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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Infinite series for a birth death chain problem

I'm afraid I can't deduce a simple expresion for the following infite serie $\sum_{i=0}^{\infty} p^{i(i-1)/2} r^i,$ with $p,r <1$. Since $p^{i^2} < p^i < 1$, I think the serie converges. ...
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83 views

Limit of Wiener processes

Let $W_t$ be Wiener process. I am trying to evaluate the following limit $$\lim\limits_{n \to \infty}~{\sum\limits_{i=1}^{n}W_{\frac{i-1}{n}+\frac{1}{2n}}\left( W_{\frac{i}{n}} - W_{\frac{i-1}{n}} ...
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Applying Ito to semi-group of Brownian motion in $\mathbb{R}^d$

Using Ito, I am trying to show that $M_t$ = $\mathbb{E}[(f(X_1))|F_t]$ is a martingale. (I know that it's a martingale by definition of it, but this is an exam question, which stipulates use of Ito.) ...
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A Brownian motion starting from 0, it becomes -1 once reach -1, what is its expectation?

$X_t$ is a Brownian Motion, it reaches becomes -1 forever once it reaches -1. Mathematically, $T = \inf\{t: W_t = -1\}$ is a stopping time. When $t<T$, $X_t = W_t$ ,while $t\geq T$, $X_t = -1$. ...
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Linear transformation of Levy processes

Here is a question about linear transformation of Levy processes. It is stated in my reference (Cont and Tankov's Financial modelling with jump processes, Theorem 4.1) that a linear transformation of ...
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Hitting time of Brownian Motion with a drift

Let $X_t =x+bt+\sqrt{2}W_t$, where $W_t$ is a standard Brownian motion. Let $T=\inf\{t: |X_t|=1\}$. I am trying to find $\mathbb{E}[T]$ for the case $b\neq0$. Firstly, I am going to apply Girsanov to ...
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Problem with applying Ito

I want to show that the following expression is a local martingale: $$M_t=V(X_{t\wedge T})+\int_0^{t\wedge T}f(X_s)ds,$$ where $T$ is some stopping time (there are more conditions, but they are not ...
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Polarisation in proving Kunita-Watanabe identinty

Kunita-Watanabe identity: Let $M,N$ be local martingales, $H$ be a locally bounded previsible process, then $$[H\cdot M,N]=H\cdot[M,N],$$ where $[M,N]$ is covariation. I am going though the proof, ...
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Why do we always assume waiting time has exponential distribution?

In many continuous models, like waiting for a car, we always assume the waiting time $t$ to have an exponential distribution. Why is such an assumption appropriate?
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$\sigma$-algebra generated by Brownian motion

Let $(B)_{t \geq 0}$ be a standard Brownian motion. Then $B$ is adapted to its natural filtration $(\mathcal{F}^B_t)_{t\geq 0}$. Often, we want to consider a slightly bigger filtration, ones ...
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265 views

Random-Tree simulation

I wanna simulate a Galton-Watson Tree to a maximum of n generation given a reproduction law P. I use Maple but I am unable to create the edges of the tree whenever there are more than two vertices in ...
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116 views

MA process ACF proof - don't understand it

I've got the proof but I don't understand a small detail. As you know for an MA process: $X_n = \sum _{i=0} ^q \beta_i Z_{n-i}$ where $Z_n$ is WGN (pure Gaussian random process). Then the ACF is: ...
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166 views

The variance of bilateral filtered random variables

I am glad to have found this great site. There is a problem I am trying to solve for a while. I want to analyze the noise attenuation behavior of the bilateral filter. So given the unnormalized ...
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53 views

For a stochastic process how is the probability affected by increasing number of realisations?

I am implementing a stochastic version of logistic equation in MATLAB. Keeping track of distribution of time it takes to for the population to reach 500 I experiment with various number of ...
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425 views

Is a Markov process a random dynamic system?

A random dynamic system is defined in Wikipedia. Its definition, which is not included in this post for the sake of clarity, reminds me how similar a Markov process is to a random dynamic system just ...
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239 views

Is every killed Markov process still a Markov process?

