A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Are these statements of my professor about periodicity of harmonic processes in time series analysis correct?

Assume $X_t$ is a harmonic stochastic process, i.e., $$X_t = \sum_{j=-k}^k A_j \exp(i \lambda_j t)$$ where the frequencies $\lambda_j$ are given and $A_j$ are uncorrelated random variables with zero ...
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14 views

Restarting a Markov chain

I'm reading an article and having difficulty understanding some basic stochastic processes (I'm new to the subject so please pardon my wording of the question). Let $S$ be a set of states and $Q$ be a ...
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1answer
13 views

Solutions to a stochastic birth-death-immigration process

A population is undergoing a birth-death-immigration process. That is, the population size can increase by virtue of birth and immigration, and can decrease by virtue of death. The birth rate is ...
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1answer
25 views

Decomposition of Poisson process

If $N(t)$ denotes the total number of visitors in the interval $[0,t]$. We suppose that $\lbrace N(t),t > 0 \rbrace $ is a Poisson process with rate $\lambda = 10$ per hour, and that we have $2$ ...
3
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1answer
236 views

Some basic questions about Stochastic Calculus

I have a transition function for a Markov process $X_t$. I want to find a density function for the stochastic process $Y_t := \int_0^t X_s \,ds$. Some questions about this: Is this the same as the ...
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2answers
61 views

Hazard function of $\min(X_1, X_2)$

Suppose I have two random variables, $X_1$ and $X_2$, that are independent (but not identically distributed) and assume both have hazard functions $\lambda_1(s)$ and $\lambda_2(s)$, for $s > 0$. ...
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57 views
+100

Poisson process and uniform random variable

Question: A single-pump petrol station is running low on petrol. The total volume of petrol remaining for sale is $100$ litres. Suppose cars arrive to the station according to a Poisson ...
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27 views

Problem with infinite product measures

Given some measurable space $\left(X,\mathcal{F}\right)$ and two probability measures $\mu$ and $\nu$ on this space one can define ...
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11 views

Conditional expectation of compound poisson process

The stochastic process $\lbrace Y(t),t > 0 \rbrace$ is a compound Poisson process defined by : $$Y(t)= \sum_{k=1}^{N(t)} X_{k} $$ and $Y(t) = 0$ if ($N(t) = 0)$ where $X_{k} $ has a geometric ...
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1answer
32 views

stochastic process with random variable as time, measurable

i have a problem with the following exercise: Let $X$ be an measurable $\mathbb{R}^d$ valued stochastic process on $(\Omega,\mathcal{F},\mathbb{P})$ and $T$ a finite $T : \Omega\to \big[ 0, +\infty ...
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1answer
5 views

What does the “empirical” autocovariance function represent?

My professor gave me the sequence ${X_n} = {1,5,5,1,5,5,...}$ and asked us to compute the empirical autocovariance function given below. $$\displaystyle \hat \rho(1) = \lim_{N \to \infty} \frac{1}{N} ...
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13 views

Is the autocovariance function of a sequence identically zero if the sequence is iid?

My professor gives the following definition for the autocovariance function. $$\rho(i,j) = Cov(X_i , X_j)$$$\\$If I have a sequence that is iid, when i compute $\rho(n,n+1)$ for $n \geq 0$, I found ...
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10 views

Clark-Ocone Formula

Why is the Clark-Ocone formula: $F = E[F] + \int_0^T E[D_t F | F_t] dW_t$ important, besides its applications to finance. That is, can you give examples of any important pieces of pure theory where ...
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47 views

i.i.d random variables

Looking at this question I. i. d distributions as best car offers I wonder about the following one: Can we find the distribution function of $X_N$ where, $$ N= \min \{ n \geq 1 \mid X_n > ...
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1answer
16 views

Showing the distribution of a poisson process

A large lump of radioactive material has a long half life. Let $D(t)$ be the total number of decays which occur in the radioactive material in the period of $t$ hours starting at noon on a particular ...
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40 views

Ito Integral of a SDE [on hold]

I would like to get help in solving the following Ito stochastic equation: $dY_t=-dW_t \, (Y_t^2+1)$ The process $W_t$ is the standard Brownian motion. Is it possible to get a path solution ...
2
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23 views

Covariance between brownian bridge and its max.

