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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Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the ...
3
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1answer
135 views

When Superposition of Two Renewal Processes is another Renewal Process?

When superposition of two renewal processes is another renewal process? If you merge (superpose) two Poisson processes with parameters $\lambda_1$ and $\lambda_2$, the outcome is another Poisson ...
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3answers
1k views

Wiener Process $dB^2=dt$

Why is $dB^2=dt$? Every online source I've come across lists this as an exercise or just states it, but why isn't this ever explicitly proved? I know that $dB=\sqrt{dt}Z$, but I don't know what ...
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1answer
31 views

Thinning a Renewal Process - Poisson Generalization

If we have a Poisson point process with rate $\lambda$ and we keep each of its point with probability $p$, we obtain another Poisson point process with rate $\lambda p$. Does this result holds for a ...
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2answers
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Time until a consecutive sequence of ones in a random bit sequence

This a reformulation of a practical problem I encountered. Say we have an infinite sequence of random, i.i.d bits. For each bit $X_i$, $P(X_i=1)=p$. What is the expected time until we get a sequence ...
15
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Random walk on $n$-cycle

For a graph $G$, let $W$ be the (random) vertex occupied at the first time the random walk has visited every vertex. That is, $W$ is the last new vertex to be visited by the random walk. Prove the ...
9
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950 views

Difference between Modification and Indistinguishable

Would someone be able to offer a layman's explanation of what is means when two stochastic processes are a Modification of each other and when they are Indistinguishable? My Stochastic Analysis notes ...
3
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1answer
1k views

Angle bracket and sharp bracket for discontinuous processes

The question is quite simple actually. I am trying to understand the differences between the angle bracket $\left<X,Y\right>$ of two processes with jumps $X,Y$, and the sharp bracket of $[X,Y]$. ...
5
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2answers
497 views

Using Recursion to Solve Coupon Collector

I read a brilliant answer by Mike Spivey on one of the questions and I was wondering how I could use it to solve a coupon collectors problem. The problem is : There are coupons labelled 1,2,3...,10 ...
10
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1answer
7k views

equilibrium distribution, steady-state distribution, stationary distribution and limiting distribution

I was wondering if equilibrium distribution, steady-state distribution, stationary distribution and limiting distribution mean the same, or there are differences between? I learn them in the context ...
19
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1answer
4k views

Interpretation of sigma algebra

My question is how to interpret sigma algebra, especially in the context of probability theory (stochastic processes included). I would like to know if there is some clear and general way to interpret ...
12
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1answer
7k views

Distribution of hitting time of line by Brownian motion

I came across the following question: Let $T_{a,b}$ denote the first hitting time of the line $a + bs$ by a standard Brownian motion, where $a > 0$ and $−\infty < b < \infty$ and let $T_a ...
12
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3answers
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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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2answers
2k views

What is meant by a continuous-time white noise process?

What is meant by a continuous-time white noise process? In a discussion following a question a few months ago, I stated that as an engineer, I am used to thinking of a continuous-time wide-sense-...
13
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3answers
432 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 \right)...
5
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2answers
594 views

Logarithm of a Markov Matrix

Start with a Markov matrix $\mathbf{M}$, whose elements are all between $0 \le \mathbf{M}_{ij} \le 1$ and each row sums to one. There is a natural connection with this matrix and the rate matrix $\...
3
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1answer
452 views

Random walk $< 0$

Suppose ${X_t}$ is a random walk with mean zero. (either discrete or continuous time) Fix a time $T$. What is: $P[X_t < 0 \text{ for all } t \leq T]$? In words, what's the probability the random ...
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1answer
67 views

Asymmetric Random Walk / Prove that $T:= \inf\{n: X_n = b\}$ is a $\{\mathscr F_n\}_{n \in \mathbb N}$-stopping time

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \...
64
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4answers
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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 ...
21
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3answers
2k views

What is the importance of the infinitesimal generator of Brownian motion?

I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have ...
7
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2answers
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Prove the time inversion formula is brownian motion

Let $B=(B_t)_{t\geq 0}$ be a brownian motion. Show the time inversion formula $\hat{B}=(B_t)_t\geq0$ is a brownian motion, where for $t \geq 0$ we set $\hat{B}=0$ for $t=0$ and $\hat{B}=tB_{1/t}$ for $...
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2answers
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How to characterize recurrent and transient states of Markov chain

According to Wikipedia with a little rephrasing: A state $i$ is transient if and only if $P(T_i < \infty) <1$, recurrent if and only if $P(T_i < \infty) =1$, where $T_i$ is the ...
4
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1answer
289 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 ...
9
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886 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 ...
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2answers
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Prokhorov metric vs. total variation norm

Let $(S,d)$ be a metric space and let $\mathcal P(S)$ denote the space of Borel probability measures on $S$ endowed with the Prokhorov metric $\pi:\mathcal P(S)\times \mathcal P(S)\to \mathbb R_+$ ...
3
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1answer
3k views

Relation between independent increments and Markov property

Independent increments and Markov property.do not imply each other. I was wondering if being one makes a process closer to being the other? if there are cases where one implies the other? ...
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1answer
1k 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 ...
4
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0answers
246 views

6-digit password - a special decoding method

Consider the situation of decoding a 6-digit password that consists of the symbols A to Z and 0 to 9, where all possible combinations are tried randomly and uniformly. Consider the following ...
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1answer
1k views

Is continuous L2 bounded local martingale a true martingale?

