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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Ito integral for simple stochastic process

I need for $l=1,2......$ prove that $E[W^{2l} (t)]=$ $\frac{(2l)!}{2^l l!}$ and $E[W^{2l+1} (t)]=0$ I know that Ito integral for simple stochastic process satisfies $E[I^2 (t)]=E\int_0^t\Delta^2(u)...
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martingale square integrable

Let $X_t=\int_0^te^{W_s}dW_s$ and $Y_t=\int_0^tW_sdX_s$. How to show that $X$ and $Y$ are martingale square integrable? ($W_t$ - Wiener) It it enough to show that $\mathbb{E}X_t^2<\infty$, $\...
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Can we model this set of experiments as an stochastic process and estimate the sample size?

I have an image with the size 5575x9440 and I'm implementing a modified version of the algorithm used in this paper on it, but because the code performance is low ...
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28 views

Compound poisson process invariant measure

Let $\rho$ be a probability measure in $\mathbb{R}$, $(N_t)$ a standar Poisson process and $(X_i) \stackrel{\text{i.i.d.}}{\sim} \rho$. Then $$Z_t = \sum_{n=1}^{N_t} X_n $$ is call a compound poisson ...
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conditions in which the repair shop process is recurrent (null\positive) or transient

here's the Story: Let $\epsilon_1.\epsilon_2,... $ be i.i.d numbers of machines for repair to the repair shop on mornings of days $1, 2,...$ . Assume that the shop is capable of repairing exactly K ...
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21 views

Conditions for limiting distribution to equal stationary distribution of SDE

I have SDE of the form $$dX_t=a\mathopen{}\left(X_t\right)dt+b\mathopen{}\left(X_t\right)dW_t,$$ where $W$ is Brownian motion. If the stationary distribution of $X$ exist is it equal to the limiting ...
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Show that $\hat{Y}$ is an optimal linear estimator of Y

Relevant Information. Let $X(t)$, $t \in T$ be a second order process. Let $M_0$ be the set of random variables of the form $a + b_1X(s_1)+ \cdots + b_nX(s_n)$ for a positive integer $n$ and constants ...
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$\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}$

Let $U$ be uniform distributed in $[0,1]$ . Show that with probability $1$ there's maximum a finite amount of $n \in \mathbb N$, so that the inequality $\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}...
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First Passage Problem, Looking for General Method

I'm trying to find a general method for solving problems like the following: Flip a fair coin repeatedly, subtracting 1 if heads and multiplying by 2 if tails. If you currently have X, what is the ...
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55 views

Quadratic covariation of two Itô processes

If $dX(t)=\Delta_x (t) + \ominus_x(t) dt$ and $dY(t)=\Delta_Y(t) dW(t)+ \ominus_Y(t) dt$, where $X(t), Y(t)$ are two Ito processes. I need show that $d[X,Y](t)=\Delta_x(t)\Delta_Y(t)dt$, where $\...
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How to compute the integral of a renewal process?

Let $\{S_n:n=1,2,\ldots\}$ be a renewal process (with the convention $S_0\equiv 0$) with $\mathbb E[S_1]<\infty$ and $S_1$ absolutely continuous with density $f$. Let $\{N(t):t\geqslant0\}$ be the ...
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22 views

Finding marginal distributions of a process

Suppose we have a process given by $S_t = S_0 \exp(\sigma W_t + (r - \frac{1}{2} \sigma^2 )t)$, and we wish to find the marginal distribution for $S_T$. (Note: $W_t$ is a $\mathbb{Q}$-Brownian Motion) ...
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I can not deduce this formula from an article…

I do not understand the calculation of a certain article published in J. Chem. Phys. 137 (2012): I have a continous time discrete state model with a total number of for example $N_0$ ...
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130 views

When is the compensated Poisson random measure a martingale ? (extensions to sets not bounded from 0)

Assume you have a Lévy process X. Let $N(t,A)$ be defined as the number of jumps in the interval $(0,t]$, such that the jumps size $\Delta X_s \in A$. It can be shown that if $0 \ne \bar{A}$, then $...
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Convergence of a process

this may be viewed as a duplicate of this post. However i have put in much effort in the shared link and donated it with reputation, to check the proof considered there. Here however i want to argue ...
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Asymptotic behavior of queueing system with a server that takes breaks

I'm working on the following problem: A single server works on an infinite supply of jobs. The amount of time it takes the server to work on a single job is exponential with rate $\mu$, ...
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183 views

Uniform bounded of Riemann-like sum and improper integral

For any $h>0$, suppose $\{(y_i,y_{i+1}]\mid i\in \mathbb{Z}\}$ be a uniform partition of $\mathbb{R}$ with mesh size $h$. I am considering under what condition for a continuous transition density ...
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38 views

Compute expectation and covariance of Brownian bridge

Let $\{X(t), t \geqslant 0\}$ be a standard Brownian motion. That is, for every $t \gt 0$, $X(t)$ is normally distributed with mean $0$ and variance $t$. Then $\{X(t), 0 \leqslant t \leqslant 1 | X(1)...
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Time scaling birth process in Poisson process

Given a birth process $\{B_t:t\geqslant0\}$ with $\lambda >0$, define $$K_t=\int_{0}^{t}B_s ds=\sum_{i=1}^{n}B_{t_{i}}(t_{i+1}-t_i)$$ if there were $n$ births in $[0,t]$ and let $t_{i}$ be the ...
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35 views

What is the distribution of the subtract of two random variables?

