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.

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Is $(B_t^2)$ Markov where $(B_t)$ is Brownian motion?

I am pretty sure $(B_{t}^{2})$ not Markov because the squared random walk is not. Showing the square of a Markov process is or isn't Markov I guess I can repeat the method since to be Markov it ...
2
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1answer
116 views

solution of SDE: $dS_t=(\alpha S_t+f(t))dW_t$

does someone know how to solve the following SDE $$dS_t=(\alpha S_t+f(t))dW_t, S_0=s$$ where $f(t)$ is a deterministic function and $W_t$ is a standard brownian motion. Is there a explicit solution ...
2
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1answer
519 views

Limiting distribution and initial distribution of a Markov chain

For a Markov chain (can the following discussion be for either discrete time or continuous time, or just discrete time?), if for an initial distribution i.e. the distribution of $X_0$, there exists ...
2
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1answer
211 views

Homogeneous Poisson processes

Suppose that $N$ is a homogeneous Poisson process with rate $\lambda$. For $0 \le s \le t < \infty$, how can we find $\mathbb E[N_s\cdot N_t]$?
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2answers
127 views

sub martingales and more

This is a problem on sub-martingales. Given : $X_n = X_0 \mathrm{e}^{\mu S_n}$, $n= 1,2,3,\ldots$, where $X_0 > 0$ and where $S_n$ is a symmetric random walk and $\mu$ is greater than zero. We ...
2
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1answer
358 views

Probability distribution of sign changes in Brownian motion

Let us consider a 1d Brownian motion. Displacements in space will be positive or negative and this is a random variable $U(t)$ that characterizes a random process and that can take just the values ...
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2answers
71 views

Expected Number of points in Point Poisson Process

Let $\lambda$ be the intensity of points, distributed as point poisson process, in a square grid of area $A$. Then, the Cumulative disributive function is given by: $$ P(r \leq R) = 1 - e^{-\lambda ...
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1answer
45 views

Simulate simple non-homogeneous Poisson proces

Consider a Poisson process whose conditional intensity is $$\lambda(t) = \alpha e^{-t}$$ starting at time $t=0$ for some parameter $\alpha>0$. I would like to simulate arrival/event/failure ...
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0answers
126 views

Differential equation with random variable

How can I derive analytically or compute numerically the solution to following differential equation $$ dy/dt = y\cdot X\cdot (y\cdot X - g(y,X))\cdot X $$ where X is a random variable (e.g. from a ...
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1answer
501 views

Sum of 2 Brownian motions

Let's say, that $B_t$, $t\geq0$ is standard Brownian motion (Wiener process). Let's define process $$X_t=B_t+B_{t^2}\text{, }t\geq0$$ I need to find its variance, covariance, find out if it's ...
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1answer
65 views

Including non-markovian processes in a birth-death process

Current model I want to model a stochastic system with a birth-death (Markovian) model. I therefore have this kind of $n$ times $n$ (where $n$ is the number of possible states) transition matrix: ...
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1answer
59 views

Brownian motion - Hölder continuity

Let $B$ stand for a Brownian motion on a finite interval $[0,1]$. If I am not wrong, I think that there exists a positive constant $c$, such that almost surely, for $h$ small enough , for all $0< t ...
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1answer
44 views

Relation between autocorrelation and mean of a stochastic process

It is said that if the autocorrelation approaches zero as $\tau$ tends to zero, then the mean of the stochastic process is also zero. I am having trouble understanding the above concept. Say we have ...
0
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1answer
66 views

Showing the square of a Markov process is or isn't Markov

Hi I am trying to show that if $X_n$ is a markov process, whether or not $X_n^2$ is a markov process. $X_n$ is a markov process if $P\{X_k = a_k|X_{k-1} = a_{k-1}, X_{k-2} = a_{k-2}, ..., X_k = a_1 ...
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1answer
619 views

Scalar product of Gaussian process

Assume that $n(t)$ is a White Gaussian Noise (WGN) process with $E[n(t)]=0$, $E[n(t)^2]=\sigma^2$ and $x(t)$ a deterministic function defined in $[0,T]$. How can I compute from first principles the ...
0
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1answer
123 views

Determining computational complexity of stochastic processes

I have an program which implements a Markov chain Monte Carlo process on a system of N bits, stopping when the process converges. Let's use T to denote the average number of steps made by the Markov ...
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4answers
3k views

Why did my friend lose all his money?

