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Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

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Probability of Seeing "X" % of Balls in "Y" Turns?

Here is a math problem I thought of: Set up: Suppose we have integers 1,2,3...99, 100 Each integer has an equal probability of being selected Game: In round=1, we pick 5 numbers randomly without ...
konofoso's user avatar
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Tightness under quenched and annealed law

If a random process in random environment $X_n$ is tight under the quenched measure, does it mean it is also tight under the annealed measure?
Mathick's user avatar
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On the equivalence in measurability of two different definitions of a playing strategy for a multi-armed bandit

Consider a $d$-armed bandit. At each time $t\in\mathbb{N}^+$ one of the arms is pulled and some reward is gained. For each arm $i=1,\ldots,d$ let $\pi_{i,t}$ be the number of times that arm $i$ has ...
Bart's user avatar
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Meaning of asymptotic in the context of stochastic process

So for a fractional Brownian motion, I define $k^{th}$ variation as $$ S_n = \frac{1}{n}\sum_{i = 1}^{n}{|B_{H}(i\times\frac{T}{n}) - B_{H}((i - 1)\times\frac{T}{n})|^k} $$ where T is fixed and $B_H$ ...
TJT's user avatar
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Markov chain question : Calculating lim[n→∞] P(X[n+1] = X[1])

i have the following matrix, and $P(X_0=i)$ i=[1,3] is also given: the question is to calculate $lim_{n\to \infty}$ $P(X_{n+1} = X_1)$ . what i did is: I've found the stationary distribution $\pi$...
CallMeDave's user avatar
1 vote
1 answer
32 views

Is a Gaussian process with covariance $C''(0)=0$ allowed?

Let $y(t)$ be a zero mean, homogeneous isotropic Gaussian process: $$ \left<y\right>=0 \qquad,\qquad \left<y(t_1)y(t_2)\right>=C(t_2-t_1) $$ where $\left< \dots \right>$ denotes ...
Sal's user avatar
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For $X_t$ a continuous-time Markov chain, show $M_t := \frac{f(X_t)}{f(X_0)}\exp\left(-\int_0^t \frac{Gf(X_s)}{f(X_s)} \; ds\right)$ is a martingale

Let $(X_t)_{t \geq 0}$ be a continuous-time Markov chain on finite state space $E$ with generator $G$ and let $f: E \to \mathbb{R}_{> 0}$ be a function. Then I would like to show that \begin{...
chessman's user avatar
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Covariance function of discrete filtered fractional Brownian motion

Consider a filter of such property: $$\sum_{j=0}^{\ell} j^r a_j = 0 \quad \text{and} \quad \sum_{j=0}^{\ell} j^p a_j \neq 0$$ and filtered standard fBm motion as such: $$V^a \left( \frac{i}{N} \right) ...
TJT's user avatar
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For any row stochastic matrix M, is $P_n=\frac{1}{n}\sum_{i=1}^n M^i$ always converging?

I knew that in general, $M^n$ is not converging, but if $M$ is an irreducible aperiodic stochastic matrix, than one has $$\lim_{n\infty} (M^n)_{i,j}=\pi_j$$ that is, each row of $M^n$ converge to the ...
Yujie Zhang's user avatar
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Delay in timetabling with supplements and exponentially distributed disturbances

I am looking at the following problem in operations research: Suppose that a train is operated over two identical consecutive trips, where on each trip the train incurs an exponentially distributed ...
Caliondo's user avatar
15 votes
4 answers
316 views

Optimal permutation of transition probabilities in random walk to minimize expected stopping time

Background Consider the set of integers $\{1,\dots,n+1\}$ and a set of probabilities $p_1,\dots, p_n \in(0,1)$. We now define a random walk/Markov chain on these states via the following transition ...
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Running Supremum of standard Brownian motion and probability distribution [closed]

I am reading Optimal Stopping and Free-Boundary Problems by Peskir and Shiryaev and noticed a result on page 151 as follows: $\mathbb{P} (\sup_{t \geq 0} (B_t - \alpha t) \geq \beta) = \exp(-2\alpha \...
Harry Wang's user avatar
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posterior covariance of a gaussian process

I'm currently studying gaussian processes. In this framework we build "stochastic" functions f for instance $\mathbb{R}^N\mapsto\mathbb{R}$. If I've got $M$ input $X_i\in\mathbb{R}^N$ ...
Oersted's user avatar
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Convergence of element of random vector if the random vector converges weakly [closed]

Let $\left\{Z_n\right\}_{n=1}^{\infty}$ be stochastics proces, where $Z_n=\left(Y_n,X_n\right)$ for each $n\in\mathbb{N}$ and $\left\{X_n\right\}_{n=1}^{\infty}$ and $\left\{Y_n\right\}_{n=1}^{\infty}$...
Waney's user avatar
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11 votes
2 answers
276 views
+50

How long will it take for a coin to repeat a certain behavior?

