A stochastic process describes evolution of random variables/systems over time and/or space and/or any other index set. It has applications in areas such as econometrics, weather, signal processing, etc. Examples - Gaussian process, Markov Process, etc.

learn more… | top users | synonyms

0
votes
0answers
5 views

markov chain - proability question

Transition matrix has been written like that; $$\mathcal P = \begin{bmatrix} 1/3 & 0 & 2/3 \\ 1/3 & 1/3 & 1/3 \\ 0 & 0 & 1 \end{bmatrix}$$ the initial vector is that ...
1
vote
1answer
14 views

Calibrating irregularly sampled Ornstein-Uhlenbeck process

I would like to recover the parameters of a Ornstein-Uhlenbeck process from observations that are irregularly spaced. Estimation via linear regression and maximum likelihood is demonstrated here for ...
0
votes
0answers
14 views

How to show that $\pi^{X(n)}_{o}$ for $n \geq 0$ and $X(n)$ a branching process, is a martingale?

If I let $X(n)$ be defined as the size of a branching process at the $n$th generation, and $\pi_{o}$ as the probability that the process will eventually go extinct, I'd like to show that ...
0
votes
0answers
14 views

Reflection principle question [on hold]

Question: $$P(X_1\gt 0, ..., X_n\gt 0, X_n=a-b)=?$$ Its Answer: $= (1,1) \rightarrow (n,a-b) $ that meet neither touch nor cross paths. $=[(1,1) \rightarrow (n,a-b) \ \ \text{all ...
2
votes
3answers
50 views

How to show $M_n = X_n^2-n$ is a martingale?

Let $X_n, n = 0, 1, 2, . . .$ denote an unbiased Normal Random Walk. $X_0 = 10$, and $X_{n+1} = X_n + Y_{n+1}$, with $\{Y_n\}$ are i.i.d. $N(0, 1)$. Then how can I show that: A) $M_n = X_n^2-n$ is a ...
0
votes
0answers
430 views

please clarify the solution? [closed]

I'm studying Problem5.3 and its solution. However, its solution is not clear for me. Please explanatorily show this answer . I need to learn such type of questions. Please help me. Thank you.
0
votes
1answer
49 views

How can I find the expected time until a random variable is greater than some constant?

I have random variables $X_1, X_2, X_3, ....$ that are i.i.d. with the same distribution F. If I define $k$ to be a constant and $T$ to be the time until any $X_i$ is greater than $k$, what would be ...
0
votes
0answers
36 views

Does a renewal process have the Markov property?

I have seen three proofs for the renewal equation as used in studying renewal processes. The one that has bothered me the most is the one on wikipedia. It states the following: Wikipedia The ...
2
votes
0answers
36 views

Does a stationary process necessarily have to be mean-reverting?

I wonder about if a stationary process is by definition mean-reverting too. I know the formal definition of a stationary process, but I'm not sure about the definition of a mean-reverting process. ...
0
votes
0answers
20 views

Limiting distribution of a Markov chain?

I have the problem below. There are n identical machines. They are all operational at time 0. The lifetime of each one is an exponential random variable with rate L. There are r repairmen (1 ≤ r ≤ ...
0
votes
0answers
18 views

2-dimensional density of Brownian bridge?

I know that a $1$-dimensional Brownian bridge $B(t)$ just follows a normal distribution with mean $0$ and variance $t(1-t)$. But how do I compute the 2-dimensional density? I mean, $\{B(s), B(t)\}$ ...
0
votes
0answers
13 views

Improvement of Minimum description length (MDL) estimate

I earnestly request apology if this question is inappropriate for the forum. The question has two parts one technical and the other is not technical. I would appreciate any response. Let me consider ...
-1
votes
1answer
20 views

Proof of Markov Chain property

Suppose that $X_n$ is a Markov Chain.Then for $m,n \in N$ such that $m<n$ $Pr[X_n=j_n|X_m=j_m,X_{m-1}=j_{m-1},...=X_0=j_0]=Pr[X_n=j_n|X_m=j_m]$ When proving for n=3,m=1 case we have to show ...
3
votes
2answers
32 views

Finding $b$ such that $e^{5B_t - bt}$ is a martingale

I have $X_t = e^{5B_t}$ and Where $B_t$ is brownian motion at time $t$. $M_t = X_t \cdot e^{-bt}$ I need to find a value for $b$ such that $M_t$ is a martingale. I am encountering difficulty, ...
0
votes
0answers
22 views

Finding the best predictor Brownian motion

I want to find the best predictor of $(B_3-B_2)(B_4-B_{\pi})$ given an observation of $B_1$ Where $B_t$ is brownian motion for time $t \geq 0$. I am not sure how to approach this. I know it will be ...
0
votes
0answers
81 views

Show $∫_0^t X(t,s)dB(s)$ is a Gaussian random variable $Y(t)$ [duplicate]

Show that if $X(t)$ is non-random (does not depend on $B(t)$) and is a function of $t$ and $s$ with $\int_0^t X^2(t,s)ds<\infty$, then $\int_0^t X(t,s) dB(s)$ is a Gaussian random variable $Y(t)$. ...
1
vote
0answers
59 views

Fit and evaluate a second order transition matrix (Markov Process) in R?

