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

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Martingale Transform counterexample

I am studying discrete time martingale theory and came across the classical "You can't beat the system" theorem: given a martingale $M$ and a previsible process $C=(C_n)_{ n \ge 1}$ such that $C_n$ is ...
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7 views

Modelling the ballot theorem as a martingale.

The page 19 in the link http://www.imada.sdu.dk/~jbj/DM839/FL15.pdf provides the explanation of what a ballot theorem is and how we can prove that it is a martingale. It takes a random variable $S_k$ ...
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18 views

Stopping time and the Martingale stopping theorem.

According to the book that I am reading, A nonnegative, integer valued random variable T is a stopping time for the sequence {$Z_{n},n\geqslant0$} if the event T = n depends only on the value of ...
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15 views

A clarification on $L_{loc}^2$ process and stochastic exponential

In the book by A. Pascucci (PDE and Martingale Methods in Option Pricing) I have found the following definition of $\mathbb{L}^2_{\text{loc}}$ process. Later (pp. 329-330) for a process ...
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23 views

Conditional expectation and Radon Nikodym derivative.

Assume $(\Omega,\mathcal{F}, \mathbb{P})$ is a probability space, $\{F_t\}_{t\leq T}$ is an adapted filteration and $M_t$ is a martingale with respect to that with $M_0=1$. We can define another ...
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22 views

Existence of compensator process under the assumption of local integrability and finite variation

I am reading a proof regarding existence of compensators under the assumption of local integrability in which I don't quite understand: Definition: The compensator of a cadlag adapted process $X$ ...
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12 views

Proving that $X^2- [X]$ is a local martingale given that $X$ is a cadlag locally square-integrable martingale

Suppose that $X$ is a cadlag locally square-integrable martingale. Let $[X]$ denote the quadratic variation of $X$. My textbook claims, by Ito's formula that $$ X^2 _t = X^2_0 + [X]_t + 2 ...
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What kind of decomposition is $X_{t \wedge L}=\tilde{X}_t+\int_0^{t \wedge L} \frac{d \langle X, M^L \rangle_s}{Z^L_{s^-}}$?

In one of the papers I was reading for my masters thesis I came across a theorem with no references. Theorem: If $(X_t)$ is an $(\mathcal{F}_t)$ martingale then there exists a $(\mathcal{F}^L_t)$ ...
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19 views

How can you model the quicksort algorithm as a martingale sequence?

A sequence of random variables $X_0 , X_1,....,X_n$ is called a martingale sequence if: $E[X_i | X_0,X_1,....,X$i-1$]$ = $X$i-1 The text that I am reading from says the following: Let us analyse ...
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Quadtratic variation of a martingale and its derivative

If I have a martingale $M_t$ and I know its quadratic variation $A_t$. Is there a connection between $\frac{d}{dt}M_t$ and $A_t$? Could you give me a book reference? Thank you very much
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33 views

Martingale under conditional prob. measure (definition)

Suppose we are given a probability space $(\Omega, \mathcal{F},P)$ s.t. r.v.s $X$ and $(Y_i)_{i=1}^\infty$ are $\mathcal{F}$-measurable. The relevant filtration is given by ...
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16 views

Locally square integrable (local) martingales

I'm reading Protter and sometimes he says "locally square integrable martingale", and sometimes he says "locally square integrable local martingale", and I wonder if these two are the same. Protter's ...
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15 views

How to show a hitting time is finite almost surely?

A one-dimensional symmetric simple random walk starts at $S_0 = 1$. How to show with probability one it passes $x = 0$ (or I guess equivalently, the stopping time of hitting $x = 0$ at the first time ...
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19 views

show that $X_n$ is a supermartingale $\Rightarrow$ $Y_n=\text{min}(X_n,x)$ is a supermartingale

I have to show that: if $X_n$ is a supermartingale then $Y_n=\text{min}(X_n,x)$ is a supermartingale; ($x\in R$) This what I did: since we can write : $\text{min}(X_n,x)=X_n{1}_{X_n \leq x} + ...
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Martingale Convergence Theorem for non-negative supermartingales: Why is limit non-negative?

Consider a supermartingale $(X_n)$ that is non-negative. The martingale convergence theorem states that $X_n \rightarrow X$ P-a.s. with $X \geq 0$ and $E[X] \leq E[X_0]$. Why can we conclude that $X ...
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19 views

When $X_{n\wedge N}$ converges to $X_N$ in probability for martingale $X_n$ and stopping time $N$?

Suppose $\sigma$-algebras $\{\mathcal{F}_n\}$ is a filtration and random variables $\{X_n\}$ are adapted to $\{\mathcal{F}_n\}$. $N$ is a stopping time w.r.t $\{\mathcal{F}_n\}$. If $(X_n, ...
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35 views

Application of Doob's inequality

Suppose that $X_n$ is a martingale with $X_0 = 0$ and $EX^2_n < \infty$. Show that $$P\left(\max_{1\leq m \leq n} {X_m} \geq \lambda\right) \leq \frac{EX^2_n}{EX^2_n+\lambda^2}$$ by using ...
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28 views

Is the following modification of a martingale still martingale?