Definition) A stochastic process $\{X(t), t \geqslant 0\}$ is said to be Brownian motion process with drift coefficient $\mu$ and variance parameter $\sigma^2$, if it satisfies that $X(0)=0$. $\{X(t)...
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Integral of non-Gaussian distribution, random walk?

I would like to evaluate $$ F = \frac{\mathbb{E} \left\{\left(\int_0^T x^3(t) dt \right)^2\right\}}{\mathbb{E} \left\{\left(\int_0^T x(t) dt \right)^2 \right\} } \approx \frac{\mathbb{E} \left\{\left(...
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Trace term in the Itō formula

I'm reading Stochastic Differential Equations in Infinite Dimensions and don't understand what the authors do in Chapter 2.3.1. Let me introduce the necessary objects: Let $K$ and $H$ be real ...
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Please verify the solution about Brownian motion process.

Problem Let $Y(t)$ denote the amount of time by which the racer is ahead when $100t$ percent of the race has been completed. $\{Y(t), 0 \leqslant t \leqslant 1\}$ is modeled as a Brownian motion ...
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predicting the future in a stationary stochastic process

Let's say I have a strictly-stationary stochastic process with known PSD (power spectral density). The process has been running, and I have all the data from time $t=-\infty$ to $t=0$. I want to ...
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19 views

Conditioning on invariant sigma algebra with respect to ergodic measure

So this question arose to me while applying the Ergodic theorem. If $X$ is a finite state (in $ \{1,\dots,d\}$) continuous-time Markov chain, which is ergodic, then $X$ has a unique invariant ...
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148 views

Limit theorem for changed time

This post seems long, but its almost everything proofed in this post. Only one step seems to be left, for the desired proof. I would be very gratefull for any help. The setup Given a Levy-Process $U_{...
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57 views

Stopping times of Poisson process

Given a Poisson process $N$, and let $S_n$ be the $n-$th jump time, i.e. $$S_n = \inf\{t\mid N_t = n\}$$ Question: is there a way to characterize all stopping times? especially, can all (or at least, ...
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How to Prove the Stochastic Fubini Theorem? (Exercise 2.19 in Chapter IV of Revuz and Yor)

Here is the theorem statement: Let $B$ and $C$ be two independent standard Brownian motions. If $\phi$ is square integrable on the unit square ($\phi \in L^2([0,1]^2)$ ), by suitable filtrations, ...
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Uniform convergence of the action of a Feller semigroup in one variable.

Assume we have two subsets of the some euclidean spaces $X\subset \mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ and a a Feller semigroup $(Q_t)_{t\geq 0}$ on $Y$. Suppose also that we have a continuous ...
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Is this transformation of a Markov process again Markovian?

Let $(X_t)_{t\in\mathbb{N}_0}$ be a stationary Markov process valued in $\mathbb{R}$ and $c\in\mathbb{R}$. Is the process $(Y_t)_{t\in\mathbb{N}_0}$ defined by $$ Y_t={\bf 1}{(X_t<c)} $$ again a ...
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75 views

Esscher Transform extended

This problem is almost solved, dont get scared by the massive text The Esscher-transform is a well know tool in the financial section. I posted this in statistics also, since it relates to continuous ...
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44 views

Solving Langevin equation

In a past exam paper that I am looking at, there is the following question: Given that the displacement, $\mathbf{x}$, of a particle in $3$-dimensional Brownian motion is given by: $$m\ddot{\...
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23 views

Why is the probability of extinction given by the probability generating function applied to 0?

