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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Expected time to failure

A machine needs two types of components in order to function. We have a stockpile of $n$ type-$1$ components and $m$ type-$2$ components. Type-$1$ components last for an exponential time with ...
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4th order correlations of a delta-correlated random process

Say I have a complex random variable A(z) that is $\delta$-correlated, i.e. I have: $ \begin{align}\langle A(z) \rangle &= 0 \\ \langle A(z) A^*(z') \rangle &= \delta(z-z') \end{align} $ ...
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20 views

When does an uncountable collection of random variables define a stochastic process?

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $( \mathcal{X}, \mathcal{B})$ be a measurable space. Let $\{X_t\}_{t\in [0,1]}$ be an uncountable collection of random variables such ...
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Brownian bridge sde

The SDE for the Brownian bridge is the following: $dX_t = \dfrac{b-X_t}{1-t}dt+dB_t$ with the solution $X_t = a(1-t)+bt+(1-t)\int_{0}^t \dfrac{dB_s}{1-s}$. The expectation and covariance are: ...
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30 views

Probability of the same order

Let's consider a set $A$ and another set $B$ where $B \subset T$ . Conside another set $C= T \backslash B$(exclude set B from T). Now, We are given a stochastic process $X(t)$ such that $P(X(t)_{t \in ...
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Prove that a succession of random variables is a martingale

I've been working on the following problem: Let $\{{Y_n:n\in \mathbb{N}}\}$ be independent identically distributed random variables with mean $\mu$ and variance $\sigma^2>0$. Define ...
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31 views

Independent Poisson process

Suppose that $\{N_1(t),t\geq0\}$ and $\{N_2(t),t\geq0\}$ are independent Poisson Process with rates $\lambda_1$ and $\lambda_2$. Show that $\{N_1(t)+N_2(t),t\geq0\}$ is a Poisson process with ...
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Expectation and Poisson process

Let {$N(t),t\geq0$} be a Poisson process with rate $\lambda$. Calculate $E[N(t).N(t+s)]$ I know that $N(t)\sim Poisson(\lambda t)$ and $N(t+s)\sim Poisson(\lambda(t+s))$ I can assume that ...
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32 views

Poisson Process proof that

For a Poisson process show, for $s<t$ that $$P(N(s)=k\mid N(t)=n)={n\choose k}\left(\frac{s}{t}\right)^k\left(1-\frac{s}{t}\right)^{n-k},\space > k=0,1,\dots,n$$ I tried a few things but ...
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Effective inter-arrival time converge to mean

I am fairly new to statistics and just recently encountered queueing theory. I have programmed a simulation for a $M/M/1$ queue in which I specify the inter-arrival times and service times. I input ...
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17 views

Quadratic Variation of Stochastic Integral of Simple Predictable process

Take $H\in S$ to be a simple process defined as: $$H_t:=\sum_{i=1}^{n-1} H_i1_{(T_i,T_{i+1}]}(t),\ \ H_i\in \mathcal{F}_{T_i}, \ (T_1\leq...\leq T_n \ stopping\ times),$$ and $X$ a Good Integrator. ...
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20 views

Probability of random walk visit in nonameanable graphs

Consider a vertex-transitive nonameanable graph. Consider a site $x$ having a graph distance $d$ from the origin and let $X(n)$ be a random walk starting from $x$. Is there a general upper bound as a ...
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30 views

Potential measure of the product of (independent $\alpha$-stable) subordinators

For a nondecreasing Levy process $\mathbf{X}$ with values in $[0,\infty)$ (i.e. a subordinator) Jean Bertoin defines the potential measure of $\mathbf{X}$ in his book "Levy processes" as follows (p. ...
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16 views

When is an Ornstein-Uhlenbeck process equivalent to Brownian motion up to a given time lag

Background The expected displacement under the assumption of Brownian Motion for time step $\tau$ is given by $$\gamma(\tau) = D|\tau|$$ where $D$ is the diffusion coefficient. If one assumes a ...
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Brownian motion needs to be defined continuous for every $\omega$ to be jointly measurable.

