2
votes
0answers
31 views

A Lemma in the book “ Mathematical Method for financial markets” (Chapter 5, Section 5.7)

In page 307, Section 5.7, Chapter 5 of the book "mathematical methods for financial markets" by Jeanblanc, Yor and Chesney, Lemma 5.7.1 is given as follows: Lemma 5.7.1.1 Let $W$ be a Brownian ...
7
votes
2answers
173 views

A planar Brownian motion has area zero

I'm looking for proofs of Paul Lévy's theorem that a planar Brownian motion has Lebesgue measure $0$. I know of only two proofs: one is in Lévy's original paper (Théorème 12, p. 532) and the other is ...
0
votes
0answers
23 views

Learning resources for Probability Distributions/Models

I've a good background in basic probability. I need to learn and get a good grip on the probability distributions and stochastic processes, counting processes, and other related topics. I am already ...
3
votes
1answer
58 views

When are stable continuous time Markov chains Feller? Always?

This is a question is similar to this 2 year-old one that never got answered (truthfully it's pretty much the same question except that I'm adding a bit more detail and the assumption that the $Q$ ...
0
votes
1answer
37 views

Book suggestions alongside Adventures in Stochastic Processes by Resnick

I am currently taking a SP course following Resnick's book. Are there any other books with exercises (and possibly solutions) I could also look at?
0
votes
0answers
27 views

optimization problem in mathmetical finance using convex duality

I'm interested in the application of stochastic processes and stochastic calculus in mathematical finance. In my lecture I often see a certain optimization problem usually of a convex function. ...
0
votes
0answers
16 views

Estimating the magnitude of a change in a non-stationary stochastic process

In this paper by Adams and MacKay, they present an algorithm for the online detection of change-points in a stochastic process subject to some hypotheses. Their algorithm gives both the predictive ...
1
vote
1answer
38 views

References for numerical stochastic differential equations

I am currently working on a topic in physics which requires me to solve stochastic differential equations (specifically stoch. Schrödinger equation). I am a physicist and have not had any ...
0
votes
1answer
86 views

“Multivariate” Markov Chains

I am interested in estimating regime-switching VAR models to a regime setup I don't know the name of. I am hoping that someone can help me out with some references, or if there exists a name for it ...
1
vote
1answer
35 views

What is a good book for multivariate stochastic processes?

I'm looking for a good book that introduces (preferred without measure-theoretic proofs though that may have to do) multivariate stochastic processes. So suppose you have $\{\mathbf{X}_n : n \in ...
2
votes
4answers
33 views

Reference request for stochastic process and applications

I am looking for a text book that will cover the following topics I hope someone could suggest me a good text book that will provide me a good guidance regarding the following; Generating functions, ...
1
vote
0answers
20 views

What is the name of this type of stochastic processes?

I've seen someone briefly define a continuous time stochastic process $X$ on $\mathbb{N}$ as the (a?) solution to $$X(t)=X(0)+Y\left(\int_0^t f(X(s))ds\right)$$ where $Y$ is an inhomogeneous ...
1
vote
1answer
28 views

Introduction to stochastic control

I'm looking for an introductory text on stochastic control. Any suggestions?
2
votes
1answer
110 views

What are some good books about martingales?

I'm looking for suggestions for well written books dealing with martingale theory, not necessarily exclusively. I'm also looking for a nice compilation of problems, preferably with answers, on this ...
2
votes
1answer
70 views

Text on Probability Theory applied to Actuarial Science

I am a senior undergraduate who has passed the first three actuarial exams on probability (P), financial mathematics (FM), and models for financial economics (MFE). I am working on passing the life ...
3
votes
1answer
52 views

A question about extensions of Markov semigroups

I've cross-posted this to MO, if a reply appears on that post I'll update this one. Suppose that $\{T(t)\}_{t\geq 0}$ is a Markov semigroup on the space of continuous bounded functions defined on ...
3
votes
0answers
47 views

Absolute continuity of quadratic variation of continuous local martingales

I am interested to know if there are any simple sufficient conditions on continuous local martingale to have absolutely continuous quadratic variation. In general , we know only that quadratic ...
1
vote
1answer
49 views

Strong Markov property given transition functions

Suppose we are given family of transition functions satisfying Chapman-Kolmogorov equation, what conditions will ensure that there exists a continuous or cadlag Markov process with given transition ...
1
vote
1answer
50 views

Fake Brownian Motion

Does there exist a martingale which has Marginal distributions same as Brownian Motion marginals but the process itself not being Brownian motion? Any references are highly appreciated. Thanks.
1
vote
1answer
49 views

Do densities of invariant distributions satisfy the Fokker Planck equation?