I am trying to understand branching processes and can't find a good explanation for why solving for the probability of extinction at time $n$ is given by $p^{(n)}(0)$ with the superscript ...
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A Simple Stochastic Integral Asymptotics

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ $(\mu,\sigma)$ obeys the linear growth condition $...
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Find (a,b) such that aX+bY is a Brownian motion

Let $$\begin{cases} dX_t = \mathrm{sin}(X_t+Y_t) dW_t \\ dY_t = \mathrm{cos}(X_t+Y_t) dV_t \\ X_0=Y_0=0 \end{cases}$$ Where $(W,V)$ is a two-dimensional Brownian motion and $(X,Y)$ be a strong ...
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12 views

condition for recurrence of semi-stable process

Let $(X_t)$ be a nontrivial $\alpha$-semi-stable process on $\mathbb R$. I want to prove that if $1\le\alpha\le2$, $(X_t)$ is recurrent if and only if it is strictly $\alpha$-semi-stable. I want to ...
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27 views

Levy process measurable past

For a Levy-process $(X_t)_{t\geq 0}$ with stationary indepedent increments which is a markov process, we know that its law is defined by its one dimensional distribution. This is so because for its ...
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From brownian bridge to brownian motion proof

Let $B_t$ be a brownian motion. and let $\{W_t=B_t-tB_1:0\le t\le 1\}$ be a brownian bridge. Now let $Y_t=(1+t)W_{t\over 1+t}$. Proof that $Y_t$ is a brownian motion in $[0, \infty)$ My attempt: 1) $...
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24 views

in a M/M/1/K Queue, the ratio of losses when the incoming rate is combination of two rates

Generally, In a M/M/1/K system, the incoming rate is $\lambda$, effective incoming rate $\lambda_e$ is equal to $\lambda(1-P_k)$, where $P_k$ is the probability that queue waiting space is full. This ...
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48 views

Doubly stochastic matrix positive recurrent?

Let $\{X_n, n \ge 0\}$ be a discrete-time markov chain with a doubly stochastic transition matrix $P$ and a finite state space $S$. Prove that all states in $S$ are positive recurrent. My work: It is ...
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37 views

Verifying a Brownian motion through the Laplace transform

Let $X(t)$ be a continuous stochastic process and $\mathcal G(t)$ be the $\sigma$-algebra generated by $\{X(\tau) : \tau\leq t \}$. Suppose that for any $0\leq s\leq t$ and $\lambda\in\mathbb C$ ...
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Find the value of $\lim_{n \to \infty}\Pr[\max(X_1,X_2, …,X_n) <a+\ln n ]$ [closed]

Let $X_1,X_2,\dots,X_n$ be independent and $\operatorname{Exp}(1)$ distributed. Calculate the limit $$\lim_{n \to \infty}\Pr[\max(X_1,X_2,\dots,X_n) < a+\ln n].$$ I have tried several things ...
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calculate the $P(B(1)\leq 0,P(B(2)\leq 0))$, $B(t)$ is the standard brownian motion.

denote $W(1)$ by $(B(2)-B(1))$. then $P(B(1)\leq 0, B(2)\leq 0)$ = $P(B(1)\leq 0, B(1)+(B(2)-B(1))\leq 0)$ =$P(B(1)\leq 0, B(1)+W(1)\leq 0)$ =$P(B(1)\leq 0, W(1)\leq -B(1))$. by conditioning by ...
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Nonanticipativity constraint (filtration/measure theory)

I am trying to show that stochastic process must attend the nonanticipativity constraint using filtration in measure theory. Adaptability of a stochastic process tell us that: $$\sigma(X_t)\subset \...
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23 views

Brownian Motion maximum process intuition

I am studying the maximum value of a Brownian Motion (BM) on an interval of time (as explained here between boxes 28 and 40) and I am having an issue aligning intuition with the mathematical result. ...
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Find function $h$ so that $h(U,V)$ equals density of $f(a) da$ for $f(a)=\frac{1}{2}e^{-\small|a|}$, $a \in \mathbb R$

Let $f(a)=\frac{1}{2}e^{-\small|a|}$, $a \in \mathbb R$ and let $U,V$ be independant and uniform distributed on [0,1]. Now I want to find a function $h$ so that $h(U,V)$ is equal to the density $f(a)...
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Expected hitting time of a stochastic differential equation with jumps (neuroscience example)

The basic model I'm working with is a neuron that receives input from other neurons which cause instantaneous spikes in the voltage. In a nutshell, I have a differential equation that describes the ...
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25 views

Negative binomial distribution?

We throw a coin with success probabilty $p$ and $Y$ is the amount of coin tosses we need untill we have $n$ successes. Now I want to show that $P(Y=n+i)=\begin{pmatrix}n+i-1\\ i \end{pmatrix}p^{n}q^{i}...
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32 views

Brownian motion independent RVs

Let $(W_t)_{t\in\lbrack 0,T\rbrack}$ be a standard Brownian motion. Does there hold that $W_s(W_t-W_s)$ and $W_k(W_l-W_k)$ for $0\leq s<t\leq k<l\leq T$ are independent RVs?
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Transition functions and Markov processes

I am wondering whether there is a one-to-one correspondence between transition functions and homogeneous Markov processes? We say that $(X_t,\mathcal{F}_t)_{t\geq 0}$ is a Markov process if $\mathbb{...