Questions tagged [martingales]
For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.
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Martingales and stopping rules [closed]
\textbf{Question:}
Let (X_1, X_2, \ldots, X_n) be a sequence of independent and identically distributed (iid) uniform variables on the interval ([-2, 2]), and define (S_n = X_1 + X_2 + \ldots + X_n). ...
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Convergence of a specific martingale
Let $(S_n)_{n\geq 1}$ be a martingale s.t. $\mathbb{E}[{S_n}^2]<K<\infty$ for all $n$. Assume that $Var(S_n)\to 0$ as $n\to\infty$. Now I would like to prove that $S_n$ converges almost surely ...
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$M$ is a continuous local martingale. Prove the consequences
$M$ is a continuous local martingale. Show that:
a) If $M_0=0$ and $\mathbb{E} (< M >) _t <\infty$ for $t\ge 0$ then $M \in \mathcal M^{2,c}$
b) There is a sequence of stopping moments $\...
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How can I check that the following process is a martingale?
Let $(\Omega, \mathcal{F}, (\mathcal{F}_t), \Bbb{P})$ be a filtered probability space. Net $N$ be a poisson process with parameter $\lambda>0$. Let $h$ be a bounded measurable function and define $...
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Simple random walk, Martingales, stopping time
Suppose $S_n = X_1 + \dots + X_n $ is a simple random walk starting at 0. For any K, let $$T = \min \{n: | S_n| =K\}. $$ $\bullet$ Explain why for every j, $$ \mathbb{P}\{T \leq j +K | T > j\} \geq ...
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P and Q martingales
Let $\mathbb{P},\mathbb{Q}$ be two equivalent probability measures. Could someone come up with an example of a $\mathbb{P}$-martingale, which is not a $\mathbb{Q}$-martingale and another example of a ...
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Is the following equation correct? [closed]
Let $N$ be a Poisson process and $t>s$, is the following equation correct?
\begin{align*}
&\mathbb E[N_s(N_t-N_s)|\mathcal F_s] = E[N_s|\mathcal F_s]E[(N_t-N_s)|\mathcal F_s]
\end{align*}
1
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2
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expected sum after rolling dice until getting 6 five times
We are rolling dice until we get 6 exactly five times. What is expected value of total sum? My approach: Lets denote $X_i$ - number rolled in $i$-th roll, and $S_n = \sum_{i=1}^{n}X_i$. So now i want ...
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Wald's identity proof
Let $Y_1, Y_2, \ldots$ be iid. random variables in $\mathcal{L}^1$, $\mu:=\mathbb{E} Y_i$, and $\tau \geq 1$ a stopping time (w.r.t. the natural filtration), $\mathbb{E} \tau<\infty$. Let $S_n=\...
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Extending Doob’s optional stopping time
Let $\tau \geq 0$ be a stopping time, $\mathbb{E} \tau<\infty$.
(b) Based on the identity
$$
\left|X_{\tau \wedge n}-X_0\right|=\left|\sum_{k=1}^n\left(X_k-X_{k-1}\right) \cdot \mathbf{1}\{\tau \...
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Optional stopping time of a simple symmetric random walk on the square lattice
Let $S_n$ be a simple symmetric random walk on the square lattice $\mathbb{Z}^2$ with $S_0=(0,0)$. That is, the walker starts from the origin and at each step independently, she steps one unit to East,...
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1
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Computing expected value of hitting time for a Feller process
We consider a Feller-Dynkin Markov process $X$ with generator $G$, which, when restricted to $C^2$ functions with compact support, is given by $Gf(x) = \frac{c(x)}{2}f''(x)$, where $c$ is a positive ...
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book recommendations for martingale theory (studying measure theory)
Any recommendations for someone studying martingale theory. My course recommends Probability with martingales - David Williams and Probability-2 Albert N. Shiryaev but I've found both to be very dry, ...
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Is continuous, uniformly integrable martingale indistinguishable from BMO martingale?
Definition:
BMO: Let $M$ be a martingale in $\mathcal{H}^2$. $M$ is said to be in BMO if there exists a constant c such that for any stopping time T we have
$$
E\{(M_\infty-M_{T_-})^2 \mid\mathcal{F}...
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Show that $Z(t)=\exp{\sigma X(t)-(\sigma \mu + \frac{1}{2} \sigma^2)t}$ is a martingale
I'm trying to show that $Z(t)=\exp{(\sigma X(t)-(\sigma \mu + \frac{1}{2} \sigma^2)t)}$ is a martingale.
Attempt:
I want to show that $E[Z(t)|\mathcal{F}(s)] = Z(s)$
$E[Z(t)|\mathcal{F}(s)] = E[Z(t)/ ...
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Conditional Expectation of random walk square $E^{F_n}X_{n+1}^2$
$X_n$ is simple random walk
It seems like $E^{F_n}[X_{n+1}^2]=X_n^2+1$
And $E^{F_n}[X_{n+1}^4]=X_n^4+6X_n^2+1$
I think the first expression is due to $X_n^2$ being positve
But how does it reach the ...
