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Questions tagged [stochastic-processes]

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 does the solution $U∗z(t)$ work for renewal equation?

This post discusses Example 3.5.4 in Resnick's book on page 203. However, when I try to understand the next example 3.5.5, I still have some doubts. Example 3.5.5 If $F(dx) = xe^{-x}dx$, then we ...
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Subscript on Filtration of a Probability Space

I have encountered a notation from Delia Coculescu's paper entitled "A Default System with Overspilling Contagion". It is this: \begin{equation} X_{t+} \end{equation} which is a filtration. What does ...
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Are Ito integrals equivalence classes or conrete random variables?

Technically speaking, are Ito integrals of stochastic processes $S$ with respect to Brownian motion $B$ $$ \int_{0}^{T} S_s dB_s $$ random variables or equivalence classes of almost surely equal ...
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Distribution function of the process

Let $N$ is homogeneous Poisson process with $λ > 0$. $$ A(t) = t − T_{N(t)} , t > 0,$$ is the age process of restoration moment, $$B(t) = T_{N(t)+1} − t, t > 0,$$ is the moment of rest of ...
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Cramér-Lundberg model for on-demand insurance

I am looking for inspiration and perhaps guidance on the following as I’ve been stuck for a while now: Context: I am working on a practically oriented project to adjust the Cramér-Lundberg model ...
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Can a stochastic process be neither adapted to filtration nor previsible?

The idea behind the question arises from my intuition about the concepts of 'adapted to filtration' and 'previsbility'. If a process is adapted, it essentially means that the evolution of the ...
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Simulating a continuous-time, discrete-state Markov chain in fixed time step

To simulate a continuous-time, discrete-state Markov chain with known transition probabilities, we can generate exponentially-distributed waiting time according to current total transition rate, and ...
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Proving a that the arrival times of a Poisson process is uniformly distributed

I am stuck on this question for a very long time and I can't figure out the solution: "Suppose $N_t$, $t\ge0$, is a Poisson process with rate $\lambda$ and $N_t = 0$. $T_0$ and $T_1$ are the first ...
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Are we allowed to modify the order of the states within a Markov chain?

For instance, if I were to switch the states such that the rows are in the order (0,1,2,3) but the columns are (1,2,0,3), can we still do the relevant analysis? By observation, I see that the row sum ...
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Conditional Expectation with Respect to an Initial Condition

$\require{begingroup}\begingroup\newcommand{\dd}[1]{\,\mathrm{d}#1}$When studying diffusion processes, I often see the notion of expectation conditioned on an initial value. By this I mean the ...
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What does non-degeneracy of of the diffusion coefficient in the context of a SDE mean?

In the introduction of a paper I was reading the author writes without elaborating that " In the case of a non-degenerate diffusion coefficient, Stroock and Varadhan , proved the existence of a ...
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Fokker planck equation, simple change of variables

I’m studying this fokker planck equation $\frac{\partial}{\partial t} p + a \frac{\partial}{\partial x} p = D \frac{\partial^2}{\partial x ^2}p $ where p is a fuction of x and t. a, D constant. Now ...
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Simple process in Itô calculus

For the definition of Itô integral, one uses simple stochastic processes. I have found two definitions for simple stochastic process, given a filtration $(\mathcal{F}_t)_{t\geq0}$, an interval $[0,T]$ ...
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Stochastic process with measurable set of indices

Let $\zeta_t$ be a random variable from probability space $(\Omega, \mathcal F, P)$ to some measurable space $(X, \mathcal X)$. Assume that indices set $T$ for process $\zeta_t$ is also a measurable ...
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Convergence stopping time

If I have a sucession of continuous time stochastic processes, such that $L_t^N \xrightarrow{a.s} L_t$. Where $L_t^N$ is a jump process and $L_t$ is continuous (with respect to t). If $\tau^n = \inf\{...
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Non-markovian random walks and their applications in machine learning

I'm searching applications of random walks in machine learning. In particular, applications of random walks with long memory. An example of this kind of processes is the so called ELEPHANT RANDOM WALK....
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expected value of Poisson process using Laplace transform and deduction of Laplace function of Poisson process

I want to get the expected value of Poisson process using Laplace transform, and also want to know how to derive the Laplace function of Poisson process.
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Definition of a family of probability measures for Ito diffusions

I have a question concerning the definition of a family of probability measures for the solutions to an Ito diffusion $$X_t^x = x + \int_0^tb(X_s^x)ds + \int_0^t \sigma(X_s^x)dB_s$$ as it's given in ...
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Do jumps in Levy processes need to be independent of the process itself?

