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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How to model my not knowledge of distributions. Example form Optional Stopping.

I'm working on a problem related to the secretary problem. Let me give a short overview on the topic I research: You are supposed to choose the best item presented to you in a row of n items. Any ...
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27 views

Why are Optional Stochastic Processes Important?

I understand to some degree why adapted processes, progressive processes, and predictable processes are important. EDIT: I am referring only to the continuous time case, NOT discrete time. But why do ...
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Should I ask about math books? I.e. scan of index page or references page?

I mean it's not harmful to anyone if I ask about certain page of certain book? I need page 263 from this book ...
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23 views

Do Optional and Progressive Processes Have Counterparts in Discrete Time?

We know that predictable $\implies$ optional $\implies$ progressively measurable. Source Predictable processes have obvious/simple counterparts in discrete time. Do optional processes and ...
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Radon-Nikodym on a Process wrt to filtration

Given a probability space $(\Omega,\mathcal{F},P)$. Let $(X_t)_{t\geq0}$ be a stochastic process defined on it with cadlag paths, lets say on $(\mathcal{X},\mathcal{B}(X))$. Let be $\mathcal{F}_{t}$ ...
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Indistinguishable Processes under local Lipschitz Condition

Let $a,b, \rho, \sigma$ be locally Lipschitz functions on $\mathbb{R}^d$, G an open subset of $\mathbb{R}^d$ and assume that on $G$ we have the equalities $a=b$ and $\rho=\sigma$. Let $\xi \in G$ and ...
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1answer
34 views

What is the probability that a given $ n $ event trains match the beginning of a Poisson process?

Here is my question with which I'm confusing myself: Assume that some event times $ \{\tau_i\}_{i \in \mathbb{N}} $ are a point process with rate $ \mu $ such that number of events that occurred ...
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1answer
23 views

Finding the mean given the probability

I'm doing some work on branching processes and would like to know where the process becomes extinct. If $X$ is the number of offspring of an individual, then the process goes extinct when ...
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1answer
26 views

Expected Square Distance from Origin of Random Walk in $\mathbb{Z}^2$

I'm trying to find the expected value of the squared distance from the origin of a simple symmetric random walk in $\mathbb{Z}^2$ at time $n$. So far, I have calculated that if $(X,Y)$ is the ...
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23 views

How do linear operators acting on paths of Gaussian processes influence the covariance function?

It is well-known that applying a linear transformation $A$ on an $n$-dimensional centered Gaussian distribution with covariance matrix $\Sigma$ results in another centered Gaussian distribution with ...
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14 views

System of SDEs and independence

I am recently reading a paper that seems to claim the following fact without justification: $Y^1_t, \ldots, Y^n_t$ are stochastic processes defined on $\mathbb{R}$. Let $b: \mathbb{R}^2 ...
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88 views
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Stochastic Integral with respect to Compensated Poisson Process

Proposition: Let $N_t$ be an $\mathcal{F}$-Poisson process and $M_t=N_t-\lambda t$ its compensated process. Then for any $\mathcal{F}$-predictable bounded process $H_t$, the stochastic integral ...
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38 views

Why Are Semimartingales the Largest Possible Class of Stochastic Integrators?

I am trying to understand why semimartingales are the most general possible class of stochastic integrators. (I was hoping that this question would give me my answer, but it didn't.) I thought at ...
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1answer
124 views

One-Dimensional Jump-Diffusion Ito’s Formula

Let, $N_t$ be a Poisson process and let $X_t$ solve the SDE $d{X_t}=a_t dt +J_t dN_t$. Then, what is the correct Ito´s fórmula: i)$df(t,X_t)=(\frac{\partial f}{\partial t} + \frac{\partial ...
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41 views

How to prove that the stochastic integral process is gaussian?

