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Questions tagged [martingales]

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

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Martingales - cyclists

The cycling race, in which $100$ cyclists participate, is played according to the following rules: after the k-th stage all riders who managed to fit in the top ten at all k in stages receive $10^k$ $....
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Martingales related to the square of a compensated Poisson process. [duplicate]

Let $N_t$ be a homogeneous Poisson process with rate $\alpha>0$ and $M_t=N_t-\alpha t$. Show that $M_t^2-\alpha t$ and $M_t^2-N_t$ are martingales. I computed \begin{align} \mathbb E[M_t^2\mid \...
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first-hitting-time and conditional probability

I have been struggling with the following problem. Let $\{X_{n}\}_{n\geq 1}$ be i.i.d. random variables with common distributional measure $\nu$. Let $B\subset\mathbb{R}$ be any Borel measurable set ...
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Quadratic martingale bound

I know that if $a_1,a_2,\dots$ are random variables and {$\mathcal{F}_{t}$ } is a filtration such that $$\mathbb{E}[a_i \mid \mathcal{F}_{i-1}] \leq K$$ for all $i$, then for any stopping time $\...
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prove that $(Y_n)_n$ is a submartingale

If $(X_n)_{n \in \mathbb{N}}$ is a nonnegative submartingale for the filtration $(\mathcal{F}_n)_n$ and F is an increasing and convexe function. We suppose that F has a differentiable function f. (F ...
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If $\mathbb{E}[\underset{0 \leq s \leq t}{\sup} \rvert X_s \rvert] < \infty$ for $X_t$ a local martingale, is $X_t$ a proper martingale?

Let $X_t$ be a continuous local martingale. Suppose $\mathbb{E}[\underset{0 \leq s \leq t}{\sup} \rvert X_s \rvert] < \infty$ for any $t > 0$. Is it true that $X_t$ is a proper martingale? We ...
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Does $\mathbb{E}[\langle X_{\tau} \rangle^{\frac{1}{2}}] < \infty$ imply $X_t$ is a martingale, given that it is a continuous local martingale

Let $X_t$ be a continuous local martingale. Suppose $\langle X_{\tau} \rangle$ satisfies $\mathbb{E}[\langle X_{\tau} \rangle^{\frac{1}{2}}] < \infty$ for any stopping time $\tau$. Is it true that $...
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How do we need to apply the martingale convergence theorem here?

Let $(E,\mathcal E,\mu)$ be a measure space $E_0\in\mathcal E$ with $\mu(E_0)\in(0,\infty)$ $n\in\mathbb N$ $B_1,\ldots,B_n\subseteq\left.\mathcal E\right|_{E_0}:=\left\{B\cap E_0:B\in\mathcal E\...
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Find the compenstor of the standard ito integral

Let $B_t$ be a Brownian motion and let $\{\mathcal F_t : a<t<b\}$ be a filtration such that for each $t$ we have that $B_t$ is $\mathcal F_t$ measurable and for and $s<t$, the random variable ...
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Show this Sequence of Random Variables converges almost surely to 1

Let $(X_{n})_{n\geq1}$ be a sequence of independent random variables on a probability space $ (\Omega, \mathcal{F}, \mathbb{P})$ taking non-negative integer values. Suppose that for each $n$ and each $...
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If $M$ is a martingale, why $M\mapsto \left<M\right>$ is a quadratic form?

In the book of Schilling and Partzsch (Brownian motion, an introduction to stochastic process), page 206, they say that $$M\mapsto \left<M\right>,$$ is a Quadratic form where $M$ is a martingale ...
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Martingale exercise

I'm struggling with an exercise from a martingales theory book: Let $M$ be an $F$-martingale and $Z$ an adapted (bounded) continuous process. Prove that: $$ \mathbb{}E\left(M_t\int_s^tZ_u \, du \...
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Uniqueness of Solution to Stochastic Integral Equation

Suppose that $N$ is an $(\mathcal{F}_{t})$-continuous local martingale, with $N_{0}=1$, $N_{t}\gt0$ a.s. for $t\geq0$ and $N$ satisfies: $$ N_{t}=1+\lambda\int_{0}^{t} N_{s}dB_{s} $$ Applying Ito's ...
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Equivalent conditions for submartingales (Problem 3.19 in Karatzas and Shreve)

I have a hard time understanding the proof of the following result in Karatzas and Shreve (Problem 3.19, page 18). Proposition The following three conditions are equivalent for a non-negative ...
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convergent process in finite time

Let $(p_t)_{t\in\mathbb{N}}$ be an stochastic process on a countable (probability measure) space. Supose it has the Markov and the Martingale properties. It converges almost surely to a random ...
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convergence of submartingale

We know that if $(X_n)_n$ is a submartingale such that $\sup_nE[X^+_n]<+\infty$ then $(X_n)_n$ converges a.s to an integrable random variable. I am searching for a proof which doesn't use Doob's ...
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Find the quadratic variation process of $\int f(s) \, dB_s$

Let $f \in L^2[a,b]$ and let $\displaystyle M(t)=\int_a^tf(s)dB(s)$. Find the quadratic variation process, $[M]_t$ , of $M(t)$. Here the quadratic variation process is the limit in probability of $\...
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Every local martingale with respect to the Brownian filtration has its continuous version.