I have a very basic question about Levy processes. Is the process of the form $$ X_t=\sigma B_t + \sum_{i=1}^{N_t}\eta_i(X_t) $$ a Levy process? Here $B_t$ is a standard Brownian motion and $N_t$ is a ...
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Do finite dimensional distributions determine the law of a stochastic process?

For $i=1,2$, let $\{X_t^i\}_{t\geq 0}$ be $\Bbb R^d$-valued stochastic processes adapted to $\{\mathscr F^i_t\}_{t\geq 0}$ on the probability space $(\Omega^i,\mathscr F^i,\Bbb P^i)$. Suppose the two ...
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How do I evaluate the following combination of random variables? Is it martingale?

I'm about to analyse the following expression $$Z_n:=\prod_{k=1}^n \left(\frac{\frac{Y_k}{\prod_{i=1}^k X_i}}{\sum_{j=1}^k \frac{Y_j}{\prod_{i=1}^j X_i}} \right),$$ where $Y_j$ for all $j\in \mathbb{...
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Gaussian Process| marginal likelihood

Suppose I do gaussian process regression and calculate the log likelihood of observing the samples $y$ as: $\mathrm{log}\, p(y | x, \theta_{\Sigma}, \theta_{\mu}) = - \frac{1}{2} \mathrm{log} |\Sigma|...
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Solving a Stratonovich SDE

I am trying to solve the following Stratonovich SDE $$dN_t=rN_tdt+\gamma N_t\circ dB_t$$ In my notes, the Stratonovich integral is defined as $$\int^t_0 N_s\circ dB_s=\int^t_0 N_sdB_s+\frac{1}{2}\...
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to show how a relation containing summation of products leads to a compact relation using generating function

I have faced with the relation $$I_k(x,t)=\sum_{n=0}^{\infty}x^n\sum_{n_1+\dots+n_k=n}\prod_{j=1}^{k}p(n_j,t)\tag{1}\label{eq1}$$ in some books. In these books, it is said that the above relation ...
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Linking Markov Chain with Renewal Process

GIVEN: $X_0,X_1,...$ irreducible, recurrent Markov chain with transition matrix $P$ Starting state $X_0=x$ $g(m)=P\{X_m=y\}$ for some fixed state $y$ I know that the renewal process is $g(m)=b(m)+\...
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Is this stochastic Picard iterator well-defined?

Preliminaries Let $x_0 \in \mathbb{R}^d$. Let $T \in (0, \infty)$. Let $$ \sigma: \mathbb{R}^d \rightarrow \mathbb{R}^{d \times d}$$ and $$\mu: \mathbb{R}^d \rightarrow \mathbb{R}^{d}$$ be affine ...
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1answer
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Is this random Lebesgue-integral well-defined?

Let $$ X : [0,T] \times \Omega \rightarrow \mathbb{R} $$ be an almost-surely continuous stochastic process. Then how is the random Lebesgue-integral $$ \omega \mapsto \int_{0}^{T} X_t(\omega ) dt \...
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Laplace transform gives a wrong result when finding Renewal function

In page 188 of the book Adventures in Stochastic Processes, it shows that if $F(dx) = xe^{-x}dx$, then the renewal function $U(x)$ will have the following expression $$U(x) = \frac{3}{4}+\frac{x}{2}+\...
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Multidimensional Correlated Geometric Brownian Motion, finding exact form of the matrices

My goal is to understand the dimensions of the matrices involved, so I am initially writing things as column vectors, and defining all the dimensions. I am working with the following setup: ...
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Proving standard Brownian motion [closed]

Suppose X_t is a standard Brownian motion. Assuming that Y_t = a^-(1/2) * X_at and that a > 0. Any advice on how I can start this? The concept of Brownian motion is very confusing to me. Thanks!
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Stochastic Processes: Independent Poisson RVs and Limits

This is a question from my Stochastic Processes class that I'm having a hard time figuring out. Does anyone know how to solve? Let $X_{n1},....,X_{nn}$ be independent Poisson random variables with ...
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Prove periodicity is a class property

Prove that if state $i$ in a class has period $p$ then all states in that class have period $p$. The proof is given on this answer is this: One way to define the period of state $i$ is as the ...
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Closed form solution for the difference of two poisson processes

I'm interested in whether there is a closed-form distribution of the time it takes two Poisson processes to output counts to have a fixed difference. For example, let $k_1$ ~ Poisson($\lambda_1t$) $...
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Markov Chain upper bound on the probability of hitting time

