1
vote
1answer
28 views

Central limits without replacement in a finite population.

"Everybody knows" that there are lots of variations on the theme of the central limit theorem. The most frequently seen form seems to be this: Suppose $X_1,X_2,X_3,\ldots$ are i.i.d. random variables ...
1
vote
0answers
24 views

Locate proof of Second Fundamental Theorem of Asset Pricing

Where can I find a $\textbf{rigorous}$ proof of the Second Fundamental Theorem of Asset Pricing. That is, A market is complete if and only if it has a unique risk neutral measure. Please do not ...
2
votes
1answer
36 views

Markov processes on function spaces

Is there any reference on Continuous time Markov process whose state space is infinite dimensional function spaces, such as the space of continuous functions $C(R^d)$? It seems Dirichlet Form is a ...
2
votes
0answers
132 views

Changing a queueing processes

Situation Consider a general queueing system $\mathscr{S}$, whose customer arrival times are independent, and whose service times are independent; both of these are allowed to have general ...
0
votes
1answer
18 views

Multi-dimensional Feynman Kac Theorem

I am trying to understand how to prove the multi-dimensional version of the Feynman-Kac formula. The single-dimensional version is proved on this page: en.wikipedia.org/wiki/Feynman–Kac_formula ...
0
votes
1answer
14 views

Quadratic Variation of Diffusion Process and Geometric Brownian Motion

I'm looking to find out the stochastic differential equation satisfied by the quadratic variation of Geometric Brownian Motion, Diffusion Process. For example, for a diffusion process that ...
0
votes
1answer
33 views

Wasserstein metric: conditions for the existence of minimizer and duality

Let $(X,d)$ be a metric space and let $\mathcal P(X)$ be the set of all Borel probability measures on $(X,d)$. The Wasserstein distance on $\mathcal P(X)$ is given by $$ W_d(\mu,\bar\mu):=\inf_{M\in ...
2
votes
0answers
48 views

Fractional Brownian motion---construction via Hilbert space?

The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms: Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, ...
1
vote
1answer
46 views

Literature on Sabermetrics in baseball

For my bachelor's thesis, I would like to study the use of Sabermetrics in baseball. I was fascinated by the book 'Moneyball: The Art of Winning an Unfair Game' by Michael Lewis, and to me, it ...
1
vote
2answers
52 views

Convergence time of a Markov chain

We know that a regular Markov chains converges to a unique matrix. The convergence time maybe finite or infinite. My interest is in the case where the convergence time is finite. How can we accurately ...
0
votes
1answer
37 views

How big a Brownian bridge can get? Confidence band.

If we know the endpoints of the Brownian path, is there any theorem telling us if it can be contained within a ball a.s. (with probability one)? For example contained in two big enough balls (call it ...
0
votes
0answers
55 views

Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail. Just to avoid misunderstanding, ...
2
votes
2answers
31 views

Markov property for a Stochastic Process

My question: Every Stochastic Process $X(t), t\geq 0$ with space states $\mathcal{S}$ and independent increments has the Markov property, i.e, for each $\in \mathcal{S}$ and $0\leq t_0\leq< ...
1
vote
0answers
17 views

Changes in the transition matrix of a Markov chain

In most or all Markov chain theories that I know of assumes that the transition matrix does not change over time. But what if certain changes are expected to occur at certain times in the transition ...
5
votes
1answer
54 views

Resource for Stochastic Calculus and Ito processes

May someone please recommend a book or website where one can learn Stochastic Calculus and Ito processes from scratch.
1
vote
0answers
38 views

References for time-inhomogeneous Markov jump processes?

In some central models in life insurance mathematics, the state of the insured is modeled using a continuous-time time-inhomogeneous Markov process with finitely many states. While many results for ...
3
votes
1answer
41 views

Reference request for positive matrices

I would much appreciate someone suggest me a text book which covers stochastic matrices in depth with all relevant theories.Thanks
2
votes
0answers
51 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 ...
9
votes
2answers
232 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
29 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
73 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
119 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
49 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
19 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
48 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 ...
1
vote
1answer
110 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
42 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 ...
3
votes
4answers
51 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
24 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
37 views

Introduction to stochastic control

I'm looking for an introductory text on stochastic control. Any suggestions?
2
votes
1answer
193 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 ...
3
votes
1answer
121 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
60 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
70 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
53 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
60 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
61 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 ...
4
votes
1answer
258 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
14 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
139 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
40 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
21 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
71 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
124 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
114 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
204 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
209 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
65 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
86 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
73 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 ...