I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...) I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
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1answer
264 views

Markov processes driven by the noise

Let $\xi_n\in \Xi$ be a sequence of iid random variables with $n \in\mathbb N\cup\{0\}$, which we call a noise process. Construct a process $$ Z_{n+1} = f(Z_n,\xi_n)\quad(\star) $$ with $Z_0\in E$ ...
4
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1answer
110 views

Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?

Suppose we have an sde of the form: \begin{eqnarray} dX_t=b(X_t)dX_t + \sigma (X_t)dB_t \end{eqnarray} where $b$ and $\sigma$ are Lipschitz. Then we have existence and uniqueness of the solution $X_t$...
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1answer
299 views

When random walk is upper unbounded

Consider a random walk $S_n = a_1+\dots+a_n$ where $a_n$ are iid random variables with $Ea_1 = a$ and $E|a_1|<\infty$. I am interested in the case when $\sup\limits_n S_n>M$ for all $M$ a.s. ...
14
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4answers
2k views

Showing that Brownian motion is bounded with non-zero probability

How do you show, that for every bound $\epsilon$, there is a non-zero probability that the motion is bounded on a finite interval. i.e. $$\mathbb{P} (\sup_{t\in[0,1]} |B(t)| < \epsilon) > 0$$ I ...
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3answers
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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 ...
8
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1answer
204 views

When does $\sum_{i=1}^{\infty} X_i$ exist for random sequences $\{X_i\}_{i=1}^{\infty}$?

Suppose $\{X_1, X_2, X_3, \ldots\}$ is an infinite sequence of random variables such that $E[X_i]=0$ for all $i$, and $E[X_iX_j]=0$ whenever $i \neq j$. Further suppose the variances $\sigma_i^2 = E[...
6
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3answers
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What is the difference between all types of Markov Chains?

I have been looking for some good material covering Markov Chains but everything seems so difficult to me... After reading about the subject, I figured out that there is basically three kinds of ...
14
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3answers
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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 $$\mathbb{E}[\exp(-\...
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1answer
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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 ...
5
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1answer
499 views

Fubini's theorem for conditional expectations

I need to prove that if $E \int_a^b |X_u|\,du = \int_a^b E|X_u|\,du$ is finite then: $$E\left[\left.\int_a^b X_u\,du \;\right|\; \mathcal{G}\right] = \int_a^b E[X_u \mid \mathcal{G}]\,du.$$ I just ...
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2answers
4k views

Equilibrium distributions of Markov Chains

I often get confused about when a Markov chain has an equilibrium distribution; when this equilibrium distribution is unique; which starting states converge to the equilibrium distribution; and ...
5
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3answers
531 views

How to calculate $E[(\int_0^t{W_sds})^n], n \geq 2$

Let $W_t$ be a standard one dimension Brownian Motion with $W_0=0$ and $X_t=\int_0^t{W_sds}$. With the help of ito formula, we could get $$E[(X_t)^2]=\frac{1}{3}t^3$$ $$E[(X_t)^3]=0$$ When I try to ...
3
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1answer
755 views

Hermite Polynomials and Brownian motion

I am asked to prove the following : Let $B_t$ be a standard brownian motion. The $n$th Hermite polynomial is $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
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2answers
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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 ...
27
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9answers
15k views

What's the difference between stochastic and random?

What's the difference between stochastic and random? I've read in the Portuguese Wikipedia that there's a difference, but I still didn't see this point on English Wikipedia.
8
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2answers
615 views

Laplace transform of integrated geometric Brownian motion

Is there any closed form of the Laplace transform of an integrated geometric Brownian motion ? A geometric Brownian motion $X=(X_t)_{t \geq 0}$ satisifies $dX_t = \sigma X_t \, dW_t$ where $W=(W_t)_{...
7
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4answers
5k views

Sum and product of Martingale processes

Given two Martingale processes $(X_t)$ and $(Y_t)$, are their sum $(X_t+Y_t)$ and their product $(X_t \times Y_t)$ also Martingale? If not, will the two $(X_t)$ and $(Y_t)$ being independent grant ...
3
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1answer
2k views

Calculating stationary distribution of markov chain

I am asked to compute the stationary distribution of the markov chain with state space $E=\{0\dots,n\}$ and transition matrix below: \begin{bmatrix} 0 & 1 \\ \frac{1}{n} &...
3
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1answer
664 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 result....
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1answer
588 views

Is a Markov process a random dynamic system?

A random dynamic system is defined in Wikipedia. Its definition, which is not included in this post for the sake of clarity, reminds me how similar a Markov process is to a random dynamic system just ...
6
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1answer
460 views

Right continuous version of a martingale

This is an exercise in chapter 2 of the book "Continuous Martingales and Brownian Motion" by Revuz and Yor: Consider the probability space $([0,1], \mathcal{B}([0,1]), dx)$, where $dx$ denotes ...