Let $B=(B_t)_{t\in[0,\infty)}$ a Brownian motion (BM) and $(\Omega,\mathcal{F},P)$ be the probability on which $B$ is defined. Some define BM as a.s. continuous, e.g., Brownian motion is almost surely ...
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50 views

Expectation and waiting time

There are three jobs that need to be processed, with the processing time of job $i$ being exponential with rate $\mu_i$. There are two processors available, so processing on two of the jobs can ...
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32 views

Failure time and exponential distribution

One hundred items are simultaneously put on a life test. Suppose the lifetimes of the individual items are independent exponential random variables with mean $200$ hours. The test will end when ...
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52 views

Expectation with exponential random variable

If $X_i$, $i=1,2,3$ are independent exponential random variable with rates $\lambda_i$, find $$E[\max(X_i) \mid X_1<X_2<X_3]$$ I really did not understand this exercise, because if ...
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22 views

Proving Ito's product rule

From Wikipedia the multidimensional Ito lemma is: If $\mathbf{X}_t = (X^1_t, X^2_t, \ldots, X^n_t)^T$ is a vector of Itō processes such that $d\mathbf{X}_t = \boldsymbol{\mu}_t\, dt + ...
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Cross variation

I have a question about the following argument. I see in my book a claim that given 2 stochastic integrals : \begin{align}X_1&:=\int_{0}^{t}f_s\mathsf dM_s\\ X_2&:=\int_{0}^{t}g_s\mathsf ...
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40 views

Probability with exponential random variable

Machine $1$ is currently working. Machine $2$ will be put in use at time $t$ from now. If the lifetime of machine $i$ is exponential with rate $\lambda_i=1,2$, what is the probability that ...
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27 views

Showing a process satisfies an SDE

The example of Ito and Watanabe in the following notes http://www.stat.uchicago.edu/~lalley/Courses/391/Lecture12.pdf is an SDE without unique solutions. $$dX_t = 3X_t^{1/3} dt + 3X_t^{2/3} dW_t$$ ...
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DRIFT MATRIX in Ornstein Uhlenbeck Process

The Weiner Process was unable to explain Brownian Motion and then there was the need of Ornstein-Uhlenbeck Process. The Ornstein-Uhlenbeck Process describes the Brownian Motion in the presence of ...
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Stochastic Integral of Simple Predictable Process is a Martingale

Take $H\in S$ to be a simple process defined as: $$H_t:=\sum_{i=1}^{n-1} H_i1_{(T_i,T_{i+1}]}(t),\ \ H_i\in \mathcal{F}_{T_i}, \ (T_1\leq...\leq T_n \ stopping\ times),$$ and $X$ a Martingale. I ...
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30 views

Is $E[E[Y_tZ_t|Y_t]|\mathcal{F}_{t-1}] = E[Y_tZ_t|\mathcal{F}_{t-1}]$ where $\mathcal{F}_t$ is the natural filtration process

As the questions is stated in the topic. Let $Y_t$ and $Z_t$ be discrete time-dependent random variables, and $\mathcal{F}_t$ is the natural filtration such that we know everything that has happened ...
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Weakening mean integral requirements of stochastic integrals.

Consider the Ito integral. It is well known that adapted and measurable processes $f(s,\omega)$ that satisfy \begin{align} E \Big[ \int_0^T \big| f(s,\cdot) \big|^2 ds \Big] < \infty \end{align} ...
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Continuous dependence on an initial condition (SDE)

Let's say I have a (one-dimensional) diffusion process $$dX=\mu(X_t)dt+\sigma(X_t)dW.$$ Assume we have fixed $\epsilon > 0$ and $t >0$ Under what conditions is $\mathbb{P}^x(X_t < ...
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104 views

Exponential distribution and poisson process

Consider a post office with two clerks. Three people, A, B, and C, enter simultaneously. A and B go directly to the clerks, and C waits until either A or B leaves before he begins service. What ...
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Name for a constrained Poisson-like bridge process

I have a sequence $t_i$ for $i=0,2,\cdots,n$ of integer jump times with $t_0=0$ and $t_n=n$ such that the waiting time $t_{i+1}-t_i$ has distribution density $f_i(t)$. So it's kind of like a Poisson ...
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34 views

Probability that a cow gets hit if cars that travel along a certain road follow a Poisson Process

I solved the following problem and just want to make sure my answer is correct. Cars that travel along a certain road follow a Poisson Process with rate 9 cars/min. If a cow takes 12 seconds to ...
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Density of subset with nonlocal boundary condition

I am having difficulty proving that $E=\bigcap_{n\geq 0} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a dense subset of: $F=\{f\in C^2 (\mathbb{R}) : ...
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51 views

Stochastic Exponential - Protter

I am trying to understand the proof of Theorem 37 at page 84 of the book Stochastic Integration and Differential Equations by P. Protter. In the proof there is the following statement, referred to ...
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23 views

