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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Brownian motion

Verify that $E( X(t) X(s) | X(0)=0 ) = min (t, s)$, where $X(t)$ is standard Brownian motion. I don't know where to start. Thanks!
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1answer
739 views

Splitting a Poisson process according to time-dependent probabilities

Let $X_t$ be a homogeneous Poisson process of rate $\lambda$. Suppose we define functions $p_1(t)$, ..., $p_k(t)$, such that for all $i$ and $t$, $p_i(t)\in [0,1]$ and $\sum\limits_{i=1}^kp_i(t) = 1$. ...
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1answer
206 views

Poisson arrivals followed by locking

following is my problem: Pulses arrive at a processor according to a Poisson process of rate λ. Suppose each arriving pulse that is processed by the processor locks the processor for a fixed time T, ...
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1answer
289 views

Limiting distribution of alternating renewal process

Consider an alternating renewal system that can be in one of two states: on or off. Initially it is on and it remains on for a time $Z_1$, it then goes off and remains off for a time $Y_1$, it then ...
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0answers
102 views

Measurability of a point process or random measure at a measurable subset

Suppose $\xi$ is a point process on $(S, B(S))$, where $S$ be locally compact second countable Hausdorff space equipped with its Borel σ-algebra $B(S)$. I was wondering if $\xi(A), \forall ...
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1answer
539 views

Confusions regarding the concept of a stopping time for a martingale

I am studying martingales and I have a few conceptual questions regarding why we need stopping times. My book (Probability and Computing by Mitzenmacher and Upfal) defines a martingale as follows: A ...
6
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2answers
677 views

Expected travel time for regularly departing trains

I'm going to ask a very simple practical question, but I believe it has some interesting mathematical properties. The simple variant is: trains depart every $x$ minutes and take $y$ minutes to arrive ...
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1answer
253 views

Question for understanding definition of point process

I am trying to understand the definition of point process when reading its Wikipedia article: Let $S$ be locally compact second countable Hausdorff space equipped with its Borel σ-algebra ...
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1answer
1k views

Stochastic integral with a Poisson process

I have a Poisson process $X_t$ for $t\ge0$. How I can find a process $b_t$ such that $$\exp ({\alpha X_t})=1+\int_0^t b_{s^{-}}dX_s$$ where $\alpha\in\mathbb{R}$ and what would be the expectation of ...
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1answer
109 views

Backward representation of the general Markov process

One usually deals with a discrete time Markov process in the following form: given a state space $E$ the Markov process is defined by transition kernel $T(B|x)$ such that $$ \mathsf{P}(X_1\in B|X_0 = ...
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1answer
101 views

optimized upper bound; stochastic

Let $I_t=\int_0^t f_tdB_t,$ where $(f_t,t\ge 0)$ is a bounded process, $|f_t|\leq M$ almost surely for all $t \ge 0.$ Show that $$\mathcal{P}\left[\sup_{0\leq t\leq T}|I_t|>\lambda\right]\leq ...
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1answer
81 views

1-dimensional diffusion process

Let $(X_t, t ≥ 0)$ be a 1-dimensional diffusion process with generator $Af(x) =\frac{1}{2}a(x)f''(x)+b(x)f'(x), \mathcal{D}(A)=C^2({\mathbb{R}})$ where $b$ and $a=\sigma^2$ are continuous ...
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2answers
3k 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 ...
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2answers
743 views

Characterization of two-step 2x2 stochastic matrices

Show that: A 2 x 2 stochastic matrix is two-step transition matrix of a Markov chain if and only if the sum of its principal diagonal terms is greater than or equal to $1$.
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2answers
162 views

modified stochastic process

Is there any study of stochastic processes where the probability matrix (for a finite state process) is time dependent? For example, probability I go from school to home is higher at night as ...
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276 views

A sequence of order statistics from an iid sequence

Given a sequence of iid random variables $X_i$ (without loss of generality from $U(0,1)$), an integer $k \ge 1$ and some $p \in (0,1)$, construct the sequence of random vectors $Z^{(j)}$, $j=0,1,...$ ...
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2answers
502 views