I have a following question. Let $Z$ be a Geometric Brownian motion, $\frac{dZ(t)}{Z(t)} = \omega dt + \sigma dW(t) $ For $\omega = -\frac{1}{2}\sigma^{2}$ one can proof that $Z$ is a martingale. ...
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66 views

Proof that process is martingale, exponential distribution

Let $X_1,X_2,\dots$ be i.i.d. random variables with exponential distribution with parameter $1$ and define $$Y_m= \sup{\{k\ge1:X_1+\dots+X_k\le m\}}$$ Prove that $Y_m-m$ is martingale and ...
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28 views

Find a Martingale in a game of exchanging hats

$n$ people play a game of exchanging hats, with the following two rules: --They throw their hats in to a pile and everyone chooses one uniformly at random, those who got back their own hat are out of ...
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43 views

Supermartingale-like property : does convergence still obtains?

A super-martingale $\{X_n\}$ in discrete time is usually represented as having the defining property $$X_n \geq E[X_{n+1} \mid \mathcal F_n] ,\;\; \forall \,n \tag{1}$$ where $\{\mathcal F_n\}$ ...
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23 views

Martingale Poisson [closed]

Can somebody help me with working out: $$E[(N_{t}-\lambda t)^2\mid F_{s}]$$ where $N_{t}$ is a Poisson process and $F_{s}$ the $\sigma$-algebra generated by $N_{s}$, $0 \leq s < t$.
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Showing martigale, submartingale or supermartingale with log

Can somebody help me with determining whether $Z_{n}=\log(2n+S_{n})$ is a martingale, supermartingale or submartingale with $S_{n}=\sum_{i=1}^{n}X_{i}$ and the are i.i.d. random variables with $P(X_i ...
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31 views

Why does there exist a right continuous version of the supermartingale $\{P(L >u \vert F_u),u \geq 0)\}$

Why does there exist a right continuous version of the supermartingale $\{P(L >u \vert F_u),u \geq 0)\}$ where $L$ is a measurable random variable Its is clear that not all supermartingales have ...
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27 views

Counterintuitive result on quadratic variation

I will describe an example that seemingly contradicts the following Theorem For a local martingale $M$, let $[M,M]_t$ be its quadratic variation at $t$. For any $t$, if $E[[M,M]_t]<\infty$, then ...
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49 views

Convergence in $L^{1}$ of martingale

I have problems with the following task: Let $X_{n} $ for $n=1,2,3...$ be independent random variables with distribution $B(n,\frac{1}{n})$ and define $Y_n=X_1...X_n$. Is $Y_n$ convergent in $L^1$? ...
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34 views

Are martingales actually continuous?

There are strong theorems like the martingale convergence theorem giving us the existence of a continuous limit for $t \rightarrow \infty$ of a martingale $(X_t).$ But I was wondering under which ...
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31 views

Uniform integrability of “exponent martingale”

Suppose that $X_n$ is an iid sequence of random variables with $P[X_i=1]=p$ and $P[X_i=-1]=q:=(1-p)$. Then, $S_n=X_1+\cdots+X_n$ (with $S_0=0$) is a simple random walk. We can easily check that ...
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41 views

Uniform Integrability and relation to $L^p$ for $p>1$

Let $X_n$ be a martingale. Then we know that for $p> 1$ the conditions $\sup_n E[|X_n|^p] < \infty$ and $E[\sup_n |X_n|^p] < \infty$ are equivalent. For $p=1$ this does not hold, because ...
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28 views

Application of Dynkin's Lemma: $L^p$-limit of martingale

Let $X_n, n \geq 1$ be a $(\mathcal{F}_n)$-martingale on $(\Omega, \mathcal{A},P)$. Suppose that $X_n \rightarrow X_{\infty}$ in $L^p$ for some $p > 1$ and that there is some $X \in L^p$ such that ...
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57 views

Showing that $P(W_{t}/\sqrt{t \log(t)}>1+\epsilon)\to0$ when $t\to\infty$, where $(W_t)$ is a Wiener process

I have a question about the martingales $\dfrac{e^{W_{t}^2/(1+2t)}}{\sqrt{1+2t}}$. With use of this martingale I want to show that $P(\dfrac{W_{t}}{\sqrt{t log(t)}}>1+\epsilon)$ goes to $0$ if $t$ ...
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17 views

An application of Ito's formula

I am reading a proof in which I don't understand how to use Ito's rule to derive the following: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space such that $M^{(i)}$ and $M^{(k)}$ are ...
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28 views

Doob's $L^p$ inequality - Case $p = 1$

I have found in the wikipedia page following generalisation of Doob's so-called $L^p$ inequality, for general nonnegative submartinagles $X_s$: $$E[\sup_{0 \le s \le T} X_s] \le \frac{e(1 + ...
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69 views

