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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Can every filtration be written as $\mathcal F^X$ for some process $X$

Given a stochastic process $\{X_t: t\in R^+\}$, which takes value in $R$, there is always a natural filtration $(\mathcal F^X_t)$ induced by $X_t$, i.e. $\mathcal F_t^X = \sigma(\{X_s^{-1}(A): s\le t, ...
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33 views

Gaussian Process Through Random Width Filter

I'm having a hard time with this issue: I have a stationary gaussian process $\{X(t)\}$ ($\mu_X=0$ for simplicity), with known PSD: $$ S_X(f)=\begin{bmatrix} 1 , |f|<b \\ 0 , |f| > b ...
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28 views

Characterization of point process, given the number of points

For a point process with independent and identically distributed inter-renewals, with distribution $p(x)$, we observed $N$ points on $[0,T]$. What is the probability distribution function of the ...
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34 views

Stochastic processes closed under truncation

I'm currently studying some properties of general stochastic processes, and am having some issue understanding how to prove this (probably simple) example. First, let me introduce the notation & ...
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23 views

Question about $X_n$ and $N(t)$ in Counting and Renewal process.

I am studying renewal theory. In the text book, $\{X_n, n=1, 2, \cdots\}$ denotes a sequence of non-negative i.i.d. with a common distribution $F$, and to avoid trivialities suppose that $F(0)<1$. ...
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45 views

Why does this generated $\sigma$-algebra contain all the sets that we have information about?

Assume that you have a collection of random variables $\{Y_\gamma: \gamma \in C\}$. Where each is a function $\Omega \rightarrow \mathbb{R}$. We define the $\sigma$-algebra: $\sigma(y_\gamma: ...
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28 views

the relationship of the counting process and the renewal process?

The textbook wrote like the following: A natural generalization is to consider a counting process for which the interarrival times are independent and identically distributed with an arbitrary ...
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34 views

Are martingales actually continuous?

There are strong theorems like the martingale convergence theorem giving us the existence of a continuous limit for $t \rightarrow \infty$ of a martingale $(X_t).$ But I was wondering under which ...
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1answer
24 views

Relative entropy for wiener measure/wiener measure with girsanov change of drift

I've read an article on relative entropy properties that gives a result for the relative entropy of two equivalent measures as they are found in applications of girsanovs theorem. For two measures P, ...
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1answer
47 views

Central Limit Theorem for Lévy Process

I am reading a book, which uses the Central Limit Theorem of Lévy Processes $X_{t}$ without mentioning the exact theorem. Due to the infinite divisible property I can write $X_{t}$ as a sum of $N$ ...
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27 views

How can I prove the equivalence of these two Ito's lemma notations?

Let $X_t=(X_1, \dots , X_T), t \in [0,T] $ be a continuous semimartingale and $f$ a function of class $C^{1,2}$ (continuous and differentiable). Then, $f(t,X)$ is a semimartingale and we have, ...
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61 views

$dX_t/X_t=\mu+\sigma \, dZ_t$, does this notation make sense?

I understand that the notation $$dX_t=\mu X_t \,dt + \sigma X_t \,dZ_t,$$ where $Z_t$ is Brownian Motion, is a shortcut to $$X_t-X_0=\int_0^t\mu X_s \, ds+\int_0^t \sigma X_s \, dZ_s, \tag{*}$$ ...
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24 views

Reference request: correlation and spectral analysis of stochastic processes

I'm wondering if anyone knows of a reasonably rigorous text on stochastic processes that discusses specifically things like the autocorrelation, spectral density, and other "correlation and spectral" ...
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1answer
23 views

How to discretely stochastically simulate a continuous-time Markov chain?

A continuous-time markov chain describes a continuously varying process, such that future state only depends on the current state. A sampling of a continuous markov chain can be described in terms of ...
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1answer
43 views

If one stochastic process is a modification of another, then they have the finite probability distribution.

On page 2 in Karatzas: Brownian Motion and stochastic calculus it is said that a stochastic process Y is a modification of X if for all t: $P(X_t=Y_t)=1$. If both are stochastic processes into ...
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35 views

Brownian motion, harmonic functions and the Dirichlet problem

I am having trouble understanding one detail of the standard use of Brownian motion to solve the Dirichlet problem, I will write the statement and proof and then point to the detail I don't ...
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1answer
20 views

An issue of dependent and independent random variables involving geometric Brownian motion.

Let $X(t)=X(0)e^{\mu t + \sigma Z(t)}$ be a geometric Brownian motion (GBM) where $Z(t)$ is the standard Brownian motion with drift $0$ and the variance rate per unit of time is $1$. Now, let $s<t$ ...
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24 views

What is exactly meaning of The notion of Sample path and Stochastic Process?

