Questions tagged [martingales]
For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.
357
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Prove $\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$
Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$
tends to infinity. I now want to prove that for every $x>0$
$$ P\left( \sup_{t \geq 0 } M_t > x \mid \mathcal{F}_0 ...
user155670
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3
answers
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Criteria for being a true martingale
Could you kindly list here all the criteria you know which guarantee that a continuous local martingale is in fact a true martingale? Which of these are valid for a general local martingale (non ...
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3
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Proving Galmarino's Test
Galmarino's Test gives a condition equivalent to being a stopping time. It says:
Let $X$ be a continuous stochastic process with index set $\mathbb{R}_+$ (i.e. each sample path is a continuous ...
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Expectation stopped Brownian motion with drift
Let $\{X_t:t\geq 0\}$ be a Brownian motion with drift $\mu>0$ and define a stopping time $\tau$ by $$\tau=\inf\{t\geq 0:X_t=a\}.$$ Now I want to show that $$\mathbb{E}(e^{-\lambda\tau})=e^{(\mu-\...
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Sum and product of Martingale processes
Given two Martingale processes $(X_t)$ and $(Y_t)$, are their sum $(X_t+Y_t)$ and their product $(X_t \times Y_t)$ also Martingale?
If not, will the two $(X_t)$ and $(Y_t)$ being independent grant ...
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2
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The Laplace transform of the first hitting time of Brownian motion
Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is
$$\mathbb{E}[\exp(-\...
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Prove a.s. convergence of $(X_n)_n$ satisfying $E(X_{n+1} \mid F_n) \leq X_n+Y_n$ for $\sum_n Y_n<\infty$
I am have a problem with proving a convergence of a sequence of random variables in the given context:
Let $F_n$ be a filtration. Assume that $X_n$ and $Y_n$ are non-negative and integrable $F_n$-...
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Submartingale convergence (Durrett 5.3.1)
Exercise 5.3.1 in Durrett's "Probability Theory and Examples" states
Let $X_n$, $n\ge 0$, be a submartingale with $\sup X_n < \infty$. Let $\xi_n=X_n-X_{n-1}$, and suppose $E(\sup \xi_n^+)<\...
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Proof of identity about generalized binomial sequences.
I was going through this old question about a wealthy gambler:
Gambler with infinite bankroll reaching his target. The answer relies on the following identities from Concrete Mathematics by Graham, ...
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2
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Prove X is a martingale
Prove $X = (X_n)_{n \geq 0}$ is a martingale w/rt $\mathscr{F}$ where X is given by:
$X_0 = 1$
and for $n \geq 1$
$X_{n+1} = 2X_n$ w/ prob 1/2
$X_{n+1} = 0$ w/ prob 1/2
and $\mathscr{F_n} = \...
- 12.7k
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How to show the following process is a local martingale but not a martingale?
I am confronted with the same problem in this thread, which hasn't had a complete answer yet.
In the sequel, we denote by $Y^T$ the stopped process $Y^T_t = Y_{T \wedge t}$. Consider the following ...
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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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Probability that a sequence of random variables converges to 0 or 1
Fix $\alpha , \beta > 0$ with $\alpha + \beta = 1$ and consider the sequence of random variables defined by $X_0 = \theta \in (0,1)$ and
$$P(X_n = \alpha + \beta X_{n-1} \mid X_{n-1} , ... , X_0) =...
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Why is stopping time defined as a random variable?
I've been given a crash course in stochastic processes and martingales for the purposes of a semester project on them. The guy I'm working with has been, I feel, a little vague in the definition of ...
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Bounded (from below) continuous local martingale is a supermartingale
Suppose $M(t)$ is a continuous local martingale. That is, there exists a sequence of stopping times $T_n$ which almost surely increase to $\infty$, and such that $M(t\wedge T_n)$ is a martingale for ...
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Can you make money on coin tosses when the odds are against you?
The strategy
Given an initial investment $n$ dollars and a "bet buffer" $b$.
Calculate the bet size $x=\left\lfloor\frac{n}{2^b-1}\right\rfloor$ dollars.
Wager $x$ dollars on random variable $C$ ...
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Relation between Hermite polynomials and Brownian motion (on martingale property) [duplicate]
Let us define Hermite polynomials as $H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n} e^{-x^2/2}.$ One can prove that $e^{\theta x-\frac{1}{2} \theta^2}=\sum_{n=0}^{\infty} \frac{1}{n!}H_n(x)\theta^n \quad (*)...