Theorem 4.14 Brownian Motion and Stochastic Calculus

I have been reading the proof of Theorem 4.14 of Karatzas' book. I wonder whether there is a typo in the description of the process $\eta^{(n)}_{t}$ as $\xi^{(n)}_{t+}-\min({\lambda,A_{t} })$ ...
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20 views

Why does this “wlog” make sense: $L^p$-norms of random variables

Let $$\overline{X_n}:=\max_{0 \leq m \leq n} X_m^+$$ for a sequence of random variables $X_i, i \geq 1$ (in fact, it is a submartingale), where $X_m^+:=\max(X_m,0)$. We want to show that ...
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36 views

A simple symmetric random walk is adapted

$\newcommand{\ee}{\mathbb{E}}$The fact that for all $n$ we have $\ee[S_n \mid \mathcal{F_{n-1}}]=S_{n-1} ~\text{a.s.}$ and $\ee[ |S_n|]<\infty $ is usually shown explicitly when showing something ...
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A stochastic process $X$ with values in a separable Banach space $E$ is a martingale iff $f(X)$ is a martingale for all $f\in E^\ast$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space and ...
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How to generalize a fact (convex function of a mtg is submtg) about martingales to multivalued martingales?

It's known that a convex function of a martingale is a submartingale. What about martingales with values in $\mathbb{R}^{n}$? Is is true that a subharmonic function of such a martingale is a ...
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48 views

exponential martingale inequality

I stumbled across a claim I couldn't verify. Let $M_t$ be a continuous local martingale, $M_0=0$ a.s. and $\lambda>0$. Then $$ \mathbb{E}\left( \exp \left( \lambda M_t \right) \right) \leq ...
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1answer
27 views

Is there a generalized Chernoff bound for submartingales

The extension of Markov inequality for submartingales is the Kolmogorov submartingale inequality. For a non negative submartingale $\lbrace Z_m, m \geq 1 \rbrace$ \begin{align*} Pr\left[\max_{1 \leq i ...
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30 views

Solving equation with Wiener process

I want to show that if $E(f(X_{t}))=E(f(W_{t})e^{\lambda W_{t}-0.5*\lambda^2*t})$, where $W_{t}$ is a Wiener Process, then $X_{t}\sim N(\lambda t,t)$. Does anyone have a clue how to solve this ...
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72 views

Showing martingale for a Brownian motion $(W_t)_{t \geq 0}$

I want to show that $\dfrac{e^{W_{t}^2/(1+2t)}}{\sqrt{1+2t}}$ is a martingale with respect to $F_{t}$. We can use that $$E(e^{\alpha ...
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51 views

Is this process a martingale?

Given $X_t=\int_0^t s W_s dW_s$ and the process $M_t=X_t^3-\int_0^t X_sY_s ds$. Find $Y_t$ such that $M_t$ is a martingale. I started thinking that $X_t$ can be seen as: $dX_t=tW_tdW_t$ then by ...
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1answer
26 views

Calculating a (Forward Measure) Martingale

Above is my question. I am, unfortunately, stuck on part (a)! Below are my workings. I've just spotted a typo -- at one point, an "$\exp$" is missing, but it's fairly obviously supposed to be there. ...
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1answer
35 views

Sequence of Martingales convergent in $L^1$-norm

Suppose $X^n_t$ is a sequence of martingales on a filtered probability space $\left(\Omega,\mathcal{F},\mathbb{P},\left(\mathcal{F_t}\right)_{t\in\left[0,T\right]}\right)$, that is for $\Delta>0$ ...
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1answer
91 views

Distribution in Polya's Urn / existence of mgf / Stolz–Cesàro alternative / dominated convergence theorem

I know this has been asked elsewhere, but I think the values or random variables are different or something. From Williams' Probability with Martingales: I proved that $M_n$ is a $\sigma(B_1, ...
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1answer
29 views

Does local martingale have the same mean value as well? [duplicate]

We know that if $\{M_n\}$ is a martingale, we know from definition of martingale that $E(M_n) = M_0$ for all $n \geq 0$. However, if we only know that a sequence of random variables $\{X_n\}$ is a ...
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1answer
37 views

Upper bound for random walk to show stopping time is bounded

I have a simple symmetric random walk (SSRW), and a stopping time: $\tau=\inf\{ n \geq 0 ~:~ |S_n|=N\}$. I am showing that $\newcommand{\ee}[1]{\mathbb{E}[#1]}$ $\newcommand{\pp}[1]{\mathbb{P}[#1]}$ ...
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1answer
30 views

Martingale linearity property

Let $X_1, X_2,\ldots$ be i.i.d r.v's with mean $\mu$. Let $$S = \sum_{i=1}^n X_i$$ Let $F_n$ denote the information contained in $X_1,\ldots, X_n$ Show that $$E[S_n \mid \mathcal{F}_m] = S_m + ...
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40 views

Martingales: Expectation of almost-sure limit

Let $X_n, n\geq 0$ be a martingale. We know that $E[X_n]=E[X_m]$ for all $m, n \geq 0$. Moreover suppose that $X_n \rightarrow X$ P-a.s. What do we know about $E[X]$? Is it clear that $E[X]=E[X_0]$? ...