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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0answers
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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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1answer
23 views
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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0answers
23 views
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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0answers
22 views
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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1answer
18 views
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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1answer
26 views
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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1answer
64 views
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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1answer
38 views
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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0answers
34 views
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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2answers
43 views
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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1answer
31 views
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:
$$
\mathbb{}E\left(M_t\int_s^tZ_u \, du \...
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22 views
Uniqueness of Solution to Stochastic Integral Equation
Suppose that $N$ is an $(\mathcal{F}_{t})$-continuous local martingale, with $N_{0}=1$, $N_{t}\gt0$ a.s. for $t\geq0$ and $N$ satisfies:
$$ N_{t}=1+\lambda\int_{0}^{t} N_{s}dB_{s} $$
Applying Ito's ...
2
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1answer
28 views
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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16 views
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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0answers
20 views
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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2answers
49 views
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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0answers
44 views
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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0answers
26 views
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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1answer
11 views
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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0answers
12 views
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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0answers
16 views
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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1answer
16 views
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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1answer
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$ ...
1
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0answers
15 views
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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1answer
41 views
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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1answer
52 views
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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1answer
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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1answer
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,
$$
for all continuous, bounded functions $h$, ...
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0answers
22 views
Density of Maximum brownian motion
Why is it that for $M(t):=\max\limits_{0\leq s\leq t}B(s)$
$$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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1answer
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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1answer
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\...
2
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1answer
34 views
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 ...
2
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1answer
41 views
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 ...
1
vote
1answer
18 views
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$...
4
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1answer
49 views
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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1answer
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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0answers
21 views
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 ...
2
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1answer
22 views
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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3answers
74 views
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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0answers
12 views
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} - ...
1
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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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2answers
25 views
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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0answers
25 views
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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1answer
16 views
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 $\mathbb{P}\left[W\left(n+1\right) = W(n)-1\right)] = 1/2$ ...
1
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0answers
27 views
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, \...
1
vote
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-...
3
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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....
3
votes
0answers
37 views
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 ...
0
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1answer
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 ...