I am wondering what Stochastic Process is exactly meaning. Let me talk about what I understood. I will give an example. $\Omega_i$ is noise of my robot's circuit on July the $i$-th day. The ...
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19 views

Help understanding proof of connection between conditional expectation and PDE

This is a TeX'd repost of Connection of expectation and PDE; help understanding step in proof, which did not get much attention. Some preliminaries: $\mathcal{A}$ is the generator of an Ito ...
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1answer
134 views

How should I understand the probability space $(\Omega, \mathcal{F}, P)$? What does “hidden” mean?

I thought I understood the measure theoretic concept of a probability space, but yesterday I realized that I really don't. Here's what I mean: I thought we could think of $\Omega$ as the set of ...
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1answer
38 views

Iterated logarithm law for difference (supremum(W) - infimum(W) ) is it 2srt(2/pi) sqrt(t loglog(t))?

Law of iterated logarithm says that $$\sup(W(t)) \sim \sqrt{2 t \log(\log(t))}.$$ Consider $\sup(W(t)) - \inf(W(t))$ my guess based on numerics that it should be $$2\sqrt{\dfrac 2\pi} \sqrt{t ...
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13 views

Brownian Motion Hitting Time?

So my problem is the following. Take a 2D Brownian motion $(W_{1t}, W_{2t})$ such that it starts at $(1,1)$. With probability 1 it will hit the x-axis. What is the probability that it will hit the ...
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23 views

Does Ito isometry hold pointwise?

It is known that the stochastic integral satisfies the following property $$ \mathbb{E}\left[\left\langle \int_0^{\cdot}X(s)\,dM(s) \right\rangle_t\right]= \mathbb{E}\left[ \int_0^t X^2(s) \, ...
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18 views

how to use and make sense of generating function method to solve coupled master equation?

I am trying to solve a two dimensional continuous time and discrete state master equation. The master equation is linear and looks as follows, $\frac{\partial P_A(x,y,t)}{\partial t} = k_{11} ...
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26 views

Simulation of brownian motion and fractional brownian motion

It's easy to simulate a path of a brownian motion with the method explained in Wiener process as a limit of random walk: ...
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18 views

Is continuous the same as uncountable in “continuous-time stochastic process”?

Is the word "continuous'' as used in "continuous-time stochastic process" or "continuous-time Markov chain'' synonymous with uncountable? Or is there a difference? I'm wondering because I have seen ...
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1answer
57 views

Markov Chain: flip coin 8 times and get 3 consecutive heads

I have confusion while reading the following example in the course material. Q: In a sequence of independent flips of a fair coin, let N denote the number of flips until there is a run of three ...
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15 views

Question about chapter “the conditional distribution of the arrival time” in the Poisson Process.

I am studying stochastic process written by Sheldon M Ross. I have a question about the conditional distribution of the arrival time. Theorem 2.3.1 $$S_1, S_2, \cdots, S_n |_{N(t)=n} \sim U_1, ...
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50 views

Uniform integrability of process with bounded conditional expectation

Let $[0, T]$ be a finite time horizon, i.e., $T < \infty$. Consider a complete filtered probability space $(\Omega, {\cal F}, {\mathbb F}, P)$, where ${\mathbb F} = \{ {\cal F}_t \}_{t \in [0, T]}$ ...
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17 views

Finding probability density function (PDF) from autocorrelation function

Given an autocorrelation function for a random process $R_{XX}(\tau)$, how can I find the first-order probability density function $f_{X}(x)$ for this process?
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1answer
30 views

Find the transition function of the Markov chain (Xm)

I haven't taken a probability/statistics course in years and I'm trying to make my way through an Introduction to Stochastic Processes book. The question reads as follows: Suppose we have two urns ...
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23 views

Time-scale law of Bernoulli-stopped process

This post is quite long, but the problem stated carries no computational burden. Consider the equally spaced partition $t_{i}^n=\frac{i}{n}$ with $i=0,...,n$ of the interval $[0,1]$ into $n$ ...
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1answer
43 views

What is a pure-jump process?

