Tagged Questions

3
votes
1answer
65 views

Optimal probability measure

Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
1
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2answers
78 views

How will studying “stochastic process” help me as mathematician??

I wish to decide if I should take a course called "INTRODUCTION TO STOCHASTIC PROCESSES" which will be held next semester in my University. I can make an un-educated guess that stochastic processes ...
-3
votes
1answer
121 views

Reference Request: Video Lectures for Stochastic Processes

It is difficult to learn Stochastic Process by self-reading. Can you provide some video lectures on Stochastic Process?
1
vote
1answer
149 views

What is some books at the level which including this inequality and its proof?

I always wanting to looking into harder random variable/probability/stochastic process/statistics books that are harder than the intro one and have multiple random variable but easy enough to have ...
2
votes
1answer
90 views

Books at similar levels as Kallenberg' Foundation of Modern Probability?

Thanks to many people who have mentioned it to me and others on this site before. I was just able to peek into Kallenberg' Foundation of Modern Probability. It is more comprehensive, deep and thorough ...
0
votes
0answers
37 views

Good book on (linear and nonlinear) continuous time stochastic filtering

I'm trying to learn about stochastic filtering at the moment, but I'm at breaking point with the textbook I've been using up till now (Fundamentals of Stochastic Filtering), as a great deal of the ...
3
votes
1answer
71 views

Stochastic geometry, point processes online lecture

Does any of you know where to find online lecture/podcast introducing stochastic geometry and/or point processes? Thank you! Riccardo
1
vote
3answers
337 views

What is more elementary than: Introduction to Stochastic Processes by Lawler

I have trouble to reading this book! What book is more elementary/preliminary than this book: Introduction to Stochastic Processes by Lawler
3
votes
2answers
130 views

Generalization of a product measure

Let $(X,\mathfrak B(X))$ and $(Y,\mathfrak B(Y))$ be measurable spaces and further let $\mu$ be a measure on $\mathfrak B(X)$ and let $K$ be a kernel, i.e. for any $x\in X$ we have $K_x$ is a measure ...
1
vote
0answers
38 views

Efficient random number generation for sojourn times in semi-Markov processes

I'm doing a self-study of semi-Markov processes and was wondering if there are efficient methods for generating random numbers for sojourn times. For example, generating a bunch of random numbers from ...
1
vote
0answers
57 views

general semimartingale theory

Last semester I attended a course about stochastic calculus. There we constructed the stochastic integral with respect to continuous semimartingales. We restrict ourselves to the continuous case. ...
0
votes
0answers
25 views

Discretized Jump-Diffusion Model

Disclaimer: I'm not sure if this is a valid question for this site, if it isn't please disregard and delete. I'm looking for any literature on a discrete time jump-diffusion model that is ...
1
vote
0answers
51 views

Existence and Unicity Weak, Strong, Pathwise, In Law, etc… for SDE's

I feel always confused with weak, strong, pathwise unicity and or existence for Stochastic Differential Equations. It is mainly my own fault and I should do something about it. But it would be much ...
1
vote
1answer
105 views

Good substitutes for Ross's book on Probability Models

I was wondering if there are any FREE good alternatives to Sheldon Ross's Probability Models which are more succinct? Are there any free online resources (websites/PDFs/course notes) which cover more ...
0
votes
0answers
47 views

Process that is not harmonizable

Harmonizable processes include stationary time series, periodically correlated processes, and some transient time series. Zero-mean harmonizable processes also have an autocovariance function that ...
0
votes
1answer
45 views

Name of this discrete stochastic process

Suppose we have $n$ blocks of wood. At each step, we choose one of these boxes uniformly at random and paint it red (so at later steps, we may be re-painting an already-red box). Let $X_t$ denote ...
0
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0answers
127 views

Processus stochastiques et mouvement brownien by Paul Lévy

Does anybody know if there is an English or German translation of the book Processus stochastiques et mouvement brownien by Paul Lévy? If not, can someone recommend a text covering similar contents ...
-2
votes
2answers
95 views

Covariance 's relationship with pure math and probabilty? [closed]

I've been looking up a lot of statistical books and cannot find out mathmatical insight behind it, but my math level wasn't allow me to read the mathmatical statistics books and get the math behind ...
1
vote
1answer
411 views

Questions and Solutions in Brownian Motion and Stochastic Calculus?

I am currently studying Brownian Motion and Stochastic Calculus. I believe the best way to understand any subject well is to do as many questions as possible. Unfortunately, I haven't been able to ...
4
votes
2answers
117 views

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 ...
3
votes
2answers
106 views

Are Ito bridges themselves Ito processes?

Let $X_t$ be an Ito diffusion process with initial condition $X_0 = x_0$. Let $T>0$ we a fixed deterministic time, and consider for $0 \leqslant t < T$ the process $Y_t = X_t| X_T = x_T$. Is ...
1
vote
2answers
266 views

Forms of the Levy-Khintchine formula

I'm writing a survey that involves Levy processes and wanted to mention the different forms of the Levy-Khintchine formula found in literature. The most common version seems to give the Levy symbol ...
0
votes
1answer
752 views

Markov Models simple introduction

We're studying Markov models (still at the basis: transitory states, periodic states, etc..) but the professor isn't very good at teaching and I feel I'm getting lost soon. I'd love to have a simple ...
2
votes
0answers
68 views

Scale invariance and $1/f^2$ power spectrum

In the paper Occlusion Models for Natural Images : A Statistical Study of a Scale-Invariant Dead Leaves Model; Lee, A. B. Mumford, D. B. Huang, J.; International Journal of Computer Vision I read ...
2
votes
1answer
136 views

Reference request for Optimal Stopping (Stochastic Analysis)

I would like to start and get into the habit of reading some publications in different areas of mathematics, to get used to the writing style / mathematical sophistication etc. that is expected. In ...
2
votes
0answers
130 views

Does every continuous time minimal Markov chain have the Feller property?

