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.

learn more… | top users | synonyms

3
votes
0answers
88 views

Discontinuous local martingales with finite variations

I would like to know if a discontinuous local martingale with paths of finite variations almost surely is a martingale. I feel that it should be the case but can't find a straightforward argument. As ...
2
votes
0answers
312 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 ...
1
vote
1answer
340 views

Composition of stochastic processes with independent increments? stationary increments?

It is a standard fact that subordination of a Lévy process yields another Lévy process. That is, if $\{X_t\}$ is an $\mathbb{R}$-valued rcll process with independent stationary ...
2
votes
0answers
234 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
99 views

Hermite rank of an $L^2$ function

Let $(H_k)_{k\in\mathbb{N}}$ be the sequence of hermite polynimials, $Z\sim N(0,1)$ and $G\in L^2(\mathbb{R},\phi)$ with $\operatorname{E}\left[Z\right]=0$. By $\phi$ we denote the density of the ...
1
vote
0answers
65 views

Do uniform convergence and pointwise convergence to normality imply convergence to a gaussian process?

A function $\widehat{f}(x) \rightarrow f(x)$ uniformly over a compact interval $[\underline{x},\overline{x}]$, in addition, for some sequence ${a_{n}}$, $\forall x\in [\underline{x},\overline{x}]$, ...
0
votes
1answer
113 views

Probability of Specific event occuring between 2 events?

Forgive me beforehand for what may be a question with an obvious seolution, but I havent had statistics courses in quite some time. I have an Excel File of approximately 3000 Events, each event has a ...
0
votes
1answer
161 views

First Order Stochastic Dominance

I am reading up on stochastic dominance(http://en.wikipedia.org/wiki/Stochastic_dominance) and have some questions: PDF and CDF of Gamble A and B look like this. Since the CDF of A is always less ...
0
votes
1answer
52 views

Adding a constant in “equal in distribution”

Why is that adding a constant in this equation, $x_B \overset {d}{=} (x_A+y)$ is equivalent to pushing some of the probability mass to the left if $ y \le 0$. Should it be pushing it down ...
2
votes
1answer
197 views

Square root of a stochastic process

i need help with the following problem. how can i derive d√v using Ito's lemma for the following process: d√v=(α−β√v)dt+δdX The parameters α, β, δ are constant. Using Itô's lemma show that dv = ...
1
vote
1answer
106 views

Is the Poisson distribution the only one where the mean and variance of the variate are equal?

Or is this possible in another probability distribution?
3
votes
3answers
1k views

Is a square-integrable continuous local martingale a true martingale? [duplicate]

I am wondering, if the following is true without any other assumptions and if so, how to prove it: Let $(M_t)_{t \geq 0}$ be a continuous local martingale on a filtered probability space $(\Omega, ...
1
vote
1answer
82 views

What sort of mathematical object is a stochastic process?

My introductory stochastic processes course uses $\{Q(t); T>=0\}$ (in particular, please note the semicolon) to refer to a random process, where $Q(t)$ is a time-dependent variate. Conceptually, ...
0
votes
2answers
116 views

A basic question on Martingale and betting games

I am new to Martingales. Why the betting strategy where if I loose then I double the amount (so, with the first win I get whatever I lost plus the amount of initial bet) is called the "martingale ...
2
votes
0answers
90 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; ...
0
votes
0answers
76 views

Steady state probabilities [closed]

I studied that there is steady state probabilities if and only if the Markov Chain is irreducible, positively recurrent and aperiodic. Why all the states should aperiodic? Is there any importance of ...
1
vote
0answers
30 views

Coverage probability in a Spatial Boolean Process

Consider a Poisson Boolean Process (X,$\lambda$,1) where $\lambda>0$ is the Poisson intensity of a two dimensional Poisson process. The Boolean process is such that at the center of each Poisson ...
2
votes
1answer
95 views

Extending a supermartingale over larger index set

The following problem comes up reading some notes on utility maximization in mathematical finance. Since it is a purely stochastic process problem, I place it in this forum. The time index is covering ...
3
votes
1answer
166 views

Probabilities of uncountable intersection of events

In order to determine a probability for some event $A\in\Omega$, I ended up with $$ \mathbb{P}\left(X_t>f(t),\quad \forall [0,T]\right)≤ \mathbb{P}(A)≤\mathbb{P}\left(X_t≥f(t),\quad \forall ...
1
vote
0answers
77 views

Question about canonical markov process with strong markov property.