I would like to prove that for a $C^1$-function f and a Wiener process W, the integral process defined by $$ Y_t:= \int_0^t f (s)dW_s := f (t)W_t -\int_0^t W_s f'(s)ds $$ Is a centered gaussian ...
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Functions of a random walk and martingales

Let $\xi_1,\xi_2,\ldots$ be a sequence of iid random variables, such that $$\mathbb{P}(\xi_i=1)=p\ne \frac{1}{2},\,\mathbb{P}(\xi_i=-1)=q=1-p.$$ Consider the corresponding random walk ...
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23 views

Uncorrelated but not independent uniform distribution

Let $X = (X_1, X_2)$ be uniform distributed on $\{(-1,0), (1,0), (0,-1), (0,1)\}$. First of all I want to show that $X_1$ and $X_2$ are uncorrelated but not independent. Secondly I thought about ...
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Convergence of a sequence over supremum

Given a cadlag-process $X_{t}$ with stationary independent increments (Levy process) for which $E\left[\sup_{s\in[0,t]}\left|X_s\right|\right]<+\infty$ for all $t>0$. For $n\in \mathbb{N}$ the ...
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28 views

Integral equation with a random variable [on hold]

Suppose we have an integtal of this kind: $$\displaystyle Q(\tau)=\int_0^\tau dt K(w(t)+\nu(t))$$ where $\nu(t)$ is a gaussian noise $\nu(t)=\mathcal{N}(0,\sigma)$ Because $w(t)$ is a known function, ...
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74 views

Hard Question in Stochastic processes - variance Martingales

I got some hard challenge to solve and I am looking for a small clue/help. My question goes like this: 10 Englishmen are trying to leave a pub in a rainy weather. They do it in the following ...
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1answer
68 views

Why do we need progressive measurability in the approximation of process by simple functions?

I am following Karatzas and Shreve- Brownian Motion and Stochastic Calculus. In the context of Stochastic integration one defines the martingale transform for simple functions: Now we would like to ...
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21 views

Distribution of Double Stochastic Integral

Assume that $f(s)$ is a $C^\infty$ univariate function and that $\{ (W_{1,t}, W_{2,t})\}_{t \geq 0}$ is a two-dimensional, correlated Wiener process. Then, does the random variable $X_T \equiv ...
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$2D$ random walk stopping time

A $2D$ random walk starts at $(X_0, Y_0) = (k, k)$ where $k>0$ is an integer. At each step $(X_{n+1}, Y_{n+1}) = (X_{n}-1, Y_{n})$ or $(X_{n+1}, Y_{n+1}) = (X_{n}, Y_{n}-1)$ with the same ...
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Version of the CLT

It is well known, that for a sequence of i.i.d. rv. $X_i$ with $E[X_i]=\mu$ and $Var[X_i]=\sigma^{2}$ that $$ ...
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1answer
41 views

Conditional independence of stopping times from i.i.d. stochastic processes

My question is somewhat arbitrary but I was thinking about independence of processes and stopping times. Say that we define two processes $X,Y$ on different probability spaces ...
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1answer
30 views

Poisson Process independent Wiener Process using singular measures

I was reading some stochastic calculus of Jump processes and saw the following result: If $W_t$ is Brownian and $N_t$ is Poisson both adapted to the $W_t$'s natural filtration then these processes ...
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24 views

Compute the Mean of the Following Process

Given the following process: $\Delta \ln(St+1)= \mu - (\sigma2/2) + \sigma(\varepsilon(t+1))$ (where both $\mu$ and $\sigma$ squared are of $S$) How does one calculate the mean of $S(t+1)/S(t)$? ...
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1answer
34 views

What is the difference between an adapted process and a predictable process?

As the footnote on page 1 of this document mentions, even most experts in the field of stochastic processes don't seem to know rigorously what the difference is. However, since I don't have any idea ...
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1answer
30 views

Conditional Probability in Poisson Process

Suppose that $N(t), t\ge 0$ is a Poisson process such the $E[N(9)]=6$. (i) Find the mean and variance of $N(8)$. (ii) Find $P(N(2)\le3)$ (iii) Find $P(N(4)\le5|N(2)\le3)$ - How do I solve this? ...
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59 views

Indicator Functions - Can someone check my working?

This is a very easy question but since some of my codes aren't coming out properly I thought I should check my theory to see if everything's okay. Say we have two values $K_{1}$ and $K_{2}$ and that ...
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67 views

Integral of Wiener Squared process

I don't have a background of stochastic calculus. It is known fact that definite integral of standard Wiener process from $0$ to $t$ results in another Gaussian process with slice distribution that ...
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1answer
11 views

Riesz theorem and $L^p$ norm in expectation

I am reading a paper that uses the following fact, which claims to be from the Riesz's theorem: For a continuous stochastic process $\{ X_t \}$, let $u_t$ be its density function at each time ...
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21 views

Esscher-Transform/ Levy-Process: Measure induced by trajectory

For a Levy-process $X_t$ w.r.t. to a measure P we define $\Theta$ as the set, for which $E[exp(\theta X_t)]$ is defined and finite. Note $\Theta$ is independent of $X_t$. Define ...
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1answer
33 views

To test whether a process is a Martingale (Stochastic calculus)?