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Minimum submartingale inequality
I have a problem that goes like this:
Let $(X_n)_{n\geq 0}$ be a submartingale and the constant $\lambda>0$. Show that $\lambda\mathbb{P}(\min_{0\leq k\leq n}X_k<-\lambda)\leq\mathbb{E}[X_n]-\...
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Expected number of coin flips to get THTHTT
Suppose I am flipping a fair coin. What's the expected number of coin flips to get THTHTT?
I know how to do this using a Markov chain approach. However, I end up with a system of six linear equations. ...
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A positive supermartingale is closed on the right
I would like some help with proving the following statement:
Let $(X_t)_{t \in I}$ be a positive $(F_t)_{t \in I}$-supermartingale, that is, $E[X_t |F_s] \leq X_s$ for all $s \leq t$. Then $(X_t)_{t \...
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1
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Conditional expectation of functions of a random variable
The conditional expectation $\mathbb{E}(X|Y)$ of a random variable $X$ given a random variable $Y$ on a probability space $\Omega$ is understood heuristically as our best guess of $X$ given the ...
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High-probability tail bounds for a quantitative supermartingale inequality.
Let $(Y_t)_{t \in \mathbb{N}}$ be a non-negative stochastic process and $c \in (0,1)$.
Suppose that $Y_1 = 1$ and that for all $t \in \mathbb{N}$ it holds that
$$\mathbb{E}[Y_{t+1} \mid Y_1,\dots, Y_t]...
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Under what conditions, if any, is a continuous function of a Martingale still a martingale?
Consider a martingale $M_s$ and a function $f(M_s) = \frac{1-e^{-kM_s}}{M_s}$. If $M_s$ is an Ito process, it is obvious from Ito's Lemma that $f(M_s)$ is not a Martingale. Does the restriction hold ...
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Expectation and Variance of $Y_t = \int_0^t sdW_s$ + Martingale property
Denote the process $Y_t = \int_0^t sdW_s$. I want to answer the following question:
Q) Calculate the expectation and variance of $Y_T$. Is $Y_T$ a martingale?
A) I have read somewhere that it was ...
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Relation between measurability of intensity, compensator, and martingale in Doob-Meyer decomposition
Consider a sub-martingale (e.g., a counting process) $N_t$ adapted to a filtration $\mathcal{F}_t$. According to the Doob-Meyer decomposition, we have:
$$N_t = M_t + A_t,$$
where $M_t$ is a martingale ...
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Is a martingale difference in terms of some random variables $X_t$ also a martingale difference in terms of $f(X_t)$ for a measurable function f?
The context is that in a version of central limit theorem for martingale difference (Billingsley 1961), the condition is for some random variable $u_n$ $$\mathbb{E}[u_n|u_1,...,u_{n-1}]=0\tag{1}$$
...
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Proof that the stochastic exponential is a local martingale
I'm struggling to understand the proof as to why the stochastic exponential is a continuous non-negative local martingale. My notes say the following:
where 3.6 is $Z_t = 1 + \int^t_0 Z_s dX_s$.
I ...
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Martingale theorem representation using two martingales
Suposse we have two martingales $N$ and $M$, and the hypothesis of martingale representation theorem holds. Then
\begin{equation}
N_t=\int_0^t\psi_sdW_s, \,\, M_t=\int_0^t\phi_sdW_s
\end{equation}
So, ...
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The conditions for an independent but not IID exponential random variable to converge in distribution to 0
Would anyone be able to check if I've missed anything for my answer to the following question. The requirements I've noted so far is that:
For every bounded and continuous $f: \mathbb{R} \rightarrow \...
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1
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How to solve these equations? (from 10.12.c, D. Williams, Probability with Martingales)
In Sec. 10.12 (hitting times of simple random works), eq. (c) (page 103) of Probability with Martingales (D. Williams 1991), the author said:
For (any) $\theta\in \mathbb{R}$, let $\alpha:=\mathrm{...
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class of processes guaranteed to exceed any thresholds
Is it correct to say that any martingale (e.g., a GBM or anything similar), taking values in some real state space and not converging to a degenerate random variable, has the property that, given a ...
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Polya's urn and SSRW.
I've got a question on a martingale that i've been stuck on for a good while.
An urn contains $n$ white and $n$ black balls. We draw them one by one without replacement. We pay £1 for any black ball ...
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Maximal inequality for martingales
Let $X_i$ be measurable for the $\sigma$-field $F_i$. Suppose that for some constants $a_i, c_i \in \mathbb{R}$,
$$
\mathbb{E}\left(X_i-X_{i-1} \mid F_{i-1}\right)<a_i \quad \text { and } \quad\...
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Prove that gambling strategy with stopping time on seeing consecutive sequence of coins is a martingale.
recently I am trying to solve the following question: Let $\xi_{1}, \xi_{2}, ... $ be Bernoulli Random variable with $\mathbb{P}(\xi = 1) = \dfrac{1}{2}$ and $\mathbb{P}(\xi = 0) = \dfrac{1}{2}$. ...
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Prove that $L^2$ martingales with bounded increments that converge almost surely to a finite limit has converging quadratic variation
If $X_n$ is a sequence of $L^2$ martingales with bounded increments (i.e. $|X_n-X_{n-1}|<K$ for some $K>0$) such that $X_n$ converge almost surely to a finite limit, prove that the quadratic ...
