0
votes
0answers
17 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 ...
1
vote
0answers
27 views

Poisson process: Has my book used a necessary condition, when it should have used a sufficient condition?

My book says that if we know that if we are viewing a poisson process with length $t$, and know that $n$ events happened in that interval, than the time that any of those events happened is uniformly ...
1
vote
0answers
13 views

Variance of exit time for simple symmetric random walk

For a simple symmetric random walk starting at 0 (that is, a Markov chain on the integers starting at 0 with equal probabilities of going to the left and right at each step), I want to compute the ...
1
vote
0answers
12 views

arbitrage free price in martingale measures

Consider a one-period market with $S^1_t,\cdots,S^n_t$, with $t=0,1$ the price process of $n$ assets, where $S_1$ is a risk-free asset: $S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the ...
0
votes
1answer
21 views

How are the waiting times distributed, poisson process.

I am wondering how the waiting times are distributed for the poisson process, conditioned on a number of events by time t. Look at this theorem: Here, the S's are the sum of the waiting time to ...
0
votes
1answer
31 views

Gaussian vectors and covariance matrix.

The following is a part of a question I was given in stochastic processes course. It goes like this - I am given a series of gaussian iid random variables $\{V_i\}_{i=1}^N$ , the variable $X_0 \sim ...
3
votes
0answers
28 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
1
vote
1answer
26 views

Probability of having 2 consequent 1's in a random finite sequence

Fix natural numbers $t,n,k$. Consider the following stochastic process for generation of finite sequences of elements from $\{1,\ldots,n\}$: $\sigma_0$ is the empty sequence. Suppose we have ...
0
votes
0answers
29 views

continuous time markov process - first passage time

Let $(X_t)_{t\ge0}$ is a continuous time-homogeneous Markov diffusion process such that $X_0=y$. Let $$p(x,t|y)=d\Pr(X_t\le x|X_0=y)/dx$$ be the respective transition probability density. Let ...
0
votes
0answers
17 views

Sufficient conditions for Uniform Law of Large Numbers

I would need a Uniform Law of Large numbers for $f_T(\theta)$ over $\Theta$ when $f$ is the indicator function and, thus, not continuous over $\Theta$. Do you know about any sufficient conditions?
1
vote
1answer
29 views

relation between multivariate probability generating function and univariate ones

Suppose I have two independent integer random variables $X_1$, $X_2$ (with constraint that $X_1+X_2\le N,0\le X_1\le N,0\le X_2\le N$), with probability generating functions $g_1(z)$, $g_2(z)$. Now I ...
0
votes
2answers
25 views

What is Poisson Point Process?

"Points $\{A_j\}_{j\in\Phi(\lambda)}$ are assumed to be distributed according to a homogeneous PPP with intensity $\lambda$, denoted $\Phi(\lambda)=\{X_j\}$, where $X_j$ is the location of the $j$th ...
2
votes
0answers
65 views

Stopping time and filtration

My question is as follow: Let $(\Omega,\cal{F}_\infty,\{\cal{F}_t\},\mathbb{P})$ be the filtred probability space. Further, denote $\cal{F}^*_t$ as the usual augmented filtration. Now, given a ...
0
votes
1answer
30 views

multivariate probability generating function

Suppose I have three random variables $X_1$, $X_2$ and $X_3$, with probability generating functions $g_1(z)$, $g_2(z)$ and $g_3(z)$. Now I have a joint-distribution $P(X_1-X_2,X_1-X_3)$, whose ...
0
votes
1answer
25 views

Mean time spent in transient states/Markov chain

I dont get this in my book: For transient states $i$ and $j$ , let $s_{ij}$ denote the expected number of time periods that the markov chain is in state $j$ , given that it starts in state $i$. Let ...
0
votes
1answer
29 views

Markov chains for group decision making

I am new to Markov chains since I am doing my own studying on it recently. I was doing some questions and came across this one that got me stuck. Suppose there are four employees and they need to ...
1
vote
1answer
290 views

Deriving SDE for process with two uncorrelated Brownian motions and factor

Derive SDE for the following 2 dimentional process $Y(t) = wX_1(t) + \sqrt{1-w^2}X_2(t)$ where $X_1$ and $X_2$ are brownian motions with drifts and brownian increments $dX_1(t)= \mu_1dt + ...
1
vote
1answer
38 views

Probability of Renewal Processes

Suppose that there are two brands of replacement components, Brand X and Brand Y, and that for political reasons a company buys a replacements of both types. When a Brand X component fails it is ...
1
vote
1answer
26 views

Variance of Martingale Difference Sequence

I'm having trouble understanding part of one of the examples here. This is taken from Hamilton's book Time Series Analysis, p. 194. My question is this. I don't understand why $$ E[X_t^2] = ...
0
votes
2answers
30 views

Finite in probability implies finite expectation

Let $T_n$ be a random variable with $T_n=X_1+...+X_n$ where the $X_i$'s are iid. Further we set $N(t)=max\{ n: T_n\leq n\}$. If $\Pr(N(t)<\infty)=1$, does this implies ...
2
votes
2answers
143 views

Traversing an array and counting the number of distanct number from the given elements in an array.