I came across the following question: Part 1: Suppose there is a coin that if heads, then the flip is heads with p=0.99. And if tails, the next flip is tails with p=0.99. If the first flip is random - ...
konofoso's user avatar
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1 answer
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Two processes with the same law, their supremum? [closed]

Suppose that there are two continuous-time processes $X = (X_t)$ and $Y = (Y_t)$ with $t \geq 0$, they have the same law under probability measure $P$. Could we say that $\sup_{0 \leq t \leq 1}(X_t)$ ...
Nightraidtown's user avatar
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Intuitive understanding of the definition of the $\sigma$-algebra of a stopping time $\tau$

I would like to better understand the basic intuition behind the definition of the $\sigma$-algebra of a stopping time $\tau$. Definition. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space ...
Quasar's user avatar
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1 vote
1 answer
70 views

Is there such a thing as the "Second Passage Time"?

I am learning about the First Passage Times (https://en.wikipedia.org/wiki/Hitting_time) of Stochastic Processes - how to derive the Probability Distribution for the time required for a Stochastic ...
konofoso's user avatar
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Let $\alpha_{ij}=\lim_{n\to\infty}\frac{1}{n}\sum_{m=1}^{n}~p_{ij}^m,~~i,j\in S.$ Show that above limit exist. [duplicate]

Consider an irreducible markov chain with finite state space $S$. let $P=[(p_{ij})]$ is given transition probability matrix and $p^n=[(p_{ij}]$ denote the $n-$ step transition matrix for the chain. ...
Ricci Ten's user avatar
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Factorial cumulants of stochastic process [closed]

I'm studying Van Kampen's "Stochastic processes in physics and chemistry" and stuck to some exercise(p.64): (Exercise problem image) A main point of the exercise is, given stochastic ...
Patche's user avatar
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4 votes
1 answer
54 views

Growth in sum-of-squares under random applications of $(a,b)\to (a+b/2,b-a/2)$ to $\{1\}^N$

A recent question considered the following problem: "Several (at least two) nonzero numbers are written on a board. One may erase any two numbers, say $a$ and $b$, and then write the numbers $a+\...
Semiclassical's user avatar
4 votes
1 answer
120 views

What is the mean of the stochastic differential equation $dX=K dt + \sigma X dW$ and how to find it?

I have the stochastic differential equation, $$dX=k dt + \sigma XdW,$$ which I expect to have just the mean $kt$, since taking the expectation of the SDE gives E[dX]=E[k dt] due to the brownian term ...
M. Z.'s user avatar
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+50

Transition probability density of non-intersecting Brownian bridges are independent of h function from Doob-h transform?

I've been trying to derive the transition probability of a $2$-dimensional Brownian motion $B = (B^{(1)}_{t}, B^{(2)}_{t})_{t \geq 0}$ conditioned to stay in the Weyl chamber and also conditioned to ...
tornt's user avatar
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Existence of multidimensional Gaussian processes

Given a positive semi-definite symmetric function $(\Gamma(s,t))_{s,t\in [0,1]}$ there exists a probability space and a centered Gaussian process $X$ with $\mathbb E[X_s X_t] = \Gamma(s,t)$. Is there ...
Stefan Perko's user avatar
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materials recommendation about stochastic process limit

I am trying to study the textbook Stochastic-Process Limits by Ward Whitt. I found the material a bit heavy. I would like to start with something easier. Is there any lecture videos or more concise ...
Jay's user avatar
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0 answers
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1 - Step Ahead Conditional Distribution

I started reading a paper, McNeil and Frey (2000), where they introduce the following: Let $(X_t)_{t \in \, \mathbb{Z}}$ be a strictly stationary time series (...). We assume that the dynamics of $X$ ...
Giordano Ribeiro's user avatar
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0 answers
52 views

Numerically solved PDE of Ornstein–Uhlenbeck process on 2-Simplex violates conservation of probability [closed]

Thanks for your consideration. I'm working to create a solution of an Ornstein-Uhlenbeck process with a force that takes mass towards the centre of a Simplex. I'm assuming absorbing boundaries. The ...
CRTmonitor's user avatar
3 votes
0 answers
78 views

One question about exponential martingale inequality at a paper of Ann. of Math.