Suppose that $\{X_t\}_{t\in[0,\infty)}$ is a $\mathbb{R}^n$ valued homogenous diffusion process with drift vector $b$ and diffusion matrix $A$. Is it ever true that if the process has an invariant ...
3
votes
1answer
128 views

Reference on Doob's h-transform

I am searching for a reference about conditioning a Markov process in the sense of Doob, i.e. using h-transforms. My particular concern is to condition a discrete-time Markov Process on a possibly ...
0
votes
0answers
12 views

Reference request: The use Lyapunov-type functions in the analysis of diffusion processes

I've been told that there exists a series of Lyapunov type results for diffusion processes that are used to establish things like the existence and uniqueness of solutions, existence of invariant ...
1
vote
3answers
114 views

Which is a good textbook on stochastic processes which takes measure theoretic approach?

I was looking for an intermediate-advanced textbook on stochastic process. I have graduate level probability knowledge.
1
vote
1answer
37 views

Stochastic processes with non-zero higher order variations

I'm under the impression that how non-zero quadratic variation of the Brownian motion results in Itō's lemma or in general, the creation of the Itō's calculus. I'm also aware that stochastic integral ...
0
votes
0answers
20 views

Establishing recurrence and positive recurrence of Markov processes via “barriers”?

I've been reading the book by Wentzell and Freidlin on dynamical systems with small random perturbations. On page 42 it's stated: It is possible to give stronger conditions for recurrence and ...
1
vote
1answer
59 views

Transition function is a Markov semigroup?

How does the transition function in a Markov process become a Markov semigroup in time homogeneous Markov processes? Thanks a lot.
4
votes
1answer
99 views

Kolmogorov continuity theorem for Banach space valued random processes

I am interested in the Kolmogorov continuity theorem. I would like to know if this theorem holds for Banach space valued random processes (probably separable Banach space). I cannot find a paper or a ...
1
vote
0answers
87 views

A query on Palm Khintchine Theorem's proof

I was searching for a good reference on Palm Khintchine theorem proof. When I googled it, I got the following reference (as a Google book) here. It states that a superposition of independent "low ...
2
votes
0answers
164 views

Applying a linear operator to a Gaussian Process results in a Gaussian Process: Proof

In this paper, it is stated without proof or citation that "Differentiation is a linear operation, so the derivative of a Gaussian process remains a Gaussian process". Intuitively, this seems ...
2
votes
0answers
195 views

The most fundamental papers in stochastic analysis

I have soft a question. What papers will be good to on start and allow me to make little step into research, without harm for reader. I am interested in an stochastic analysis. I am looking for ...
2
votes
0answers
55 views

almost sure convergence of sums of triangular arrays

A well known result (see for example Kallenberg Theorem 4.17) is that if $x_j$ are symmetric independent random variables, then the following are equivalent: i)$\sum x_j<\infty$ almost surely; ...
2
votes
1answer
71 views

Reference request for the law of the stopping time in the gambler's ruin problem

Suppose we have a sequence of independent and identically distributed random variables $(X_n)_{n\ge 1}$ such that $$ P(X_n=1)=p,\quad P(X_n=0)=r,\quad P(X_n=-1)=q $$ with $p,q,r\in[0,1]$, $p+q+r=1$, ...
2
votes
0answers
64 views

References for basics of Piecewise-Deterministic Markov Processes

I am looking for introductory/pedagogical material to Piecewise-Deterministic Markov Processes (see http://en.wikipedia.org/wiki/Piecewise-deterministic_Markov_process) (For the moment I am interested ...
1
vote
0answers
125 views

Good books on “advanced” stochastic analysis

Any good books suggestion for studding advanced features of stochastic analysis ? Thank's in advance
4
votes
2answers
92 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
vote
2answers
109 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 ...
0
votes
1answer
467 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
157 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 ...
3
votes
1answer
237 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 ...
3
votes
2answers
150 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
4
votes
4answers
2k 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
192 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
47 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
87 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. ...
1
vote
0answers
70 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
160 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
1answer
50 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
votes
1answer
217 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
106 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
960 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 ...