You are given an array $A[0 \ldots n-1]$ of $n$ numbers. Let $d$ be the number of \emph{distinct} numbers that occur in this array. For each $i$ with $0 \leq i \leq n-1$, let $N_i$ be the number of ...
1
vote
2answers
36 views

Expectation of a parallel system

A system consists of $n$ components in parallel. The lifetimes of the components are i.i.d. exp($\lambda$) random variables. The system functions as long as at least one of the $n$ components is ...
1
vote
1answer
40 views

If $dX_t=b_tdt+\sigma_tdW_t=\tilde{b}_tdt+\tilde{\sigma}_tdW_t$ then $b_t=\tilde{b}_t$ and $\sigma_t=\tilde{\sigma}_t$ a.s

Let $X_t$ be an Ito's process where $dX_t=b_tdt+\sigma_tdW_t=\tilde{b}_tdt+\tilde{\sigma}_tdW_t$. Prove $b_t=\tilde{b}_t$ and $\sigma_t=\tilde{\sigma}_t$ a.s Here my solution for ...
0
votes
2answers
28 views

If $u(z),$ where $Z_t=W^1_t-iW^2_t$, is a complex anlyt. fx, show $du(Z_t)=u'(Z_t)dZ_t$

If $u(z)$ is a complex analytical function, where $Z_t=W^1_t-iW^2_t$ is a complex Wiener process, show $du(Z_t)=u'(Z_t)dZ_t$.
0
votes
1answer
30 views

Quadratic Variation for $X_t= \int \sigma_s dW_s$ where $\sigma_s \in S$

Let $\sigma_s \in S$. Setting $X_t=\int^t_0 \sigma_s dW_s$ and partitioning the interval $[0,t]$ i.e. $0=t^n_0<t^n_1... $ such that $d_n=\max_i |t^n_{i+1}-t^n_i| \rightarrow 0$ as $n \rightarrow ...
0
votes
0answers
22 views

Transition matrix, stationary distribution and expected number

A company wants to operate s identical machines, but they are subject to failure according to a given probability law. To replace them, the company orders new machines at the beginning of each week to ...
0
votes
0answers
41 views

Expected time to reach 6th return time of Markov Chain

I'm having a hard time figuring out this problem from Resnick's Adventures in Stochastic Processes: Harry is negotiating a new tv show and the negotiations follow a discretely indexed Markov chain. ...
0
votes
1answer
40 views

Quadratic Variation for $X_t= \int b_s ds$ where $b_s$ is an $F_t$ adapted process.

Let $b_S$ be an $F_t$ adapted process, Borel measurable in $t$ st $\int |b_s|^2ds < \infty$ (a.s). Setting $X_t=\int^t_0 b_sds$ and partitioning the interval $[0,t]$ i.e. $0=t^n_0<t^n_1... $ ...
1
vote
2answers
59 views

Does a Brownian motion remain in any given open set for a given interval of time with positive probability?

Let $B$ be a standard $d$-dimensional Brownian motion. Given $b>a>0$ and an open ball $U$ in $\mathbb{R}^d$, I want to be able to comment on the probability that $B$ remains in $U$ during the ...
2
votes
0answers
37 views

Why is a brownian motion conditioned to stay positive a Bessel-3

I am told this result long ago but I still don't know how to prove it. Is it because that this conditioning can be turned into a Girsanov probability change? Or is there any simpler ways to see it?
2
votes
1answer
36 views

Probability distribution of Poisson process

Let $X_t$ and $Y_t$ be two independent Poisson process with rate parameter $\lambda_1$ and $\lambda_2$, respectively, measuring the number of customers arriving in stores $1$ and $2$, respectively. ...
0
votes
0answers
88 views

Expectation of a Poisson Process

Cars pass a certain street location according to a Poisson Process with rate $\lambda$. An old lady and her trusty boyscout want to cross the street at this location. They wait until they can ensure ...
2
votes
1answer
55 views

About the increasing process in the Doob-Meyer decomposition

As we know, a RCLL submartingale on [0,T], $Y$, in class D can be decomposed as: $$Y_t=Y_0+M_t+A_t,\ a.s.,$$ where $M$ is a martingale and $A$ is an increasing previsible process. In my question, I ...
2
votes
1answer
82 views