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Wiener Process Textbook or Reference Specifically Containing Ramp Intersection (or Exit Time) Analysis
I am looking for a reference (preferably a textbook so that additional preparation material is handy) that calculates the exit time of a Wiener process from a region bounded by sloped lines.
Thank you,...
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Martingale in a modified gambler's ruin problem
Let $\{Y_t\}_{t \in \mathbb{N}}$ be a random walk on $\mathbb{Z}$ defined as follows:
$\mathbb{P}(x,x+1) =
\begin{cases}
p & \text{ if } x \geq 1\\
1-p & \text{ if } x \leq 0
\end{cases}$
$\...
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How do I go from $E[Z_{n+1}\mid {\mathcal {F}}_{n}] = Z_n,\; n \in \mathbb N$ to $E[Z_{n}|{\mathcal {F}}_{k}]=Z_k,\;k<n$ using the tower rule? [closed]
How do I go from $E[Z_{n+1}\mid {\mathcal {F}}_{n}] = Z_n,\; n \in \mathbb N$ to $E[Z_{n}|{\mathcal {F}}_{k}]=Z_k,\;k<n$ using the tower rule?
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Showing that a specific stochastic process is a martingale.
Let $ X = (X_n, n \in \Bbb N_0)$ be a stochastic process with state space $\Bbb N_0$, with unitary mean value for each $n \geqslant 1$, with independent increments and such that $P(X_0 = 0) = 1.$ I ...
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Convergence theorem for bounded continuous martingale in $L^2$
I would like to be sure to understand the proof this theorem.
Consider the set $\mathcal{M}^2$ of continuous martingale bounded in $L^2$. If $(X_t)_{t\geq0}\in\mathcal{M}^2$, there exists $X_{\infty}\...
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From maximal inequality in finite time to continuous time
In class we have seen a maximal inequality about discrete sub martingale. The setting was the following : Consider $T=\{0,1,..,N\}$ and $(X_t)_{t\in T}$ a sub martingale. Then we have for all $\lambda&...
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Show that $\mathbb E[X\mid Z_0,Z_1,\ldots,Z_t]$ is a martingale, where $\mathbb E[X]<\infty$ and $Z_t$ is a martingale.
Let $X$ be a random variable such that $E[|X| < \infty$, and let $\{Z_t: :t = 0,1,\ldots\}$ be a random sequence. We define the random sequence $\{X_t: t = 0,1,\ldots\}$ by $X_t = E[X\mid Z_0, Z_1, ...
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Gambler's Ruin and the expectation of a stopping time in the case the conditions of the Optional Stopping Theorem do not hold
In these lecture slides (pages 13-21) on optional stopping theorem (OST) and martingale I have found a great example of the gambler's ruin problem and what happens when the criterions of the OST are ...
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Polya's urn, should I use martingales or LLN
I am trying to prove the following question, but I am finding it a bit tricky to determine the distribution of $X_i$ (the number of red balls drawn in the $i$-th round) and thus I do not know which ...
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Proof of a maximal inequality in a finite interval
I consider the set $ I = \left\{ 0,…, N\right\}$ and $X$ a sub martingale on this set. I would like to prove the following inequality for $\lambda>0$
$$
\lambda\mathbb{P}(max(X_0,… X_N)\geq\lambda)\...
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Brownian motion is not of bounded variation
I would like to prove that Brownian motion, denoted $B_t$, is not of bounded variation using the fact that its quadratic variation is finite.
Here is my attempt:
Consider $[t,s]\subset[0,+\infty)$ and ...
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Almost Sure Convergence (Durrett 2.4.1)
Suppose the $j^{th}$ light bulb burns for an amount of time $X_j$ and
then remains burned out for time $Y_j$ before being replaced. Suppose
the $X_j$,$Y_j$ are positive and independent with the $X_j$’...
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Limit and Convergence of Doob Martingales
Let $(\Omega, \mathcal{F}, \mathcal{P})$ be a probability space for $X\in L^1(\mathcal{P})$ and $(\mathcal{F}_n)$ a filtration. Then $$Y_n = \mathbb{E}(X|\mathcal{F}_n)$$ is called Doob Martingale. ...
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Almost sure convergence in density of infinite dimensional product measure of normal distributions unsing Kakutani's theorem
In measure theory Kakutani's Theorem is used to determine if two infinite product measures are equivalent, meaning they are both absolutely continuous with respect to each other. In one of my text ...
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Convergence of urn model with repeated binomial draws
I'm analyzing a simple urn model, defined for $N>1$ an integer.
The setup is as follows: we have an urn with $\frac{N}{2}$ white balls and $\frac{N}{2}$ black balls.
At each step $k$, we sample $N$ ...
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About Martingale Stopping Theorem
Prove that:
A right-continuous process $X$ adapted to the filtration $\{\mathcal{F}_t\}$ is a martingale if and only if for any bounded stopping time $T$, $X_T \in L^1$ and $E[X_T] = E[X_0]$.
I know ...
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