I already built 1 first order discrete state Markov Chain model. It was built with R using the function 'markovchainFit()' in ...
3
votes
0answers
58 views

simulating birth death process with random numbers from negative binomial

I am trying to generate random deviates for the population size at time $t$ for a birth-death process with constant birth and death rates per individual and initial size $N_0 \gt 0$. For the simple ...
2
votes
0answers
69 views

a question on 0-1 valued stochastic process [closed]

Consider a stochastic process $X_{t}$ taking values in the set $\{0,1\}$ according to the probability measure $\mu$. Let $$Y_{t} = \mu\left(\limsup_{T \rightarrow \infty}\frac{1}{T}\sum_{t = 0}^{T - ...
0
votes
1answer
12 views

Birth & Death process - Combining Transition rates

I think I'm missing a fundamental step in regards to how to combine two exponential distributions in the context of this problem. If we have a birth and death process where birth rate ~ ...
3
votes
3answers
88 views

Stochastic Differential Equations - A Few General Questions

I just have a few questions about stochastic differential equations. I generally did a lot of pure math but signed up for a course on probability models and stochastic differential equations because I ...
1
vote
1answer
56 views

How is $P[X_t\le x_t | X_1,\ldots, X_{t-1}]=P[X_t\le x_t]$ when $X_t\sim WN(0,\sigma^2)$?

In this slide , p.30 , p.31 , it is written that : White noise : $X_t\sim WN(0,\sigma^2)$ i.e., ${\{X_t}\}$ uncorrelated, $\mathbb E[X_t]=0, \mathbb V[X_t] =\sigma^2$ Example : i.i.d noise : ...
11
votes
1answer
169 views

What are the main differences between Granger's and Pearl's causality frameworks?

Recently, I ran across several papers and online resources that mention Granger causality. Brief browsing through the corresponding Wikipedia article left me with the impression that this term refers ...
0
votes
1answer
55 views

writing down markov chain transition matrix

Question: An experimental animal can stay in room-A until 1 minute,and it can stay in room-B until 2 minutes. There exist deadly gases in room-C. One room among these three rooms is being randomly ...
5
votes
0answers
113 views

A question related to Borel-Cantelli Lemma

Note: Borel-Cantelli Lemma says that $$\sum_{n=1}^\infty P(A_n) \lt \infty \Rightarrow P(\lim\sup A_n)=0$$ $$\sum_{n=1}^\infty P(A_n) =\infty \textrm{ and } ...
4
votes
2answers
38 views

Definition of $X_t$ in the context of Stochastic process and Time Series

In the book An Introduction to Stochastic Modeling , Stochastic process is defined as : A stochastic process is a family of random variable(s) , $X_t$ , where $t$ is a parameter running over a ...
0
votes
0answers
39 views

Detrending or Differencing in order to make a series stationary?

I got several time series for which I want to find out if they are stationary or not. So I computed for each series the kpss.test(). But before making further ...
0
votes
0answers
14 views

Class of semimartingales for which all characteristics can be estimated?

I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great. An Ito semimartingale is a martingale for which the ...
2
votes
1answer
35 views

Proof of Chapman Kolmogorov equation

In the proof of Chapman Kolmogorov Equation $p_{ij}^{(m+n)}=\sum_{k=0}^{\infty}p_{ik}^{(n)}p_{kj}^{(m)}$ Proof: $p_{ij}^{(m+n)}=P[X_{m+n}=j|X_0=i]$ By the total probability it says ...
3
votes
1answer
57 views

How can I calculate this probability: $P(W_1<cW_2$) and $c\geq 0$?

Let $(W_t)_{t\geq 0}, $ be a Brownian motion. I want to calculate the following: $P(W_1<cW_2$) and $c\geq 0$ For $c=1$ it is easy. I just write it as an increment, but how can I do it when $c$ ...
1
vote
1answer
43 views

Frequency distribution of Chinese Restaurant Process?