I saw a strange inequality about martingales: If $\operatorname{M}\left(s,t\right)$ is an continuous $L^{2}$ martingale start at $s$, $\left[\operatorname{M}\right]\left(s,t\right)$ is its quadratic ...
shanlilinghuo's user avatar
1 vote
0 answers
19 views

Global minimum of exponential distributed data fitted to Nakagami-m distribution?

I tried finding a minimum such that $$ D_{KL}\left(f_\mathrm{E}\lvert\lvert f_\mathrm{N} \right) = \int\limits_{0}^{\infty} a\mathrm{e}^{-ax}\ln \left(\frac{a\mathrm{e}^{-ax}}{\frac{2}{\Gamma(m)}\...
Faylen Gaussling's user avatar
1 vote
0 answers
49 views

Survival probability Amoeba population

I have a question regarding the already covered Amoeba problem. Here is the set up: Every minute, an amoeba may die, stay the same, split into two or split into three with equal probability. All its ...
Anton2107's user avatar
1 vote
0 answers
42 views

Finding the law of a Brownian motion triple

Let $B=(B_t)_{t\geq 0}$ be a standard Brownian motion and let $S_{s,t} = \sup_{s \leq r \leq t} B_r$. It is well known from a classic result of Lévy that $S_{0,t} - B_t$ realises a Brownian motion ...
user82832's user avatar
1 vote
0 answers
69 views

Stochastic Simulation - Simulation from the Marginal Distributions

I am reviewing some material on MCMC / simulation and I realised I never quite understood this point. Given a joint distribution $f(x_1, ..., x_n) = f(x_1) f(x_2 | x_1)...f(x_n | x_{n-1}, ..., x_1)$ ...
InvestingScientist's user avatar
2 votes
1 answer
53 views

How to show Geometric Brownian motion is not a Gaussian process?

Let's consider Geometric Brownian motion: $$ X_t = e^{\mu t + \sigma B_t} $$ where $B_t$ is Brownian motion. Question: How to prove that this process is not Gaussian? I understand that $B_t$ itself is ...
poiug07's user avatar
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1 vote
0 answers
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Does weak convergence of a sequence of stochastic processes impliy convergence of their supremum?

Consider a sequence of stochastic processes $X_n$ on $L^\infty([0,1]^d)$ and assume that $\mathscr{L}(X_n)$ converges weakly to $\mathscr{L}(W)$, where $W$ is a Brownian bridge and $\mathscr{L}(\cdot)$...
Jack London's user avatar
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3 votes
1 answer
45 views

Levy Process is a Feller Processs

I want to show that every Levy Process is a Feller Process. Let $ X=(X_t)_{t\geq 0}$ be a Levy Process and $\mu_t(dy):=P\circ X^{-1}(dy)$ . I found a proof where it is shown that the transition ...
kays44's user avatar
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1 vote
0 answers
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Is there a notion of approximation of continuous-time Markov processes by finite-valued Markov processes?

Recall that in practice, to simulate a Brownian motion on $[0,1]$, we usually use the interpolated process $X^n=(X^n_t)_{t\in[0,1]}$ between the jumps of a random walk $(S_k)_{k=1,...,n}$ with $n$-...
Jeffrey Jao's user avatar
2 votes
1 answer
28 views

Is Feller semigroup in this definition strongly continuous?