Birth and Death Process Question (Queuing)

A small shop has two people who can each serve one customer at a time. There is also space for two customers to wait. Anyone who arrives and sees that the shop is full will go to another store. ...
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2answers
351 views

Collisions in random walk in $\mathbb{Z}^n$

Given a set $S$ of $r$ points in $\mathbb{Z}^n$, $S=(p_1,p_2,p_3.., p_r)$ , each a starting point for random walk with step size 1. What is the probability they will all eventually meet at the same ...
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3answers
4k views

Distribution of compound Poisson process

Suppose a compound Poisson process is defined as $X_{t} = \sum_{n=1}^{N_t} Y_n$, where $\{Y_n\}$ are i.i.d. with some distribution $F_Y$, and $(N_t)$ is a Poisson process with parameter $\alpha$ and ...
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1answer
804 views

covariance function for Brownian motion

What is Cov[W(t),W(0)] when W(t) is t*B(1/t) and W(0) = 0 where B(t) is standard Brownian motion. The answer is min {s,t}. I am unsure how they get that because I get min{0,1}. Here is what I did: ...
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2answers
509 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 ...
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1answer
550 views

Time Until Extinction for a Pure Death Process Where the Time is Exponentially Distributed

Let $X(t)$ be a pure death process starting from $X(0)=N$. Assume that the death parameters are $\mu_1, \mu_2,\dots,\mu_N$. Let $T$ be an independent exponentially distributed random variable with ...
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1answer
183 views

Birth Process Question

The Yule process is a pure birth process with parameter $\lambda_n = n\beta$. If $X(0) = 1$, then find the probability there are no births during the time interval $(5,8]$. I was thinking of ...
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1answer
1k 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
2k views

covariance function for Brownian motion

What would the covariance function be of $V(t) = (1-t) B[t/(1-t)]$ if $B(t)$ is standard Brownian motion. Also $t$ is between $0$ and $1$. Thanks for the help! EDIT: Here is where I am stuck: I ...
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1answer
254 views

covariance function for Brownian motion

What is the covariance function for $U(t)$ if $U(t) = e^{-t}B(e^{2t})$ for $t \geq 0$ where $B(t)$ is standard Brownian motion. Thanks for the help!
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1answer
445 views

Find expectation of brownian motion

How would I do the following question. I know how to do it with two variables (just B(U) and B(U + V) but I do not know how to figure this out with 4 (or even 3) terms) Thanks for the help. E ...
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1answer
473 views

A random walk on the positive integers with P[+1 Step] <= P[-1 Step] and a reflecting boundary at the origin

Consider a one-dimensional random walk on the positive integers with: (1) - A fully reflecting boundary at "x = 0". (2) - Initiation of the walker at "x = 1". (3) - A P[+1 Step] / P[RHS Step] ...
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1answer
70 views

A generic question on stochastic integrals

What is the right approach to take and find the moments of the following: $$\mathcal{Z}_t=\int\mathcal{W}_t^k\,d\mathcal{W}_t=?$$ $$\mathcal{W}_t \sim \mathcal{N}(0, t),\ k=2,3...$$ ...
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1answer
137 views

posterior distribution after having partial information on some linear combinations of unknown variables (Revised)

$x_1$, $x_2$, and $x_3$ are i.i.d. normal random variables with distribution $N(0, \sigma_x^{2})$ $\epsilon_1$, $\epsilon_2$, and $\epsilon_3$ are i.i.d. normal random variables with distribution ...
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1answer
174 views

Nonuniform infinite random monkeys and Hamlet

I have a rudimentary grasp of the second Borel-Cantelli lemma and vaguely understand how finite patterns can repeat infinitely often. The one proof I am familiar with that verifies this unfortunately ...
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1answer
92 views