I have been reading some notes and they keep referring (without definition) to a "pure jump process". On wiki I can only find a reference in the Levy-Ito decomposition theorem, but still I can't find ...
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1answer
54 views

Exercise 8.12 Introduction to stochastic processes Gregory Lawler [closed]

Let $X_t$ be a standard Brownian motion starting at 0 and let $T=min \{t:|X_t|=1\}$ and $\hat{T}=min \{t:X_t=1\}$ (a) Show that there exist positive constants $c$, $\beta$ such that for all ...
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29 views

Construction of Wiener Process using integral of covariance multiplied by a function

I read in the notes of Stochastic Processes that there is a construction of Wiener Process (knowing that $Cov(W_s, W_t)=min(s,t)$ ) which going like this: consider operator $Q$ on $C([0,1])$ $$Qf(t)= ...
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2answers
70 views

Markov Chain: flip 8 coins and get 3 consecutive heads

I was reading the material and I am confused at the following example. Q: In a sequence of independent flips of a fair coin, let N denote the number of flips until there is a run of three ...
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1answer
40 views

Markov Chain: memoryless property?

I have a question about the Markov Chain. We were doing derivation of the Chapman-Kolmogorov Equations, the $n+m$ step state transition probability (please see below): where $P_{i,j}$ is the ...
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1answer
28 views

Black-Scholes formula with non-constant volatility (function of time)

Let's have the following stochastic process: $$dS_t = r S_t dt + σ(t) St dW_t$$ where $W_t$ is the Brownian motion, r the drift and $σ(t)$ the volatility, a deterministic function of the time. ...
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1answer
15 views

Independent stochastic processes

I have 2 stochastic processes that are independent.. so E [X(t)C(t)]=E[X(t)]* E[C(t)] ... now I would know if ** X^2(t) and C^2(t)** are both independent and why.. Thanks
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22 views

The expected value of a stochastic process

I am asked to calculate the expected value of the maximum of the stochastic process $X(t)=At+B,\,0\le t \le 1$, where both $A$ and $B$ are independent, normally distributed random variables with mean ...
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14 views

Branching Process, Supercritical Case: Why is $P[Z_n=0] \leq \rho$

Consider a branching procesc with $Z_n$ the number of individuals at generation $n$, offspring distribution $\nu$ with mean offspring number $m > 1$. We consider the generating function ...
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57 views

Showing that $P(W_{t}/\sqrt{t \log(t)}>1+\epsilon)\to0$ when $t\to\infty$, where $(W_t)$ is a Wiener process

I have a question about the martingales $\dfrac{e^{W_{t}^2/(1+2t)}}{\sqrt{1+2t}}$. With use of this martingale I want to show that $P(\dfrac{W_{t}}{\sqrt{t log(t)}}>1+\epsilon)$ goes to $0$ if $t$ ...
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1answer
19 views

Mean time for the trajectory. Find mean

What is the mean of time when the trajectory of the wiener process, $W_t$, is over the line $y=t$? We need to find $\Bbb{E}\tau$, where $\tau=\sum\limits_{a,b:\forall t\in(a,b) ; ...
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1answer
29 views

Exponential martingale and change of measure

$\newcommand{\qq}{\mathbb{Q}}\newcommand{\ee}{\mathbb{E}}$ Denote $Z_t= \exp( \theta B_t - \frac{1}{2}\theta^2t )$ Given the probability measure $\qq(A) := \ee[ Z_t \mathbb{1}_A ]$ I must ...
2
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1answer
54 views

Quadratic variation of semi-martingale

$X_t = e^{B_t-\frac{1}{2}t^2}$ I need to find $[X]_t$, the quadratic variation process. I have tried to solve the problem and my main question is whether this approach is correct or not. ...
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17 views

An application of Ito's formula

I am reading a proof in which I don't understand how to use Ito's rule to derive the following: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space such that $M^{(i)}$ and $M^{(k)}$ are ...
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25 views

Brownian motion and sup of a Brownian motion

I am stuck with the following problem: let $B_t$ be a standard Brownian motion and let $S_{t}:=\sup_{0 \leq s \leq t} B_s$. Prove that for every $\lambda \geq 0$ and $\mu \leq \lambda$, ...
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25 views

Proof finite stopping time and Wiener process bounded

Let $T_{-a,b}=\inf\{t\geq 0: W_{t} \notin [-a,b]\}, a,b>0$. I want to show that this is a finite stopping time ($P(T_{-a,b}<\infty)=1$) and that $|W_{\min(T_{-a,b},t)}|$ is bounded by a ...
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34 views

Connection of expectation and PDE; help understanding step in proof

The following is taken from a set of lecture notes available online by Nizar Touzi. Some points before reading the proposition: $\mathcal{A}$ is the generator of a stochastic process $X_s^{t,x}$. The ...
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69 views

Theorem 4.14 Brownian Motion and Stochastic Calculus

I have been reading the proof of Theorem 4.14 of Karatzas' book. I wonder whether there is a typo in the description of the process $\eta^{(n)}_{t}$ as $\xi^{(n)}_{t+}-\min({\lambda,A_{t} })$ ...