I'm really stuck on this bit, maybe someone can help be along: Let $X$ be a canonical, right-continuous Markov process with values in a Polish state space $E$, equipped with Borel $\sigma$-algebra ...
0
votes
1answer
63 views

limit of the first moment of solution of stochastic differential equation

Suppose $X^x$ is solution of $$d X_t = X^3_t dW_t, \quad X_0 = x>0.$$ In the above, $W$ is a Brownian motion in a given filtered probability space. Such an equation has unique strong solution, ...
3
votes
4answers
116 views

On which measure space is $S_n = X_1 + \dots + X_n$ considered?

A common setting in law of large number theories is letting $X_1, X_2, \dots$ be independent indentical random variables on probability space $(\Omega, \mathcal{B}, P)$. Let $S_n = X_1 + \dots + X_n$. ...
3
votes
1answer
486 views

If $P$ is a regular transition probability matrix then $P^{n^2}$ has no zero element

A transition probability matrix $P\in M_{n\times n}$ is regular if for some $k$ the matrix $P^k$ has all of its elements strictly positive. I read that this can be tested by using the following ...
1
vote
1answer
775 views

Is identically distributed and uncorrelated sequence a white noise?

A white noise is defined to be a stationary process with constant power spectral density. Is a sequence of random variables, which are identically distributed and uncorrelated, a white noise? Is ...
2
votes
1answer
131 views

Dominated Convergence Thm (DCT) for Double Sequences

By a version of the Dominated Convergence Theorem (Thm 25.12 in Billingsley 86) $ E(X_n)\rightarrow E(X) $, if $X_n \overset{p}{\rightarrow} X$ and $X_n$ is uniformly integrable sequence of random ...
2
votes
1answer
86 views

Skorokhod Representation: $|X_n(t) - X_n(s)| \leq |t-s| \Rightarrow |X(t) - X(s)| \leq |t-s|$

While working out problems from the Chapter on Skorokhod Representation from "Stochastic processes" by Richard Bass, Chapter 32, I came across this problem: Question: Suppose $X_n$ converges weakly ...
1
vote
0answers
88 views

Different definitions of an ergodic stationary process

From page 3 of a note: A stationary process is ergodic if any two variables positioned far apart in the sequence are almost independently distributed. A formal definition is the following: ...
1
vote
0answers
142 views

Binary Tree and Geometric Distribution

I have the following algorithm for "constructing" a binary tree: A probability $p_g$ for elongation, i.e. adding an edge A probability $p_b$ for branching, i.e. adding to a node two "child" edges ...
1
vote
1answer
865 views

Form of the spectral density in Wiener Khinchin theorem?

The Wiener–Khinchin theorem says the autocorrelation function of a wide sense stationary process can be written as a Stieltjes integral, where the integrator function is called the power spectral ...
2
votes
1answer
146 views

Weighted integral of random variables

Given a random zero-mean gaussian random variable $X(t)$ with parameter $t$, such that $E [X(t) X(t^\prime)] = \sigma^2 (t) \delta_{tt^\prime}$, is it possible to produce a single gaussian random ...
0
votes
2answers
168 views

Shift operator of a stopping time, what does it mean exactly?

I'm trying to figure out this question: Let $X$ be a canonical, right-continuous Markov process with values in a Polish state space $E$, equipped with Borel $\sigma$-algebra $\mathcal{E}$. Assume $t ...
1
vote
0answers
46 views

How to show that this element is predictable

Suppose I have a process $Y_t$, which is predictable. We can assume that the filtration satisfies the usual condition. Then, let $A\in \mathcal{F}_t$. I was wondering about the following. Is ...
1
vote
0answers
44 views

how to describe this case with markov-chain

I want to describe this case in markov chain: The case: Mr. Meier reads NYTimes everyday and puts the newspaper on news rack. His wife sometimes cleans the house(with prob $1/3$ each day) and throws ...
0
votes
1answer
72 views

Brownian Motion Question - Requires Verification

Suppose $Z(t)$ is a standard Brownian motion process with $Z(0)=0$, then calculate: $P(Z(3)>Z(2)>0)$ I have the following, but unsure if my rationale is correct: $Z(3)=X_1+X_2+X_3, ...
1
vote
1answer
108 views

Poisson Process with a Random Variable

I really couldn't wrap my head around this basic concept so I'm looking help for some basic calculations to solidfy my understanding: Suppose we have $T\in (0,\infty)$. Say we have $E(T)=\mu$, ...
1
vote
1answer
97 views

why the sigma algebra generated by null set and Brownian Motion is right continuous?