I would appreciate some help on the following. In class, we said that Every local martingale with respect to the Brownian filtration has its continuous version. To prove this, it is apprently enough ...
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Maximum of a zero-mean random walk

Assume $S_n=\sum_{k=1}^n X_k$ where $X_k$ is i.i.d distributed and $\mathbb{E}X_1=0$ and $\mathbb EX_1^2<\infty$. Let $M_n=\max_{1\leq k \leq n}\{S_k\}$. What is the exponential asymptotic ...
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Why is finiteness of T not sufficient for the conclusion of Optional Stopping Theorem for martingales?

What is wrong with the following reasoning? If $(X_n)$ is a martingale then $EX_n = EX_0$ for all $n \in \mathbb N$. Therefore if $T$ is a stopping time that's finite a.s., then $T \in \mathbb N$ a.s....
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Existence of $L^2$ limit of sequence of martingales

I am currently revising some martingale theory, and was trying out an old past paper question. I haven't come across anything like this before, so was wondering how one would approach it, and also in ...
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Local martingale from weak solution to SDE

Let $X,\{\mathcal F_t\}$ be a weak solution to the SDE $$dX_t=b(X_t)\,dt+\sigma(X_t)\,dWt$$ where $b_i:\Bbb R^d\rightarrow \Bbb R$, $W$ is a $r$-dimensional Brownian motion and $\sigma_{ij}:\Bbb R^d\...
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Necessity and sufficiency of martingale

Let's consider probability space $(\Omega,\Sigma,P)$ with sequence of random bounded variables $(X_n)$. We assume also that $S_n:=X_1+X_2+...+X_n$. We put filtration $ {\displaystyle {\mathcal {F_n}}}=...
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29 views

Prove that the sequence is a Martingale.

Consider an urn that initially contains b black balls and w white balls. At every iteration, we draw a random ball is chosen and the chosen ball is replaced by c > 1 balls of the same color. Let $X_i$ ...
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Bound of a scaled random walk

Let $S$ be a Markov Chain on the non-negative integers such that $S_0 = s_0 \in \mathbb{Z}$ and $S_{k+1} = S_k +1$ with probability $p$ and $S_{k+1} = S_k -1$ with probability $1-p$. For $n \in \...
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key identities in the proof of Optional Sampling Theorem

Let $(\Omega,\mathscr{F},\mu)$ be a $\sigma$-finite measure space, let $(\mathscr{F}_n)_{n\in\mathbb{N}}$ be a filtration of sub-sigma-algebras of $\mathscr{F}$, let $(u_n)_{n\in\mathbb{N}}$ be a ...
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Martingale: Proof of strong law of large numbers

Is there any book or article that give a formal proof of strong law of large numbers by using martingale?
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47 views

Convex functions of a martingale.

Let $X_1(t)=e^{B(t)}$ and $X_2(t)=e^{-B(t)}$ where $B(t)$ is the standard brownian motion and $\{G_t\}$ is the filtration generated by the brownian motion. Determine what kind of martingale $X_1(t)$ ...
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42 views

Martingale if and only if integral against continuous, bounded function is zero.

I have seen it written that the integrable process $(X_1,\ldots,X_n)$ is a martingale if and only if $$ \mathbb{E}[h(X_1,\ldots,X_k)(X_{k+1} - X_k)] =0, $$ for all continuous, bounded functions $h$, ...
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Density of Maximum brownian motion

Why is it that for $M(t):=\max\limits_{0\leq s\leq t}B(s)$ $$P(M(t)\geq a) = 2P(B(t)\geq a)=\frac{2}{\sqrt{2\pi}}\int\limits_{a/\sqrt{\sigma^2t}}^{\infty}e^{-y^2/2}dy, \ a \geq 0$$ Doesn't result in: $...
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32 views

Almost sure equality if $P(X_n = x \text{ infinitely often})=1$ presumed

Let's say $(X_n)_{n\in N}$ is a non-negative supermartingale. From the Martingale convergence theorem follows that there exists $lim_nX_n = X_\infty$ a.s. I presume that $P(X_n = x \text{ infinitely ...
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84 views

Girsanov THM and Radon-Nikodym derivative

I've been having a hard time to applicate Girsanov theorem with Radon-Nikodym derivative in the demonstration of German-El Karoui-Rochet formule. I know that $\Pi_0:=S_0\mathbb{Q}^S(S_T\geq K)-K\...
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1answer
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Revuz and Yor's Book “Continuous Martingales ans Brownian Motion” - Chapter 1 - Exercise 1.11 (again)

Context : This post is the first of a series of posts taking their origins from the exercises in the Revuz and Yor's Book "Continuous Martingales ans Brownian Motion". The reason for doing so is ...
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Show that $M_t=e^{X_t-\frac{t^3}{6}}$ is a martingale

Let $X_t=\int_0^t s\text dB_s$, show that $M_t=e^{X_t-\frac{t^3}{6}}$ is a martingale and give its distribution. First encounter with the form of integral and Brownian motion, the tool I came to is ...
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1answer
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Find the constant $a$ such that $Y_t$ is a martingale.