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If $(X_n)_{n\in \mathbb{N}}$ is a martingale s.t. $\sup_n E[|X_n|]\leq M < \infty$, then $\sum_{n\geq 2}(X_n-X_{n-1})^2<\infty$ almost surely.
I need to prove the above statement. The hint provided was to consider the stopping time $T_l=\inf\{n\in \mathbb{N}||X_n(w)|\geq l\}$ where $l\in \mathbb{N}$ and then show that $E[\sum_{n=2}^K(X_n-X_{...
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Asymmetric Random Walk / Prove that $T:= \inf\{n: X_n = b\}$ is a $\{\mathscr F_n\}_{n \in \mathbb N}$-stopping time
Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \...
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Why did my friend lose all his money?
Not sure if this is a question for math.se or stats.se, but here we go:
Our MUD (Multi-User-Dungeon, a sort of textbased world of warcraft) has a casino where players can play a simple roulette.
My ...
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2
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What are some easier books for studying martingale?
What are some easier books for studying martingale?
They are defined to be comprehensive but easier than Roger and William's martingale book. For example, to study Q and F martingales?
It should ...
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2
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Proof that a stopped continuous-time martingale is a martingale.
The proof for a stopped discrete-time martingale is shown as follows.
Let $M=(M_n)_{n\ge0}$ be a discrete-time martinglae w.r.t. the filtration $(\mathcal F_n)_{n\ge0}$, and let $M^T=(M_{n\land T})...
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Supermartingale with constant Expectation is a martingale
In my lecture notes they use the fact, that every supermartingale $(M_t)$ for which the map $t\mapsto E[M_t]$ is constant is already a martingale. Unfortunately I can't prove it. Some help would be ...
user20869
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votes
1
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Monkey typing ABRACADABRA and gamblers
Problem: A monkey is sitting at a typewriter, typing a letter (A-Z) independently and with uniform distribution each minute. What is the expected amount of time that passes before ABRACADABRA is ...
- 1,967
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1
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Does law of large numbers converge in $L^1$?
I've seen the law of large numbers stated mainly in two (or three) forms: $S_n/n$ converges in probability (weak law) and converges almost surely (strong law). Also, there is convergence in the $L^2$-...
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Martingale not uniformly integrable
I've come across a statement that implies that non-negative martingales for which $\{M_{\tau}\mid \tau \ \rm{stopping} \ \rm{time}\}$ is not uniformly integrable exist. I personally can't think of an ...
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Martingale oscillating between three values
I'm self-studying Martingales. I came accross the following exercise (exercise 4.3.1.) in Durrett's Probability Theory and Examples (5th Edition).
Exercise. Give an example of a martingale $X_n$ ...
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Example: satisfying $E(X_{n+1}\mid X_n)=X_n$ but not a martingale
I am wondering if there is such a sequence of random variables $(X_n)_{n=0}^\infty$ such that $\mathbb{E}(X_{n+1}\mid X_n)=X_n$ for all $n\geq0$ but which is not a martingale with respect to the ...
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Rigorous Book on Stochastic Calculus
I have already taken a couse in Stochastic Calculus.
Due to time constraints on many ocassions we had to skip some formalities among the proofs.
I'm trying now to fill the gaps left, and I have been ...
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$L^p$ submartingale convergence theorem
In class today, we learned about the familiar $L^p$ martingale convergence theorem: For $p >1,$ if $X_n$ is a martingale with $\sup \mathbb{E}|X_n|^p <\infty$, then $X_n \rightarrow X$ a.s. and ...
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Distribution at First Time a Sum Reaches a Threshold
Consider the following problem.
Roll a die many times, and stop when the total exceeds $M$, for some prescribed threshold $M$.
Call this time $\tau$, and call the running score after $n$ rolls $X_n$.
...
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1
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Chebyshev Inequality for Martingales
Suppose $\{X_n\}_{n \geq 1}$ is a square-integrable martingale with $E(X_1)=0$. Then for $c>0$:
$$P\left(\max_{i=1, \ldots, n} X_i \geq c\right) \leq \frac{\textrm{Var}(X_n)}{\textrm{Var}(X_n) + c^...
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Expectation of $TS_T$ where $T$ is the absorption time at $\{a,-a\}$ of a simple symmetric random walk $\{S_n\}$
I was trying to calculate the expectation of $T^2$ using some martingale and got that I needed the expectation of $TS_T$. Any idea?
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For a martingale $X$ does uniform integrability imply integrability of $\sup |X_{n}|$?