I encountered the following problem. $\{x_t\}$: Markov chain in discrete time; $\Omega$: a finite state space s.t. $|\Omega|=n<\infty$; $\tau_w\equiv\min\{t\ge 0\,|\,x_t=w\}$, $w\in\Omega$ (first ...
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Proof finite dimensional equaltiy implies equality on interval

Consider the stochastic process $\{X(t)\}_{t\in T}$ and certain specific operations $g$ and $f$. If $T$ is of finite dimension, i.e. $T=\{t_1,\ldots,t_k\}$, with $t_i\in 0,\infty)$ then I have been ...
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$3$ scorpions are chasing $1$ ant on the edges of a cube. The ant is $3$ times faster than any scorpion. Can the ant survive?

The problem: Three scorpions are chasing a single ant on the edgegraph of a cube. The scorpions have the same speed ($v$), while the ant is $3$ times faster ($3v$). They can move in any direction and ...
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Understand better stochastic integral through a.s. convergence

I know that $$\int_0^T f(B_t, t)dB_t=\lim_{n\to \infty }\sum_{i=1}^n f(B_{t_i^{(n)}},t_i^{(n)})(B_{t_{i+1}^{(n)}}-B_{t_i^{(n)}}),\quad \text{in }L^2,$$ where $\{t_i^{(n)}\}_{i=1}^n$ is a sequence of ...
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A stopped process is adapted

I am trying to understand the proof of Theorem 2.2.2(Optional Stopping Theorem) in Fleming and Harrington's Counting Processes and Survival Analysis. Let $\{X(t):0\leq t<\infty \}$ be a right-...
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1answer
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Reference : Proving something is a Brownian Motion

Given a (standard) Brownian Motion $W_t$ if we do some sort of scaling, inversion or reversal then we also get a Brownian motion. I have seen proofs but the proofs only seem to rely on showing ...
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Independence between Brownian motion and hitting time.

If given a standard one dimensional Brownian motion $B_t$ and stopping time $T = \inf\{ t : B_t = |a|, a \in \mathbb{R}\}$ We will have independence between $B_T$ and $T$ as $P(B_T = a) = \frac{1}{2} ...
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Definition of transient state

Consider the following definition Transient States It is often useful to talk about whether a process entering a state will ever return to this state. Here is one possibility. A state ...
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Compute moments of Brownian motion stopped at exit time of $[a,b]$

Given $B_t$ a standard brownian motion and $a < 0 < b$ Set $T = \inf\{ t : B_t = a \vee B_t = b\} $ For any $\alpha \in \mathbb{Z}^+$, find $EB_T^\alpha$. I know I can use optional ...
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Understanding definition of Periodicity of Markov chain

Consider the following example that is used to understand the definition of periodicity property. Why does it says that: starting in state $1$, it is possible for the process to enter ...
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Reference for simple Markov chain construction

Let $M_n$ , $n\in\mathbb{N}_0$ be a Markov chain on a general state space $X$. Fix $m\in \mathbb{N}$. ¨ My question is if there's a name / reference for this trivial Markov chain on $X^m$ defined by ...
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Looking for scientific papers about lotteries

Do you know any paper focusing on the statistical science and math behind lottery and number guessing games? I am creating a new one, where the winning numbers are known in advance, and the gain is ...
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Birth/Death Processes with constant departure rate

So, I'm looking at a Birth/Death process $Z$ with an arrival rate of $\frac{1}{n+1}$ and a departure rate of $1$. I'm trying to show that this process is positive recurrent and find the stationary ...
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Why does the unordered arrivals in Poisson process iid uniform when conditioned by $N(t)=k$?

Let $N(t)$ be the number of arrivals in the Poisson process of rate $\lambda$. I already know that the 'ordered' arrival times are uniformly distributed on the region $0<t_1<t_2<\cdots<...
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KL divergence contraction coefficient - basic question

I'm studying the Blahut Arimoto algorithm using these notes and towards the end of section 6, an interesting quantity arises. The author does not talk about how to compute it so I was hoping I could ...
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covariance of two stochastic integrals

I'm trying to evaluate the covariance between two stochastic integrals such as $$Cov(\int_0^t g_udW_u, \int_0^t h_udW_u) = \int_0^t E[g_uh_u]du $$So I am trying to prove this and I thought I would do ...
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Continuous-time Martingale and Brownian Motion Supremums

I am reading through Le Gall's book on Brownian Motion, Martingales, and Stochastic Calculus. I just read through the chapter on optional stopping of martingales, but I cannot solve the first exercise:...