Not sure if this is a question for math.se or stats.se, but here we go: Our MUD (Multi-User-Dungeon, a sort of textbased world of warcraft) has a casino where players can play a simple roulette. My ...
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1answer
2k views

What is the Kolmogorov Extension Theorem good for?

The Kolmogorov Extension Theorem says, essentially, that one can get a process on $\mathbb{R}^T$ for $T$ being an arbitrary, non-empty index set, by specifying all finite dimensional distributions in ...
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5answers
7k views

Convergence of $np(n)$ where $p(n)=\sum_{j=\lceil n/2\rceil}^{n-1} {p(j)\over j}$

Some years ago I was interested in the following Markov chain whose state space is the positive integers. The chain begins at state "1", and from state "n" the chain next jumps to a state uniformly ...
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5answers
3k views

Good introductory book for Markov processes

Which is a good introductory book for Markov chains and Markov processes? Thank you.
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3answers
461 views

Is $(B_{t}+t)^{2}$ a Markov process?

Let $B_{t}$ be a Brownian motion relative to a filtration $F_{t}$, is $(B_{t}+t)^{2}$ a Markov process? Thanks!
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3answers
2k views

The Laplace transform of the first hitting time of Brownian motion

Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is ...
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3answers
975 views

Criteria for being a true martingale

Could you kindly list here all the criteria you know which guarantee that a continuous local martingale is in fact a true martingale? Which of these are valid for a general local martingale (non ...
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1answer
1k views

What are some open research problems in Stochastic Processes?

I was wondering, what are some of the open problems in the domain of Stochastic Processes. By Stochastic Processes. Any examples or recent papers or similar would be appreciated. The motivation for ...
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1answer
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Concept of Random Walk

Reading the Wikipedia page for Random Walk, I was wondering what is the definition for a general random walk as a random process, so that all the concepts such as random walk on $\mathbb{Z}^d$, ...
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618 views

Relation of Brownian Motion to Helmholtz Equation

one can obtain solutions to the Laplace equation $$\Delta\psi(x) = 0$$ or even for the Poisson equation $\Delta\psi(x)=\varphi(x)$ in a Dirichlet boundary value problem using a random-walk approach, ...
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1answer
322 views

Definition of the Brownian motion

The way I understood the definition of a Brownian motion $B_t$ in $\mathbb R$ is that it consists of two parts: We first define the finite-dimensional distributions $$ ...
6
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1answer
255 views

Properties of Markov chains

We covered Markov chains in class and after going through the details, I still have a few questions. (I encourage you to give short answers to the question, as this may become very cumbersome ...
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0answers
119 views

In stochastic calculus, why do we have $(dt)^2=0$ and other results?

I'm doing actuarial problems of Exam MFE and it covers some of the stochastic calculus (like Ito's Lemma). One of the frequently used results are the so-called "multiplication rules": $(dt)^2=0$ ...
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1answer
172 views

Is there a standard proof for $\mathbb P(S^X_n\text{ hits }A\text{ before }B) >\mathbb P(S^Y_n\text{ hits }A\text{ before }B)$?

Let $X_i$ and $Y_i$ be two continuous random variables on $\mathbb{R}$ having distribution functions $F$ and $G$, respectively satisfying $G(y)>F(y)$ for all $y$. Let futhermore $S^X_n=\sum_{i=1}^n ...
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3answers
366 views

Limit of a Wiener integral

How to show that $$ \lim _{\alpha \rightarrow \infty } \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} $$ where $\left (B_s ...
6
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3answers
803 views

Why is stopping time defined as a random variable?

I've been given a crash course in stochastic processes and martingales for the purposes of a semester project on them. The guy I'm working with has been, I feel, a little vague in the definition of ...
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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
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1answer
1k views

Quadratic variation of Brownian motion and almost-sure convergence

Say that $W(t)$ is a Brownian motion. The quadratic variation $[W,W](t)$ is defined in terms of a partition $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = t\}$ by $$ \begin{split} [W,W](t) &= ...
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2answers
90 views

“Back to square one” problem

There's a problem I've been stuck on in preparation for junior programming contest I'm going to participate in. It is as follows: The "back to square one" problem is played on a board that has ...
5
votes
1answer
361 views

$\int_0^tB_s^2\ dB_s$ - Gaussian Process and independent increments?