Suppose we've got $X=(X(t))_{t\geq 0}$. $X$ is a strong Markov process with respect to filtration $\mathcal{F}_t$, taking values in some subset of $\mathbb{R}$. We take $\tau$ - a stopping time w.r.t ...
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Cover time and intersection time for lazy random walks on graphs

Consider a simple lazy random walk on an $n$-vertex undirected, connected graph: this is the Markov chain which transitions from $i$ to $j$ with probability $p_{ij}=1/(2d(i))$ where $d(i)$ is the ...
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Cover times and hitting times of random walks, once again.

This is a followup to my question Cover times and hitting times of random walks. Consider a random walk on an undirected graph with $n$ vertices which, at each step, moves to a uniformly random ...
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425 views

$\mathcal{F_t}$-martingales with Itô's formula?

I need a little help with a problem. I am given some stochastic processes and supposed to show that they are $\mathcal{F_t}-$martingales. The first one is this, and they all look similar: ...
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Positive recurrence of a continuous-time jump process, from its jump chain

I am looking at an irreducible, continuous-time jump process $(X_t)_{t\geq0}$ with the following jump times. Let a Poisson process $(T_i)_{i=1}^\infty$ determine the event times. With probability ...
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Natural & important probability measures on $\mathcal{C}[0,1]$, in particular the Wiener measure

Which probability measures on $\mathbf{\mathcal{C}[0,1]}$ are known? (Here $\mathcal{C}[0,1]$ is the space of continuous real-valued functions defined on the unit interval.) I'm pretty sure the ...
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Computing the stationary distribution of a markov chain

I have a markov chain with transition matrix below, $$\begin{bmatrix} 1-q & q & & & \\ 1-q & 0 & q & & \\ & 1-q & ...
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Measure theoretic question

When I have shown, for $s\le t$ and for two continuous stochastic process an inequality: $$ X_s \le Y_t$$ P-a.s. How can I deduce that this P-a.s. simultaneously for all rational $s\le t$ ? Thank ...
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Question about a step in the proof of the bracket process

Let's first state the theorem $\forall M$ continuous local martingale, there exists a unique increasing continuous process $\langle M\rangle $ zero at $t=0$ and such that $M^2-\langle M \rangle $ ...
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Distribution and continuous time markov chains

Let be a probability distribution on the nonnegative integers such that $\pi_i > 0$ for all i. Write down the transition matrix of an irreducible, aperiodic, recurrent Markov chain on the ...
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Markov chain transition matrix

Consider the Markov chain with the following transition matrix: $$P = \pmatrix{0& 0.5 &0 &0 &0 &0.5\\ 0.25 &0 &0.25 &0.25 &0 &0.25\\ 0 &0.5 &0 &0.5 ...
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178 views

Finding again the stationary distribution of a markov chain

I am asked to compute the stationary distribution of the markov chain with state space $E=\{0\dots,n\}$ and transition matrix below: \begin{bmatrix} 0 & 1 \\ \frac{1}{n} ...
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297 views

Karhunen-Loève expansion of Poisson process

Let $X_t,t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the Karhunen-Loève expansion of $X$ in interval $[0, T]$. How about the KL expansion of the centered process $X_t−\lambda ...
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Finding the stationary distribution of a markov chain

I am asked to compute the stationary distribution of the markov chain with state space $E=\mathbb{N}_0$, $q_n >0$ for all $n \in \mathbb{N}_0$ and transition matrix below: \begin{bmatrix} q_0 ...
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PDF for the integral of a Stochastic Process

My continuous-time, continuous step Stochastic Process P runs from time $t=0$ to $t=t_f$ and generates a path. I am able to observe its starting and ending position (so $P(0)=a$ and $P(t_f)=b$), but ...
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Writing a FV process as a Riemann integral

Proposition 3 (ABDL03): If a special semimartingale process $X$ is square integrable with respect to the natural filtration of a standard Brownian motion $W$, then one can write $X_t - X_0 = ...
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Continuous/discontinuous decomposition of local martingales

Continuing from my previous question Decomposition of semimartingales, In his answer, George Lowther mentioned that if $X$ is a local martingale, then $X^d$ and $X^c$ in its decomposition $X_t - ...
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Calculating stationary distribution of markov chain

I am asked to compute the stationary distribution of the markov chain with state space $E=\{0\dots,n\}$ and transition matrix below: \begin{bmatrix} 0 & 1 \\ \frac{1}{n} ...
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132 views

stochastic process and conditional density

Let a stochastic process defines as: $$X(t+1)=A X(t)+B U(t)$$ with: $X(t) \in R^n$, $U(t) \sim N(0,Q_t)$, $Q_t$ semi-positive-definite of size $n \times n$, $X(0) \sim N(0,W_0)$, $A$ of size $n \times ...
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270 views

Random walk on infinite line - can it be stationary?