A property of the expected value of a Poisson Process

Let X(t) be a Poisson Process and $f\geq0$. Show that $E[e^{-\sum_{n=1}^{\infty}f(W_n)}]=e^{-\lambda \int_0^\infty(1-e^{-f(t)})dt}$ ($W_n$ is the time of the nth ocurrence in the Poisson Process) ...
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Distribution of stochastic integral w.r. to brownian motion

Let $B=(B_t)_{t \geq 0}$ be a standard brownian motion, $T > 0$ and $f : [0,T] \rightarrow \mathbb{R}$ a continuous function. I want to determine the distribution of the following integral: ...
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48 views

Quadratic Variation of Quadratic Variation

Consider a good integrator $X$ (semi-martingale) and the relative quadratic variation process indicated by: $Y_t:=[X,X]_t$. Why is that: $$[Y,Y]_t=0 \ \ \ \ \ and \ \ \ \ \ \ [X,Y]_t=0 \ \ ?$$ ...
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29 views

Generating cross-correlated stochastic processes

I am looking for a robust way to represent and generate multiple stochastic processes that contain time and cross-correlations i.e. I am looking at stochastic processes $X_t^{1}$, $X_t^{2}$, $\ldots$, ...
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20 views

Continuous chain of correlated Brownian motions

Let $(X_{s,t})_{s\ge 0\,,t\ge 0}$ be a stochastic process such that for fixed $t=t_0$, $(X_{s,t_0})_{s\ge 0}$ forms a standard Brownian motion and for fixed $s=s_0$, $(X_{s_0,t})_{t\ge 0}$ is a ...
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27 views

Compute the expected value of the time of the 6th ocurrence in a Poisson Process

I have the following problem: Compute $E[W_6|X(4)=5]$, where X(t) is a Poisson Process and $W_n$ the time of the nth ocurrence. I know that $E[W_6|X(4)=5]=\int_4^\infty tf_{W_6|B}(t|B) dt$, where ...
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A 2-variable stochastic difference equation exhibiting 2 stable orbits with switching?

I have some social science data to which I would like to fit a stochastic difference equation in two variables. I will describe the dynamics of the system that I have observed. I am hoping someone ...
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Doubly stochastic matrix proof

A transition matrix $P$ is said to be doubly stochastic if the sum over each column equals one, that is $\sum_i P_{ij}=1\space\forall i$. If such a chain is irreducible and aperiodic and ...
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Proof that state can be reached

Prove that if the number of states in a Markov Chain is $M$, and if state $j$ can be reached from state $i$, then it can be reached in $M$ steps or less. To me it just seems the definition of ...
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Brownian Motion is almost surely unbounded, and a proof for the discrete Random Walk

If $B_t$ is a Brownian Motion, why we have $$ P(\liminf_{t\to\infty} B_t = -\infty) = 1 $$ and $P(\limsup_{t\to\infty} B_t = +\infty) = 1$? I guess for the following "discrete version" of ...
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Poisson Point Process: conditional probability

Assuming I have a homogeneous spatial PPP of intensity $\lambda$. I'm looking for the probability $\mathbb{P}( n$ points in $A \vert $ at least $k$ points in $A )=\mathcal{P}$ My approach looks ...
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86 views

How to prove a set is a core for an infinitesimal generator

I am trying to prove that the set $D=\bigcap_{n\geq 1} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a core for the infinitesimal generator of reflecting brownian ...
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Properties of Stochastic Differential Equations

Suppose I have an SDE of the form: $$dx_i = x_if(x_1,\cdots,x_n) + \sigma_ix_idW_t $$ By defining $y_i = \log x_i$, I can change the SDE to: $$dy_i = y_i g(y_1,\cdots,y_n) + \sigma_idW_t $$ Both ...
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23 views

Empirical Quantilfunction as Integral Bound

This is my first post, so please be nice ;) I'll try to outline my problem correctly and whilst keep it as short as possible! I have to deal (for my master thesis) with the integral ...
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17 views

Interpreting Ventzell Boundary conditions

I am trying to understand the article "On boundary conditions for multidimensional Diffusion processes" of A. D. Ventzell (or Wentzell). I copy the images for greater convenience: In the ...
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28 views

If $P^r$ has all positive entries, then so does $P^n$

Let $P$ be the transition probability matrix of a Markov Chain. Argue that it for some positive integer r, $P^r$ has all positive entries, then so does $P^n$, for all integers $n\geq r$ I ...
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limit of gaussian process

If I have a sequence of gaussian random process $X_{t}^{n}$ which converge in $L^2$ norm to a process $X_t$ for every $t$. can I say that $X_t$ is also gaussian process? thank you
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Proof of Itō's lemma for the Brownian motion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...