Consider a Q-matrix on a countable state space. (A Q-matrix is a matrix whose rows sum up to $0$, with nonpositive finite diagonal entries and nonnegative offdiagonal entries). As explained for ...
1
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0answers
45 views

Green kernel of Hunt processes

Let $X$ be a regular Hunt process on $R^+$ with starting at $x$ and $T_y :=\inf\{t>0: X_t=y\}, y\in R^+$. $G_q(\cdot,\cdot), q >0$ denotes Green kernel of the process $X$. We have the following ...
1
vote
0answers
41 views

Necessary condition(s) for transforms of Markov diffusion to stay Markov diffusions

I feel it always necessitates a certain amount of work before reaching the conclusion that the transform of a diffusion by a function is not a Markovian diffusion. I was wondering if there were ...
5
votes
0answers
146 views

Potential theory: discrete-time Markov processes

Recently I've found lecture notes on "Analysis on Graphs" where the potential theory methods were used to study discrete-time, time-reversible Markov chains (i.e. the state space is countable). ...
3
votes
2answers
133 views

Determining distribution of $X_t = \int_0^t W_s^2 \mathrm{d} s$

Premise Let $W_t$ be the standard Wiener process, and let $X_t = \int_0^t W_s^2 \mathrm{d} s$. I am interested in determining the distribution of $X_t$. What I did My line of attack has been to ...
3
votes
1answer
331 views

Exercises on stochastic calculus

Where I can find exercises on stochastic calculus (stochastic integration, SDE)? I'm mostly interested in medium difficulty exercises, preferably not proof-oriented, ideally with solutions. I've ...
1
vote
0answers
49 views

Is this some type of convergence of stochastic processes?

I have learned that there are several types of convergence of random variables and they are related somehow, but not yet for stochastic processes. I was wondering if the following is some type of ...
3
votes
2answers
468 views

First time passage decomposition for continuous time Markov chain

For discrete time finite Markov chain, the first passage time $T_j$ to visit state $j$, is determined from the recurrence equation: $$ p^{(n)}_{ij} = \sum_{k=0}^n f_{ij}^{(k)} p^{(n-k)}_{jj} = ...
3
votes
2answers
114 views

Is $M^2-[M]$ a local martingale when $M$ is a local martingale?

I've learned that for each continuous local martingale $M$, there's a unique continuous adapted non-decreasing process $[M]$ such that $M^2-[M]$ is a continuous local martingale. For a local ...
1
vote
0answers
72 views

Second eigenvalue of a stochastic block matrix

Considering a stochastic block matrix in the form of, \begin{equation} \textbf{$P_{}$} = \left( {\begin{array}{cc} \textbf{$A_{}$} & \textbf{$B_{}$}~; \ \textbf{$B_{}$} & \textbf{$A_{}$} ...
3
votes
2answers
228 views

Brownian motion introduction

I didn't get any answers to my previous question; so I am trying a different tack. I am familiar with a first course in probability theory using measure theory, to the extent of proving the Central ...
1
vote
2answers
120 views

Central limit theorems, Almost sure invariance principles and Brownian motion

In a paper I was reading on dynamics, I came across a proof of a central limit theorem in a certain situation using brownian motion and an almost sure invariance principle. I am not very experienced ...
7
votes
1answer
205 views

PDE - Feynman-Kac vs. finite difference methods

I've heard that in greater than three dimensions, it's more efficient to solve a second-order parabolic PDE using a Monte-Carlo method based on the Feynman-Kac formula that it is to use finite ...
8
votes
4answers
240 views

Roadmap to SPDEs

I'm trying to learn about the Kushner-Stratonovich-Pardoux equations in filtering theory. I'm familiar with Itô calculus at the level of Øksendal's book (but struggle with much of Karatzas and ...
1
vote
0answers
278 views

Books by Ross and Karlin on stochastic processes [closed]

Ross, S.M. has written some books on stochastic processes: Introduction to Probability Models (Side question: does "probability models" always mean stochastic processes?) Stochastic Processes ...
5
votes
2answers
407 views

Joint moments of Brownian motion

My approach to this SE question uses the following joint moments of Brownian motion. For $n=1,2$ they are obvious and well-known, the others are not terribly hard to work out. Is there a reference ...
7
votes
2answers
794 views

Nice references on Markov chains/processes?

I am currently learning about Markov chains and Markov processes, as part of my study on stochastic processes. I feel there are so many properties about Markov chain, but the book that I have makes ...
5
votes
1answer
478 views

Recommendation on stochastic process books

I was wondering if someone could recommend good books on stochastic processes with measure theory treatment with not much or no measure theory treatment for each, it would be nice to have some ...
13
votes
5answers
1k views

Good introductory book for Markov processes

Which is a good introductory book for Markov chains and Markov processes? Thank you.