Probability equation for case when N objects might join

I have following problem. I have set of objects, there is N objects. These object might join with each other. Two objects might join with probability $p_{join}$. In every computations step I take all ...
2
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2answers
218 views

Questions about stationarity of a Markov process

I was wondering for a Markov process: Is it true that: it is a stationary process, iff its transition probability function is invariant to time-translation (i.e. it is homogeneous) and its ...
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2answers
874 views

martingales as a sum of random variables that are not independent

All the problems I've seen treat martingales as a sum of independent rvs, $X_t=\sum_{k=1}^{n}B_k$, where $\mathbf{E}[B_{k}|\mathcal{F}_{k-1}]=\mathbf{E}B_{k}$. What is a good approach to study (e.g. ...
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1answer
552 views

Martingale problem

If $X_t$ is an $\mathbb{R}$- valued stochastic process with continuous paths, show that the following two conditions are equivalent: (i) for all $f\in C^2(\mathbb{R})$ the process $$f(X_t) − f(X_0) ...
4
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1answer
127 views

Stochastics problem: give an example

Give an example on $\Omega = \{a, b, c\}$ in which $E(E(X|F_{1})|F_{2}) \neq E(E(X|F_{2})|F_{1})$ -- Obviously X is a random variable and $F_{1}$ and $F_{2}$ are sigma-algebras... but I'm not even ...
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1answer
1k views

transient and steady state probabilities

The difference between transient and steady state probabilities is only that, transient probabilities are time dependent and steady state probabilities are not?
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1answer
172 views

Double checking solution (Poisson process)

I am just double checking to see if my solutions are correct for these two questions. Assume that for both customers arrive at rate lambda. I am unsure about the second one so confirmation would be ...
2
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2answers
270 views

Differentiate between linear death process and pure death process

I am having difficulty (and my textbook is of no help!) figuring out when to treat a problem as a linear death process with death parameters $\mu n$h vs. a pure death process with death parameter ...
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1answer
632 views

pure death process

A chemical solution contains N molecules of type A and M molecules of type B. An irreversible reaction occurs between type A and type B molecules in which they bond to form a new compound AB. Suppose ...
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1answer
1k views

Poisson Process

Customers arrive at a certain facility according to a Poisson process of rate lambda. Suppose that it is known that five customers arrived in the first hour. Each customer spends a time in the store ...
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1answer
285 views

Poisson Process Question

Let $X(T)$ be a Poisson process. What is $$ \mathbb{P}(X(t) - X(s) = 1 \mid X(t) = 4) \,? $$ I split this up into $\mathbb{P}(X(s) = 3, X(t) - X(s) = 1)$ and found the answer. However, it did not ...
4
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1answer
380 views

Finding expectation of waiting time

Can someone explain this solution to me? The question was find E[W1 | X(t) = 2] where W1 is the time until the first event occurs and X(t) is a Poisson process. V1 represents the smallest of the ...
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2answers
140 views

How can an element not be a member of its own equivalence class?

I'm working my way through these notes on stochastic calculus: The following is taken from section 2.20: In discrete probability, equivalence classes are measurable. (Proof: for any ...
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500 views

Group-Interview Secretary Problem

The secretary problem is a well-studied optimal stopping problem with a simple solution. Suppose a set of $N$ candidates are interviewed for a secretarial problem, one at a time, in random order. ...
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3answers
289 views

Choosing probabilities as to make a coin game fair

Consider a game with two players A and B. Player A throws a coin with probability $p_A$ of landing heads, in which case he wins the game. If not, player B throws another coin with probability $p_B$ ...
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3answers
475 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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2k views

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 ...
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1answer
284 views

Death process (stochastics)

From what I understand, the question is asking me to find P(X(t) = n | X(0) = N). I know that with a linear death rate this probability is (N choose n) * [e^(-alpha*t)]^n * [1 - e^(-alpha*t)]^N-n ...
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1answer
69 views

Help with summation stochastic

I don't understand how to get from the first line to the last. What happens to the summation? Please help! An important piece of information given in the question is that the transition probability ...