_n+Z_n$$ $X_n$ is $F_n$ measurable, $F_n$ is a filtration. How to prove $Z_{n+1}$ is $F_n$ measurable? I tried to prove by induction. Since $Z_1$ ...
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1answer
70 views

Martingale Transforms and quadratic variation

Let $M$ be a martingale with $\mathbb{E}M_{n}^2<\infty$ for all $n$. Let $C$ be a bounded predictable process and set $X=M\cdot C$. Show that $\mathbb{E}X_{n}^2<\infty$ for all $n$ and that ...
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2answers
68 views

Simple formula on $X_n^{(k)} = \sum_{1 \le i_1 < … < i_k \le n} Y_{i_1} \cdot \dots \cdot Y_{i_k}$ (to show $X_n^{(k)}$ is martingale)

Let $$X_n^{(k)} = \sum_{1 \le i_1 < ... < i_k \le n} Y_{i_1} \cdot \dots \cdot Y_{i_k}$$ If I take $k=2$ and $S_n = Y_1 + \dots + Y_n$ I have of course: $$X_n^{(2)} = \frac{1}{2} (S_n^2 - ...
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1answer
31 views

Girsanov Theorem Confusion

I'm getting completely bogged down by sign errors when using Girsanov's theorem. Keeping it simple, suppose $W_t$ is a standard Brownian motion under a probability measure $\mathbb{P}$, and we have a ...
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1answer
54 views

Prove that $S_n^2-s_n^2$ is martingale

Let $(X_i)$ be iid such that $EX_i = 0$ and $\operatorname{Var}X_i = \sigma_i^2$. Let $s_n^2 = \sum_{i=1}^n \sigma_i^2$ and $S_n = \sum_{i=1}^n X_i$. Prove that $S_n^2 - s_n^2 $ is martingale. My ...
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57 views

Martingales application

Let $X$ be a random variable, $X\in\mathcal{L}^2(\Omega,\mathcal{F},\mathbb{P})$, such that: $\mathbb{E}|X|<\infty$ and consider $\mathbb{F}$ a filtration. Define: ...
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1answer
17 views

Continuous martingale

I have a general question regarding the notion of "continuous martingale". Does this expression refer to a continuous time stochastic process that has the property of being a martingale or does it ...
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1answer
56 views

Ito Integral representation for bounded claims

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be Probability space and let $\{(X_t) : 0 \leq t \leq T \}$ be a continuous semimartingale on it. Let $\xi$ be $\mathcal{F}_T^X$ measurable and bounded. Does it ...
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0answers
16 views

Looking for resources: Generalizations of martingales to $\mathbb R^2$

In most introductory courses, a martingale $Y$ is defined as a stochastic process $$Y: T \times \Omega \to S$$ ,which satisfies certain conditions. ($\Omega$ is a probability space and a filtration ...
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2answers
55 views

Motivation behind study of martingales

Today I wanted to ask a question which I am sure has been answered in multiple places but for which I do not yet have a very clear understanding. Though martingales is a very well explored area of ...
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1answer
44 views

martingales, almost sure convergence

I am given a sequence of independent random variables $(X_n)$ with respective laws given by $P(X_n=-n^2)=\dfrac{1}{n^2}$ and $P(X_n=\dfrac{n^2}{n^2-1})=1-\dfrac{1}{n^2}$, and letting $S_n=X_1+...+X_n$ ...
2
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1answer
29 views

Martingale property of product of martingale and stochastic process.

$M_t$ is a martingale with respect to $\mathcal{F _t}$ for $t \geq 0$ and $Z$ is a bounded $\mathcal{F_r}$ measurable random variable. $0\leq r < s <\infty$. I want to show that $Z( M_{s\wedge ...
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11 views

Arbitrage-free market for continuous distribution

Is it true, that a one-period market say $(0,t)$ is arbitrage-free if $S_t$ is continuously distributed on $\mathbb{R}$? I.e., for continuous distributions on $\mathbb{R}$, there always exists a ...