All is in the title: if $(X_{n})$ is a uniformly integrable martingale is it true that $\sup_{n\in \mathbb{N}} |X_{n}|$ is an integrable variable ?
If I had to take a guess I'd say the answer is no, ...
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Prove $A_t := W_t^3-3t W_t$ a martingale
Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$
Let $(A_t)_{t \geq 0}$ where $A_t = W_t^3 - 3tW_t$. ...
- 12.7k
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How do I prove that a martingale has a constant expected value?
I can´t prove that a martingale has constant expected value.
$$
\mathbf{E}[M_t]=\mathbf{E}[M_0]
$$
Thanks people.
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Easy proof of Black-Scholes option pricing formula
I use this Book to read the option pricing in Black-Scholes model in pages 93-99, The proof of the formula given by
$$c(s,t)= N(d_1(s,t)- Ke^{-rT}N(d_2(s,t)))$$
where
$$d_{1,2}=\frac{\ln(s/K)+(r\pm \...
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votes
1
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NON-martingale approach to ABRACADABRA problem
The well-known ABRACADABRA problem states (see D. Williams, "Probability with martingales", for example):
a monkey is typing letters A-Z randomly and independently of each
other, each letter ...
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votes
1
answer
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Probability on first hitting time of Brownian motion with drift
I am struggling with the following problem:
Let $B$ be a one dimensional Brownian motion and $a,b>0$. Show that $$P[B_t=a + bt \text{ for some } t\geq 0] = e^{-2ab}.$$
The following hint is ...
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What are some martingales for asymmetric random walks?
Here are some examples for symmetric ones:
https://mathoverflow.net/questions/55092/martingales-in-both-discrete-and-continuous-setting/55101#55101
Is there a similar list for asymmmetric random ...
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3
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Prove this is a Martingale
Prove that
$$Z_t:=\frac{e^{W_t^2/(1+2t)}}{\sqrt{1+2t}}$$
is a $\mathscr{F}_t$-martingale.
I have tried all the usual manipulations without any success.
The only useful fact I think should be used is ...
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1
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1
answer
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On "for all" in if and only if statements in probability theory and stochastic calculus
1 In my friend's Probability Theory long test there was this question:
Let $(\Omega, \mathfrak{F}, P)$ be a probability space on which is
defined all sub-$\sigma$-algebras, events and random ...
- 12.7k
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votes
1
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Prove a thm on stopped processes given fundamental principle 'you can't beat the system'?
How does the principle below imply the thm below?
From Williams' Probability w/ Martingales:
Principle:
Thm:
What I tried:
$$E[X_{T \wedge n} - X_0 | \mathscr{F_m}] =/ \le X_{T \wedge m} - X_0 \ \...
- 12.7k
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votes
4
answers
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Example of filtration in probability theory
I'm studying Martingales and before them filtrations. Given a probability space $(\Omega, F, P)$ I define a filter $(F_n)$ as a increasing sequence of $\sigma$-algebras of $F$, such that $F_t \subset ...
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Convergence in distribution of conditional expectations
I was just reading this question, which is about how the classical central limit theorem can be interpreted as giving a rate of convergence for the law of large numbers for iid random variables. I was ...
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1
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How to get closed form solutions to stopped martingale problems?
Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It ...
user76844
11
votes
1
answer
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Random walk as a martingale?
Let $S_0$, $Z_1$, $Z_2$, $\ldots$ be independent random variables. $S_n=S_0+Z_1+\cdots+Z_n$, $n=0,1,2,\ldots$
$S_n$ is a random walk starting in a random point, $S_0$ I need to find out, when it is a ...
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1
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How to prove the Lebesgue density theorem using martingales?
The Lebesgue density theorem says that for almost every $x \in A \subset [0,1]$ with $A$ Lebesgue measurable
$$\lim_{h \to 0^+} \frac{|A \cap (x-h, x+h) |}{2h}=1 \tag{1}.$$
Here, "almost every" is ...
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1
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Some version of Itô isometry with conditional expectations
Let $B = (B_t)_{t \geq 0}$ be a Brownian motion, $ \mathcal{F}=
(\mathcal{F}_t)_{t \geq 0}$ the natural filtration associated to $B$, $u \in L^2_{a,T}$ (that is, $u$ is an stochastic process $u = (...
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1
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Martingale preservation under independent enlargement of filtration
I think this is probably a very easy question but I haven't worked with $\sigma$-algebras in depth for a long time now so am finding myself a little rusty. Would be very grateful if someone could give ...
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