I mean why the generated one satisfies the definition of right continuous?
7
votes
0answers
185 views

How well can the maximum of a Gaussian process be approximated by a finite-dimensional Gaussian variable?

Consider a compact set $K$ in $\mathbb{R}^p$, and let $W$ be a mean zero continuous Gaussian process on $K$, meaning that $W$ takes its values in the space of continuous functions from $K$ to ...
0
votes
0answers
183 views

On the second derivative of Wiener process

As we all know, continuous white noise is the derivative, with respect to time, of a Wiener process. My question is that does the second derivative of Wiener process exists? If so, what is it and how ...
2
votes
1answer
108 views

Computation of a mean (random sum)

Let $X_1$, $X_2$, ... be independent and identially distributed positive random variables and define the sum $S_n = X_1 + X_2 + ... + X_n$. Consider the first time $N$ where $S_N \ge b$ with a given ...
2
votes
1answer
459 views

Wald's second equation

We have a random walk $S_N=\sum_{i=1}^{N}X_i$ where $X_i$ are i.i.d with $0<E(X_i)<\infty$ and $N$ is a stopping time. What is the "exact" second equation of Wald ? I've seen different results ...
0
votes
1answer
72 views

question about the transformation of a Markov process

I have a question about Markov Process: Let $X_t=(X_t^1, X_t^2,..., X_t^n)$ be a Markov process with regard to the filtration $\mathcal{F}_t$, let $Y_t:=\max_{1\leq k\leq n}X_t^k$, then is $Y_t$ a ...
5
votes
2answers
516 views

Almost sure convergence of stochastic process

Suppose that we have a (almost surely) continuous stochastic process $\{ X_{t} \}_{t \geq 0}$ on $[0,1]$ with non-stochastic initial value $X_{0} = x_{0} \in [0,1]$ and exponentially decreasing ...
4
votes
1answer
115 views

Convergence in probability, continuity and uniform convergence in probability

Let $(X_i)_{i\in\mathbb{N}}$ be a strictly stationary sequences of real valued random variables with finite variance. We have the empirical distribution functions $F_{n}(u):=\frac{1}{n} \sum_{i=1}^n ...
1
vote
0answers
70 views

Uniform convergence of random distribution functions

Let $(X_i)_{i\in\mathbb{N}}$ be a strictly stationary sequences of real valued random variables with finite variance. We have the empirical distribution functions $F_{n}(u):=\frac{1}{n} \sum_{i=1}^n ...
1
vote
1answer
169 views

Expected value of a stochastic harmonic series

It doesn't seem straightforward to put this into mathematical notation, but I'll do my best to explain the setup. Consider a harmonic series of the following type. For the sake of argument, say we ...
2
votes
1answer
89 views

A continuous random walk of length 1

Suppose one starts at origo in in the plane and takes $N$ steps of length $1/N$ in a random direction, what is the distribution of the resulting distance from origo as $N$ approaches infinity? For one ...
3
votes
0answers
49 views

Decisive equivalence of collections of probability measures

Working on the optimal decision theory in stochastic setting, I've found out that the following notion of equivalence is very useful. Let $(X,\mathscr A)$ be a measurable space, and let $\mathrm ...
1
vote
0answers
50 views

Poisson distributed variable after iterative process

The value of $x$ is changed in a stochastic iterative process. Changes of $\pm1$ are possible. I am searching transition probabilities $p(x=n \rightarrow x=n+1)$ and $p(x=n \rightarrow x=n-1)$ that ...
2
votes
2answers
137 views

Continuous-time finite-state Markov chain as a subordinated Brownian motion

I think I read somewhere that every semimartingale is representable as a time changed Brownian motion (sorry, I don't have a reference). This suggests that in particular a continuous-time finite-state ...
1
vote
1answer
101 views

Why universally and not just Borel policies

In a famous book Stochastic Optimal Control: The Discrete-Time Case by Bertsekas and Shreve they use universally measurable policies that come up with some handy features: e.g. they show that every ...