Set-up I was simulating the Generalized Chinese Restaurant Process as shown on the wikipedia page [link] with a discount, $\alpha$, and concentration parameter $\theta$ For $n=5$ total customers ...
0
votes
0answers
29 views

Bayesian Ridge vs Stochastic Gradient Descent

I was running some Regression algorithms on a dataset and it just so happens, that the Bayesian ridge Regression techniques is performing not so well as the SGD (Stochastic Gradient Descent) ...
1
vote
0answers
35 views

Reorder point with stochastic lead time and demand

I'm trying to determine the optimal reorder point for some products. The reorder point must be greater than the demand during lead time a % of the times that I should determine, let's say 95%. ...
3
votes
1answer
51 views

Distribution of stochastic integral

I would like to find the distributions of the following random variables: $Z_k= \frac{1}{\pi} \int^{2\pi}_{0} cos(kt) dW_t$ $k=1,2,...$ and $(W_t)_{t\geq 0}$ is a Wiener process. What is the ...
0
votes
0answers
30 views

Best textbooks on Non-Homogeneous Stochastic Processes?

just wanted to know which are in your opinion some of the best available books on theory and applications of NH Poisson Stochastic Processes, and Non-Poisson processes out there. I've studied Parzen ...
0
votes
0answers
32 views

How to use R to get drift rate and volatility rate of stock prices changes?

I am doing a research on the historical annual stock prices changes, where I have about 30 rows of annual stock prices. How can I use R to get the drift and volatility rate?
0
votes
0answers
27 views

How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...
2
votes
0answers
55 views

Time Index of Lévy Process

Consider (for all $t\geq 0$) a linear time transformation function $\nu(t)=at+b$ with the following properties: $\nu(0)=-1$ $\nu(t)$ is an increasing function of the time index $t$ i.e. $a>0$. ...
4
votes
0answers
62 views

emails arriving in a Poisson process

Emails arrive according to a Poisson process with rate $λ=2/hour$. You check your inbox (instantly reading all new emails) at time $t=5$ hours and also at some uniformly distributed random time ...
3
votes
0answers
33 views

Electrical components failing in a Poisson process

A machine has infinitely many identical components. They fail according to a Poisson process with rate λ = 4/hour. A repairman arrives at time t and instantly repairs all of the broken components, but ...
1
vote
0answers
25 views

On sampling from all the continuous functions in [0,1]

I was watching this youtube video for motivation on measure theory and at the 7:20 mark a reason for utilizing measure theory is given; to be able to pick a random function from all the continuous ...
2
votes
2answers
312 views

Why doesn't my Wiener process simulation work?

The Wiener process at time $t=0$ is $0$. It has independent increments, so $W_t-W_0 \sim N(0,t-0)$, but wouldn't it mean that $W_t\sim N(0,t)$ for every $t$? But if I try to simulate a a ...
0
votes
0answers
40 views

Multivariate stochastic time series forecasting

I have a multivariate time series like this ...
0
votes
0answers
10 views

A good read to recommend on stochastic processes?

can you recommend a good read, ideally up-to-date, for stochastic processes? I am not afraid of math, all I appreciate is the fluency of materials. I've read about Dirichlet/Pitman-Yor/Gaussian ...
0
votes
0answers
9 views

Simulating an ODE model with non-constant parameter

I have a model, I can formulate the model using ordinary differential equation with parameter $P$. I want to simulate the model, but instead of using a fixed constant $P$ for the parameter, I want to ...
0
votes
1answer
83 views

Mathematical Modeling and Statistical Modeling

What is the difference between mathematical modeling and statistical modeling? I only know that a mathematical model is deterministic while a statistical model is stochastic. Is that all to answer ...
2
votes
1answer
48 views

Are linear processes stationary?

I am reading Soren Johansen's book on cointegration and I'm wonder about the following definition: Definition 3.1. A linear process is defined by $Y_t=\sum_{i=0}^\infty C_i\epsilon_{t-i}$, $t=0, ...
1
vote
0answers
19 views

On Kolmogorov's Theorem In Time series theory and methods (1990)

I am following Time series theory and methods, Brokwell and Davis (1990). And theorem 1.2.1 called by the text Kolmogorov's Theorem is only stated but not proven. I will rewrite it here: The ...
1
vote
0answers
17 views

Markovchains number of passages in a set

$\eta_A$ is the number of passages of a markovchain $(X_n)$ on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ in a set $A$ and $P_x(\eta_A = \infty)$ is the probability of visiting $A$ infinite often. ...
0
votes
0answers
16 views

General state space

is there a clear definition for a "general state space" in the sense of Markovchains ? Is for example $\mathbb{N}$ a general state space because it is countable infinity?