Suppose a random walk on an infinite line $[...-3,-2,-1,0,1,2,3,...]$, starting from 0. Probability to go right or left are equal. Does such a process stationary? I think that it is NOT, since the ...
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What mathematical distributions describe better traffic in a mobile network?

If it were to describe the traffic in a mobile phone network, what would be the best mathematical distributions? It would be more of a combination of distributions? Which distribution would describe ...
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optional sampling theorem

I deal a standard deck of 52 cards to you face up, one card at a time. Before any deal of any card you can shout out "NOW". If you shout out "NOW" and the next card I deal is a queen, then the game ...
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256 views

Brownian motion Convergence

If $X$ is a standard 1d brownian motion and $M_t$ $= \mbox{max}\{X_S: 0 \le s \le t\} $, what can we say about $M_t/t$ as $t \rightarrow \infty$? Mainly, what can we say about the behavior of this ...
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Show that the assumption of right-continuity in the statement of the stopping theorem cannot be omitted

In our homework assignment, we were supposed to find an example showing that the assumption of right-continuity in the statement of the stopping theorem cannot be omitted in general (cf. ...
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What does it mean for a stochastic process to be independent of a sigma algebra?

Does anybody know what it means for a stochastic process $ X = (X_t)_{t \geq 0} $ on a filtered probability space $ (\Omega, \mathcal{F}, \mathbb{F}, P) $ to be independent of a sigma-algebra $ ...
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poisson processes as a function of a parameter

Problem: The number of customers arriving at a bank in a Poisson process $\lambda = 3$ per minute. What is the probability that no customer arrives in a $T$ minute interval? Solution: This is a ...
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92 views

Two opposite events fill whole probability event space, a process selects them [duplicate]

Possible Duplicate: Probability of two opposite events Suppose there is string of eight bits, e.g.: 00100110 Bits are randomly chosen from the string. Location of bit (in the string) ...
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Homogenous Poisson process

Consider a homogeneous Poisson process $N$ with rate $\lambda$. For For $0 < s < t$, I'm trying to show that: $$P(N_t-N_s=0\mid N_t>0)= \frac{e^{\lambda s} - 1}{e^{\lambda t} - 1}$$ I'm ...
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323 views

Galton Watson Branching process

Let $X_0$,$X_1$,...be a Galton-Watson branching process. Let us denote $\epsilon$ for the probability when $X_0 = 1$ that the population eventually becomes extinct (that is, that $X_n = 0$ for all ...
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378 views

Differential equation with gaussian noise

The equation has the following form: $$x'' + w^2 x=n$$ $w=1$, $x(0)=1$, $n$ is Gaussian noise with mean $0$ and standard deviation of $1$. Without the Gaussian noise, i can easily solve the ...
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420 views

Markov Process: Have you seen this notation and do you know what it means?

Ok, I've already posted this a minute ago, but my text deleted itself while I was editing it :-( So next try: Can you help me to understand the notation my professor uses to describe Markov ...
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Stationary distribution of random walk

Let $\mathcal{X}$ be a simple random walk with barrier at zero, state space $E = \mathbb{N}_0$ and transition matrix below with $0<q<1$. \begin{bmatrix} 1-q & q & & ...
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2k views

Show irreducibility of markov chain

I need to show that the markov chain that has transition matrix written below is irreducible. \begin{bmatrix} 0.2 & 0.5 & 0.1 & 0.1 & 0.1 \\ 0.2 & 0.5 ...
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Why are persistent states of a Markov chain on a finite state space non-null?

i would like to understand the following statement about Markov chains on a finite state space S: "If S is finite, then one state ist persistent and all persistent states are non-null." It is more ...
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A question regarding the strong Markov property

In our lecture on Brownian motion & stochastic calculus we proved: If $ X $ is a canonical RCLL process having the strong Markov property and $ \tau $ is a stopping time with $ \tau < + \infty, ...