Does anyone know how to compute $\text{Cov}[\max_{s\in [0,1]}B(s), B(t)]$ where $B(t)$ is the standard Brownian bridge on the interval $[0,1]$?
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13 views

stationary probability distribution in markov chains [on hold]

What will be the stationary probability distribution of an absorbing state in a given markov chain?
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19 views

How to Maximize the Probabilty of Doubling Your Money!

This is an interesting questions I have heard mentioned a few times but don't know how to solve. Consider a geometric Brownian motion with some finite time horizon $T$ and a money market account with ...
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18 views

Stochastic PDE representation

I am trying to find a pde which $u$ satisfies when $u(x) = E^{x}[\cos(X_1)]$ where $dX_t = \sin(nX_t)\,dt + dW_t$ and $X_0 = x$. I have tried using Feynman-Kac but I can't seem to get it into the ...
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1answer
29 views

Why is chemotaxis considered an emergent behavior?

this is an applied math question. I could have posted this under a biological stackexchange, but the idea of emergent behavior or emergent properties of a system seems more appropriate to an applied ...
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2answers
296 views

birth and death processes

Suppose we have a system of N balls, each of which can be in one of two boxes. A ball in box I stays there for a random amount of time with exponential(lambda) distribution and then moves ...
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1answer
31 views

Convergence of Quantiles moments.

QUESTION: Let $F$ be an absolutely continuous distribution function whith density f, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence ...
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11 views

restarting a Markov chain

I'm reading an article and having difficulty understanding some basic stochastic processes (I'm new to the subject so please pardon my wording of the question). Let $S$ be a set of states and $Q$ be a ...
0
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1answer
14 views

Computing the PMF for N(t) in a renewal process

I'm in a stochastic processes course, and we just started on renewal theory. Unfortunately, we skipped the section on queuing theory, and nearly example in my textbook for renewal processes uses ...
2
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1answer
91 views
+50

Is there a standard proof for $\mathbb P(S^X_n\text{ hits }A\text{ before }B) >\mathbb P(S^Y_n\text{ hits }A\text{ before }B)$?

Let $X_i$ and $Y_i$ be two continuous random variables on $\mathbb{R}$ having distribution functions $F$ and $G$, respectively satisfying $G(y)>F(y)$ for all $y$. Let futhermore $S^X_n=\sum_{i=1}^n ...
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1answer
60 views

differential equations for continiuos markov processes

I'm trying to find the forward equations for birth-and-death processes with no birth, that is, when all $\lambda$ coefficients are zero. The forward equation for a birth-and-death process has the ...
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2answers
1k views

How to read transition probability matrix for Markov chain

Suppose that whether or not it rains today depends on previous weather conditions through the last two days. So if $RR$ (rained yesterday and today), then it will rain tomorrow with probability $0.7$. ...
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1answer
18 views

Branching processes extinction (homework)

This is my stochastic process course homework. I can solve (a)(b), which are easy to prove. But I have no idea about (c). Could you give me some idea?
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1answer
823 views

Quadratic variation of Brownian motion and almost-sure convergence

Say that $W(t)$ is a Brownian motion. The quadratic variation $[W,W](t)$ is defined in terms of a partition $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = t\}$ by $$ \begin{split} [W,W](t) &= ...
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22 views

Mean-value like result for stochastic integrals

I'm working through Protter's book on stochastic integration; this is problem 16 from chapter 2. I can't seem to crack it--maybe someone here can give me a hint? Let B be standard Brownian and H be a ...
2
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1answer
24 views

Is the sum of two compensated poisson processes always a martingale?

Let $M^{1}_t=N^{1}_t-t\lambda^{1}$ and $M^{2}_t=N^{2}_t-t\lambda^{2}$ be two compensated poisson processes, where $\lambda^{1}$ and $\lambda^{2}$ are the constant intensities of $N^{1}_t$ and ...
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131 views

Asymptotics of sum of binomial distributions

Definition 1: For any random variable $X$, we define $\mathrm{Bin}(p,X)$ as a variable with binomial distribution having parameters $p$ and $X$. Definition 2: For all $i \in \mathbb{N}$, define ...
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An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
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1answer
25 views

Computing joint probability [closed]

Let $X,Y\sim \text{Exp}(1)$ (exponential random variables with parameter $1$). Then prove that $$Pr(X> z_1, \frac{Y}{X} > z_2) = \dfrac{e^{-z_1 (1+z_2)}}{1+z_2}, \forall z_1,z_2>0$$
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459 views

Different versions of functional central limit theorem (aka Donsker theorem)?