Let $C_0=C_0(\mathbb R^d)$ be the space of continuous functions vanishing at infinity, equipped with the usual supremum norm $\| \cdot \|$. Let $T=(T_t)_{t\geq 0}$ be a semigroup of operators ...
Jeffrey Jao's user avatar
2 votes
1 answer
48 views

A cadlag Feller process for $\mathcal F$ is Markov w.r.t $\mathcal F_+$ (Th. 46, Chap. 1, Stochastic Integration - Protter)

In page 35 of the book Stochastic Integration by P. Protter, he defines a Feller process as follows: Then he states the following theorem. In the proof, he used the following strategy: Next, he ...
Jeffrey Jao's user avatar
2 votes
0 answers
28 views

Tail Probability of r.v. $X_t$ from a Langevin diffusion

Backgrounds: An overdamped Langevin dynamics on $\mathbb{R}$ is defined as the solution to the following SDE: $$ dX_t=-\nabla V(X_t)\,dt+\sqrt{2\beta^{-1}}\,dB_t,\qquad X_0=x_0. $$ If $V(x)=\frac{x^2}{...
Hirofumi Shiba's user avatar
1 vote
0 answers
31 views

Autocorrelation function of conditional expectation.

I am working on a problem involving stochastic processes and would appreciate some help and guidance. Given: $X(t)$ is a zero-mean wide-sense stationary Gaussian process with an autocorrelation ...
peterwei272's user avatar
1 vote
0 answers
27 views

Subsequence Process of Non-Markovian Stochastic Process

I have a problem that I haven't encountered before and would like to know if there is literature on the problem. Assume $X_t$ is a non-Markovian stochastic dynamical system and that $X_t \in S=\{1,2,...
E.S.'s user avatar
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0 votes
0 answers
18 views

Generating MA Process in matlab using gaussian nosie

I have an MA process that is: $x[n] = a * w[n] + b * w[n - 1] + c * w[n - 2]$ Where $a, b, c \in \mathbb{R}$ and $w \sim \mathcal{N}(0, \sigma^{2})$ How can I generate $x[n]$ in Matlab ? I know that ...
Daniel Cohen's user avatar
-1 votes
0 answers
78 views

Random walk with indipendent but not identically distributed increments

Suppose $\{Z_i\}_{i=1,2, \ldots}$ are normally distributed (independent but not identically distributed) random variables with mean $\mu(y)>0$ and positive variance $\sigma^2(y)$. Therefore we ...
ag_c1768918's user avatar
2 votes
1 answer
75 views

Definition of a Markov process

I found 2 Definitions for a Markov process and I am trying to understand how they are connected. Let $X=\left(X_t\right)_{t\geq 0}$ be a $\mathcal{F}_t$ adapted Process. We say $X$ is a Markov ...
kays44's user avatar
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0 answers
29 views

Bounded random walk joint distribution

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. normally distributed random variable with zero mean and variance $\sigma^2$. Consider the bounded random walk $(S_n)_{n\in\mathbb{N}}$, defined ...
Aguazz's user avatar
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1 vote
0 answers
44 views

Stochastic Integral and Ito's Isometry: Sharp-bracket process [M] or Angle-bracket process <M>?

I am learning stochastic integral, and I have noticed that the Ito's Isometry is sometimes stated using the quadratic variation process $[M]$ (e.g., pg. 47, Eq. (27.3), vol. 2 of Rogers & William),...
Mingzhou Liu's user avatar
1 vote
1 answer
56 views

Why is it harder to prove lower bounds on expected maxima (than upper bounds)?

I am currently learning upper and lower bounds for the expected maxima of random variables, e.g. bounds for the quantity: $$ \mathbb{E}\left[\max_{1 \leq i \leq n} X_i\right] $$ Most of the ...
rubikscube09's user avatar
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3 votes
1 answer
77 views

Show this truncated stochastic process is a martingale

Consider a filtered space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq 0}, P)$. Let $(M_t)$ be a uniformly integrable (UI) martinagle w.r.t. $\{\mathcal{F}_t\}_{t\geq 0}$. Let $\tau, \nu$ be two ...
Mingzhou Liu's user avatar
1 vote
0 answers
43 views

Optimal matching of Bernoulli random variables

Let $Z_1$, ..., $Z_n$ be a sequence of independent Bernoulli random variables such that for all $i\in\left\{1,..,n\right\}$ $Z_i\sim\mathcal{B}(p_i)$ where $p_i < 1/2$. Define $l(x_{1:n}, y_{1:n}) =...
Ibra's user avatar
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1 vote
0 answers
47 views

Where will a Non-Symmetric Random Walk be after time=t?

I posted this question about ranking the final position of a symmetric random walk after $t$ steps in terms of likelihood: What is the 2nd most likely value of a Random Walk after time=t? I am now ...
konofoso's user avatar
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