If $W_t$ is a standard Brownian motion, I was trying to prove $Y_t = \exp (\int_{0}^{t} s\cdot dW_s)$ is a martingale ! First I started finding $dY_t$ using Ito formula. But I am confused how to ...
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Can this approximation result for stochastic processes be modified.(p=1 instead of p=2)?

In McKeans stochastic integrals from 1969 he proves this: You have a filtered probability space $(\Omega,\mathcal{F},P)$, where the filtration is based on a Brownian motion. Assume that $X_t$ is ...
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Residual life distribution for renewal process after time T

Suppose we have a renewal process with inter-arrival times $\boldsymbol X=\{X_1, X_2, ...\}$, where $X_i$ are i.i.d variables. Assume that the CDF and PDF for $X_i$ is $F(x)$ and $f(x)$. 1) Let $A_t$ ...
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1answer
18 views

Is equality of processes stable to multiplication with an independent process.

Assume that all processes to be considered are regular (say cadlag). Assume $X^1$ and $X^2$ are stochastic processes such that $X^1_t = X^2_t$, that $Y^1$ and $Y^2$ are processes such that ...
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3k views

what does `ensemble average` mean?

I'm studying this paper and somewhere in the conclusion part is written: "Since this rotation of the coherency matrix is carried out based on the ensemble average of polarimetric scattering ...
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25 views

Integrating over random boundary

What are some correct stochastic integral notions or theories which make formal sense of the problem of "integrating a function over the boundary of random domain"?
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35 views

Gaussian process via RKHS construction: joint measurability comes for free?

Billingsley's "Probability and Measure" (and other books) show the joint measurability of the Brownian motion using the continuity of paths. Makes me wonder if we can say the joint measurability ...
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39 views
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Reformulate a SPDE parameterized by space and time as an SDE parameterized by time (as it is possible for PDEs)

Let $d\in\mathbb N$ $\mathcal V_t\subseteq\mathbb R^d$ be a bounded domain for $t\ge 0$ $\Phi_t:\mathcal V_0\to\mathcal V_t$ be bijective for $t\ge 0$ with $$\Phi(\;\cdot\;,x_0)\in C^1\left(\mathbb ...
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What is the conditional distribution function of the moment when the first event happens when the total number of event is given?

Suppose books are getting missing from a library as a Poisson process $N(t)$ with intensity $\lambda$. Suppose we know that at the moment $t$, $N(t) = n$. What is the conditional distribution ...
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A Stochastic Integral Inequality

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ Is there a pair $(\mu,\sigma)$ such that ...
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20 views

Are sojourn times independent in non homogeneous poisson process?

We know that in homogeneous Poisson process, sojourn times are independent. But are they still independent if $\lambda (t)$ varies? If so, what is the proof? Thanks the problem I have now is that ...
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1answer
390 views

Function of stationary processes

Suppose we have stationary processes $X_1(t), X_2(t),..., X_n(t)$ and let $f_t(X_1(t), X_2(t),..., X_n(t))$ be a continuous function of these stationary processes. Will $f_t(\cdot)$ also be stationary ...
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1answer
80 views

What does a steady state vector tell us if the Markov chain is irregular?

When a Markov chain is regular, the finding the steady state vector (i.e. the eigenvector corresponding to the eigenvalue $1$) will tell us the long term probability of ending up in any of the states, ...
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1answer
19 views

Intuition of Doob-Meyer decomposition ( case of totally inaccessible jumps)

I try to understand Theorem 10 on page 107 of Protter's Stochastic integration and differential equations. The proof is really long, and for now, I just want to get an intuition. Here is the theorem: ...
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1answer
27 views

Lévy-process property

I get a problem that comes up in the construction of the Lévy-Itõ decomposition. For a Lévy-process $X$ there is a independently scattered poisson random measure $N$, such that for each t, and for ...
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27 views

Information in Filtrations

Is the “information” kept track of by filtrations the same as information-theoretic “information”? If not, is there some way the two concepts can be reconciled?
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148 views

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...