I have seen several versions of functional central limit theorem (see the end of this post). I am confused, and hope someone could help to clarify their relations and differences. For example, I am ...
0
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1answer
22 views

Application of Ito's Lemma to integral expression

I have a problem applying Ito's lemma. I know that if: $dX_t= \mu_t \, dt + \sigma_t \, dB_t$ then for $f(t,x)$: $df(t,X_t) =\left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial ...
2
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2answers
178 views

Independent increments?

The questions are simple: Does the process $ X(t) = \int_0^t B(s)ds$ have independent increments? What about $X(t) = \int_{t-r}^{t}B(s)ds$? Here $B$ denotes the standard Brownian motion. ...
3
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2answers
290 views

Conditional distribution in Brownian motion

I need to prove the following: Let $X$ be a Brownian motion with drift $\mu$ and volatility $\sigma$. Pick three time points $s < u < t$. Then, the conditional distribution of $X_u$ given ...
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24 views

Probabilistic model of parallel web servers

Note: The following probabilistic model of parallel web servers is abstracted from an engineering project. I am not good at probability theory and I am seeking some evaluations and suggestions. ...
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Counting processes question

Let's say that arrivals at a counter come at times of a Poisson process with rate $\lambda$. A ball that arrives to an unlocked counter is registered and then locks the counter for an amount of time ...
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Ito's lemma on $a=\int_0^t x(q) \mathrm{d}B(q)$,where $B=$brownian motion process. [closed]

Can someone help me apply Ito's lemma on $a=\int_0^t x(q) \mathrm{d}B(q)$, where $B=$brownian motion process. I did this so far: $$\mathrm{d} a=\frac{\partial }{\partial B} \bigg[ \int_0^t x(q) ...
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14 views

Waiting time probability question

I want to solve the following problem: A dentist works 4 hours a day. Patients arrive on the average of 1 per 20 minutes and one patient spends on average 15 minutes with the dentist. Both time ...
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1answer
29 views

Poisson process and moment generating function

If we have a Poisson process $ \lbrace N(t), t > 0 \rbrace $ with rate $\lambda > 0 $ and if we have a random variable $S$ having a uniform distribution on the interval $[0,2]$. I was wondering ...
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1answer
16 views

Covariance function meaning

I have this sentence in a report but I don't quiet know what it means. I am familier with covariance and covariance matrices but not with covariance functions. $f(t)$ is a continuous-time ...
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36 views

Construct SDE with two uncorrelated Brownian motions

Using $Y(t) = wX_1(t) + \sqrt{1-w^2}X_2(t)$ as a model to construct a process where X1 and X2 are brownian motions with drifts and brownian increments $dX_1(t)= \mu_1dt + \sigma_1dW_1(t)$ ...
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9 views

Differentiability of random processes.

I know the appropriate criterions for mean-square differentiability of random processes. These criterions are connected with covariance function of a process. Are there any criterions for ...
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12 views

Does this Stochastic Differential Equation have a name?

I came across this SDE and since I am not an expert I am wondering if this SDE is known to have an closed form solution for first passage times. The SDE is $$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$ ...
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1answer
20 views

Solving the SDE $dX_t=bdt+cX_t dW_t$

I want to solve the SDE $dX_t=bdt+cX_t dW_t$, $X_0=0$ for $b,c\in\mathbb R$. I start by rewriting this as $$dX_t=(\mu_1+\mu_2 X_t )dt+(\sigma_1+\sigma_2 X_t )dW_t$$ where $\mu_1=b, \mu_2=0, ...
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20 views

Branching processes, extinction probability

Why do we assume that a branching process starts with one ancestor. What happens to extinction probability if we have more than one ancestor in generation Y(0)?