Poisson Processes and Arrival of Passengers at a bus stop

Arrivals of passengers at a bus stop form a poisson process $X = {X(t); t>= 0}$ with the rate of 4 per unit of time. Assume that $T$ denotes the arrival time of the next bus. Then $X(T)$ is the ...
0
votes
0answers
23 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 ...
0
votes
1answer
25 views

Expectation of the Product of a Poisson and an Exponential Random Variable

Problem: Consider a Poisson process with parameter $\lambda$ . Let $T$ be a random variable representing the time required to observe the first event, and $X_\frac{T}{K}:=N\left(\frac{T}{K}\right)$ be ...
1
vote
6answers
114 views

where can I take an online Second course in Linear Algebra or second abstract algebra?

Hi i am looking for an online/independent study Second course in Linear Algebra or abstract algebra I and II? Can you point me in the right direction. I need college credit. I need help ASAP Even a ...
1
vote
2answers
48 views

How to prove two stochastic processes have the same distribution

Let $C([0,\infty), R)$ be the canonical space of continuous functions. Assume $(\Omega, \mathcal{F}, \{\mathcal{F}_{t}\}_{t\geq 0})$ is a measurable space with a filtration. Let $P, Q$ be two ...
0
votes
0answers
14 views

Calculating the joint distribution of an affine stochastic process

I have a recursively defined system given by $$X_i = X_{i-1}H_i+N,$$ where $H_i$s are i.i.d. exponential random variables and N is a constant. At the $n$th iteration I have $$X_n = ...
1
vote
0answers
28 views

Jump time of a previsible process is previsible?

Here is my question: In our setups, the filtration satisfies the usual condition. $V$ is an increasing process with only jumps (between the jumps it is flat). We also know that $V$ is right ...
1
vote
0answers
23 views

Expected Sum of Weights after Drawing Without Replacement

We have an urn containing $k$ balls where for all $i:1\le i\le k$, the ball $b_i$ has the size $s_i$ that determines its probability to be drawn. For instance, a ball $b_i$ with size $s_i=3$ is ...
2
votes
0answers
16 views

Supremum of empirical process

Suppose $\hat{F}_{n}$ is the empirical distribution function based on a sample $(X_{1},\ldots,X_{n})$, where each $X_{i}$ has distribution function $F$. Also, suppose that the distribution of ...
0
votes
0answers
20 views

How to understand this equation for brownian motion

I am reading this article from the notes 'an intro to SDE'. Here I dont know why in (1) he take that integral from - infinity to infinity. I mean why we do that? I just dont know what the physics or ...
4
votes
0answers
122 views

Asymptotics of sum of binomial distributions

Definition 1: For any random variable $X$, we define $Bin(p,X)$ as a variable with binomial distribution having parameters $p$ and $X$. Definition 2: For all $i \in \mathbb{N}$, define recursively ...
1
vote
0answers
12 views

Characterizing limit of value functions in a stochastic control problem

Consider a probability space $(\Omega, \mathcal F , \mathbb P)$, $(B_t)_{t\geq0}$ M-dimentional brownian motion adapted to a filtration $(\mathcal F_t)_{t\geq0}$ over $\Omega$. In this context ...
3
votes
0answers
42 views

2-state HMM / ARMA process?

I have issues with this problem: Let $\{X_t, t\in \Bbb N\}$ be a 2-state stationnary Markov chain, with transition $M$ (and $M(1,2)\neq 0 \neq M(2,1)$), let $\{W_t, t\in \Bbb N\}$ be a strong ...
1
vote
3answers
45 views

The uniqueness of solution for stochastic differential equation involved with sign function.

When I read a paper about Levy distribution thoerem (http://www.maphysto.dk/publications/MPS-RR/1998/22.pdf). In the first page, the author mentioned the following: There is a unique strong solution ...
2
votes
2answers
63 views

Why are these two Poisson-processes independent?

I have two poisson-processes, I have seen a mathematical proof that they are independent, and offcourse they must be independent since the proof is in several textbooks. But logically I can not ...
1
vote
2answers
44 views

Poisson Process and Conditional Probability

Let $X= (X(t); t\ge0)$ be a poisson process with the intensity ($\lambda$ per hour) A) Find the conditional probability of having $m$ events in the first $t$ hours, given that there were $n$ events ...
1
vote
1answer
46 views

Markov Chain: prove that state is positive recurrent by calculating expected # of transitions to return to this state

Given the transitional probabilities below (states: 0,1,2,3), I need to prove that state 3 is positive recurrent by calculating expected # of transitions to return to this state $$P = ...