For $(B_t)_{t\ge0}$ a standard Brownian motion (Wiener process) define the stochastic process $X_t:=\int_0^tB_s^2\ dB_s$. I am currently trying to assess if $(X_t)_{t\ge0}$ is a Gaussian process and ...
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3answers
134 views

What is in clear mathematical terms the definition for a sequence of integers, to be called *random*?

Sequences of integers might be ordered, totally ordered,... For all such attributes we find definitions in clear mathematical terms. (1) But what is in clear mathematical terms the established and ...
4
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1answer
294 views

Solution to the stochastic differential equation

Let $X_o=x$, $dX_t=\frac{1}{X_t}dt+X_tdW_t$, $W_t$ is a brownian motion i am thinking of trying $Y_t=\frac{X_t^2}{2}$ and apply ito's lemma on $Y_t$
3
votes
2answers
260 views

Comparing the expected stopping times of two stochastically ordered random processes (Prove or give a counterexample for the two claims)

Information: a-) $X$ and $Y$ are two continuous random variables on $\mathbb{R}$ having continuous distribution functions $F$ and $G$ with $G(y)\geq F(y)$ for all $y$. b-) $S^X_n=\sum_{i=1}^n X_i$, ...
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2answers
270 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 ...
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2answers
549 views

Sobolev meets Wiener

Even though the Wiener process (Brownian motion) is continuous, it has no derivative at any point. Does it at least have weak derivatives?
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1answer
396 views

Bounds for submartingale

Let $x$ be a positive number and $X_n$ be real-valued submartingale such that $X_0 = x$. I am interested in upper bounds on probability $$ \psi(x) = \mathsf{P}_x\left\{\inf\limits_{n\geq 0}X_n \leq ...
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1answer
104 views

How to find $\mathbb{E}[X\mid\min(X,Y)]$?

Say I have two independent variables $X$ and $Y$ that are exponentially distributed with respective rates $\lambda_X$ and $\lambda_Y$. How do I compute $\mathbb{E}[X\mid \min\{X,Y\}]$?
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1answer
173 views

“Continuity” of stochastic integral wrt Brownian motion

I'd like to prove a nice property of a stochastic integral with respect to Brownian motion. Let $(H_t)_{t\geq0}$ be a progressive and bounded process that is continuous at $0$ and $B$ a standard ...
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1answer
4k views

Martingale that is not a Markov process

1. On the internet, it is suggested that $$ X_t=\left(\int_0^t X_s \;ds\right)\;dW_{t} $$ is a martingale, but not a Markov process. I understand that the process $$ I_t(C)=\int_0^t C_s \; dW_s$$ ...
2
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1answer
313 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 ...
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1answer
133 views

Proving that $T_t := S_t -\left| x \right| -\frac {n-1}{2} \int _0 ^t \frac {1}{S_u}~du$ is a brownian motion

Consider $B=(B_t)_{t\geq 0}$ $\mathcal F_t$ - brownian motion in $\mathbb R ^n, \ (n\geq 2)$ starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. ...
2
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1answer
933 views

Distribution of Brownian motion

How would I go about finding the distribution of $B(u) + B(u+v)$ where $u+v > u$? I know that both $B(u)$ and $B(u+v)$ are normal random variables. The sum of two normal random variables is also ...
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1answer
465 views

Ruin-time in Gambler's Ruins

Could someone give me some tips? Let $e_1,e_2,\dots$ be iid normal mean 0 variance 1. Let $X_t := e_1+\cdots+e_t$, for $t=1,2,\dots$ and $X_0 := 0$. (So we have a discrete-time random walk whose ...
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2k views

First exit time for Brownian motion without drift

I am dealing with the simulation of particles exhibiting Brownian motion without drift, currently by updating the position in given time steps $\Delta t$ by random displacement in each direction drawn ...