Let $X_t$ be the solution of SDE $\text{d}X_t=3X_t\text dt+2X_t\text dB_t$ and $X_0=1$ which $B_t$ denotes the Brownian motion with $B_0=0$. Let $Y_t=e^{at}X_t$, Find the constant $a$ such that $Y_t$...
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A new stopping time built from a stopping time

Let $T$ be a stopping time for the filtration $(\mathcal{F_n})_{n \in \mathbb{N}}.$ For all $n \in \mathbb{N} \cup \left\{+\infty \right\},$ we set $\phi(n)=\inf\left\{k \in \mathbb{N};\left\{T=n \...
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32 views

Examples of martingales goes to $-\infty$

One example I can find is that set $X_{0}=0$, Let $X_{j}=\left\{\begin{array}{ll}{j^{2}} & {\text { with probability } \frac{1}{j^{2}}} \\ {\frac{-j^{2}}{j^{2}-1}} & {\text { with ...
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Gluing together 2-dimensional martingale measures to create n-dimensional martingale measure, Strassen's Theorem

Strassen's theorem states that a necessary and sufficient condition for existence of a discrete-time martingale with a finite number $n$ of given marginals $\mu_1,\ldots,\mu_n$ is that the marginals ...
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1answer
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Expectation property of martingales

Suppose $(Y_n)_{n=0}^\infty$ is a martingale of discrete random variables. Put $A:= \{(Y_0, \dots, Y_{n-1}) = (y_0, \dots, y_{n-1})\}$ In a proof I'm reading, it is claimed that $$\mathbb{E}[...
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How do I evaluate the following combination of random variables? Is it martingale?

I'm about to analyse the following expression $$Z_n:=\prod_{k=1}^n \left(\frac{\frac{Y_k}{\prod_{i=1}^k X_i}}{\sum_{j=1}^k \frac{Y_j}{\prod_{i=1}^j X_i}} \right),$$ where $Y_j$ for all $j\in \mathbb{...
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Put-Call Parity for a stock with dividends in Black-Scholes

I want to find a put-call parity relation for European options in the Black-Scholes model where the underlying stock pays a continuous dividend. I know that the relation will be $$S_0e^{-\delta T} - ...
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1answer
36 views

A stopped process is adapted

I am trying to understand the proof of Theorem 2.2.2(Optional Stopping Theorem) in Fleming and Harrington's Counting Processes and Survival Analysis. Let $\{X(t):0\leq t<\infty \}$ be a right-...
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Why are these two definitions of Martingales equivalent?

I am recently reading books on Probability theory, in Durrett's book Probability: Theory and Examples, the definition is following: If $X_{n}$ is sequence with (1) $\mathbb{E}\left|X_{n}\right|...
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Continuous-time Martingale and Brownian Motion Supremums

I am reading through Le Gall's book on Brownian Motion, Martingales, and Stochastic Calculus. I just read through the chapter on optional stopping of martingales, but I cannot solve the first exercise:...
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Bound of the Absolute Value of a Random Walk

Let $W(n)$ be a simple random walk on $\mathbb{N}$ with $W(0) = w_0$. That is, $\mathbb{P}\left[W\left(n+1\right) = W(n)+1\right)] = 1/2$ and $\mathbb{P}\left[W\left(n+1\right) = W(n)-1\right)] = 1/2$ ...
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stopping time almost surely finite

Let $(X_n)_n$ be a sequence of independent random variables and identically distributed such that $P_{X_1}=p\delta_1+q\delta_{-1}+r\delta_0$ where $0 \leq p,q,r<1$ and $p+q+r=1.$ Let $\alpha, \...
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2answers
50 views

Càdlàg Feller process is quasi-left-continuous

I've been working in Chung's "Lectures from Markov Processes to Brownian Motion", and I got stuck at Exercise 1 from 2.4. The objective of the problem is to give a short proof of the quasi-left-...
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1answer
76 views

If $\sum_{n=1}^\infty E(X_n-X_{n-1})^2/b_n^2 < \infty$, then $X_n/b_n \rightarrow 0$ a.s.

This is Durrett Exercise 5.4.9. I'm trying to show that if $X_n$ is a martingale, and $b_m \uparrow \infty$, $\sum_{n=1}^\infty E(X_n-X_{n-1})^2/b_n^2 < \infty$, then $X_n/b_n \rightarrow 0$ a.s....
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Definition of Martingale

I'm a little bit confused by the definition of a martingale. We say that a sequence of RV's is a martingale if: $E[X_n|X_{n-1}]=X_{n-1}$. I'm confused because it seems like the LHS should be just a ...
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32 views

Stopping theorem counterexample

I have been thinking about the conventional counterexample to stopping theorems where the stopping time is not bounded. For example, flip a fair coin infinitely many times, representing heads with 1 ...