0
votes
0answers
19 views

Stochastic processes on group-valued variables

I have had this question in my head for a long time, and if I don't find out the answer I may explode. So I'm familiar with a basic Ito process, let's say: $dX_t = \mu d t + \sigma d Z_t$. There ...
0
votes
0answers
23 views

Kolmogorov zero-one law in continuous time?

Let $(X_t : t \geq 0)$ be a stochastic process. Is it necessarily the case that $$P (\limsup_{t \geq 0} X_t \leq a) \in \{ 0,1\}$$ as it is in discrete time? If some conditions are needed on the ...
1
vote
1answer
15 views

How Do I Find The Permanent of a Double Stochastic Matrix n * n size

I am reading a book on Stochastic Models, and I don't understand this practice questions: A doubly stochastic n × n matrix S has all entries equal to 1/n. The permament of a n × n matrx A is ...
0
votes
1answer
34 views

Battery lifetime as normal distribution?

I want to model battery lifetime, which decrements continuously at every epoch (i.e., work-cycle) in the following way. So it takes values such as 100, 99.7, 99.3, 99.2, ... 0 (a continuous random ...
5
votes
2answers
67 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 ...
0
votes
1answer
23 views

Integrated Brownian motion: independent stationary increments?

Let $B_t$, $t\in [0,T]$ be a $d$-dimensional standard Brownian motion. Let $\sigma:[0,T] \rightarrow \mathbb R^{d\times d}$ be a deterministic function such that $$\sigma(u) = diag( \sigma_1(u), \dots ...
0
votes
1answer
19 views

Stopping Time and Brownian Motion [on hold]

Let $B_t$ be a Brownian motion. Let $a < 0 < b$. Consider $\tau: = \min\{T_a, T_b\}$ where $T_a := \inf\{s \geq 0: B_s \leq a\}$ and $T_b := \inf\{s \geq 0: B_s \geq b\}$, namely, the first ...
1
vote
1answer
13 views

Piecewise homogeneous Poisson process

Is there a name for a Poisson process which is piecewise homogeneous? I.e. time-homogeneous but with a parameter change each increment. Any references appreciated.
0
votes
1answer
29 views

Stochastic Processes, requirement at ''source" probability space, is it always an product over $T$?

Let $(\Omega, \mathcal F, P)$ be a probability space, and let $(S, \mathcal S)$ a set $S$ together with a $\sigma$-Algebra over $S$, also let $T$ be some index set, then for each $t \in T$ let $X_t : ...
0
votes
0answers
11 views

Determining Bounds of a Generating Function of a Stopping Time [duplicate]

Consider the diffusion process $$DX_t=b(X_t)dt+\sigma(X_t)dW_t$$ where $\sigma\sigma*$ is positive definite and $b, \sigma$ are smooth and bounded. Given a one-dimensional domain bounded from 1 side ...
0
votes
1answer
32 views

How does a Nakagami Random Variable behave?

A Nakagami random variable has the following pdf $$f_{\Omega,m}= \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1}e^{-\frac{m}{\Omega}x^2}$$ I have two questions regarding this random variable, 1- Is a sum of ...
0
votes
0answers
11 views

video lectures on stochastic geometryand point process in wireless communication domain

Can you pls share any link from where i can have video lectures on stochastic geometry with the application in wireless communication. while searching on net i have found some lecture notes in pdf ...
1
vote
0answers
20 views

Derivative of stochastic process

I have a set of data of a random process (one sample path). The process is sampled every 10 min and each sample is a 10 min average from a sensor. I can compute the statistics of the random process, ...
-2
votes
1answer
66 views

Is $\theta_1-\theta_2$ independent of $\theta_1-\theta_3$ given all are uniform random variables between $[-\pi,\pi]$

I have three random variables $\theta_1, \theta_2, \theta_3$ all are i.i.d uniformly over $[-\pi,\pi]$. These in reality represent angles in my problem that I am trying to solve. I have a linear ...
1
vote
1answer
11 views

Find distribution of a bernoulli funtion of a unifrom random variable?

I have a uniform random variable $\theta \in [-\pi,+\pi]$. I also have a bernoulli function of this random variable $G(\theta)$, defined as follows, \begin{align} \begin{cases} 1 & \text{if $ - ...
2
votes
1answer
39 views

Conditional Integral of Square of Brownian Motion?

I am struggling to compute the expectation and variance of the following, where $W(s)$ is a standard Brownian motion: $$ X := \int_{0}^{A}W(s)^2ds$$ $$ Y:= \int_0^AW(s)ds $$ $$E[X\mid Y] = \space ?$$ ...
0
votes
1answer
36 views

A way to check the accuracy of a Markov chain?

I am not sure whether I should post this question on MSE or SSE. I will post it here 1st to see if I can get some feedback. Say I have a finite discrete Markov chain constructed maybe using some data ...
1
vote
1answer
125 views

The Brownian motion process in Sheldon M. Ross

Today I study Brownian Motion and Geometric Brownian Motion using textbook: An Elementary Introduction to Mathematical Finance, Third Edition by Sheldon M. Ross but I missed the class because I was ...
6
votes
1answer
117 views

Integral of Wiener Process and Central Limit Theorem

I am trying to solve the following exercise: (1) Given $W$ is a Wiener process, find a constant $M$ such that $\lim\limits_{t\to\infty} \frac{1}{t}\int_{0}^{t}\sin^2W_s ds=M$ (2) Then show ...
0
votes
1answer
22 views

Can anyone help me find the variance of this expression?

I have a vector of the form \begin{align} {\bf a }= \frac{1}{\sqrt{N}}[1, e^{jA}, e^{j2A},\cdots, e^{j(N-1)A}]^T \end{align} where A and N are constants. I also have a vector N of i.i.d ...
1
vote
0answers
27 views

Stochastic process using Markov chain (thief on the run!!)

I'm given an exercise where we are to simulate a thief escaping from an officer. The thief (let's call him/her T for simplicity) and an officer (O) have four cities to be in. Let's call the cities A, ...
1
vote
1answer
42 views

Tips for evaluating $P(X\gt Y\gt Z)$

Does anyone know of any references for how to evaluate stochastic inequalities? Surprisingly, I can't find any good references for general problems. For example, suppose we have three random ...
0
votes
1answer
38 views

What is the distribution of a Brownian motion evaluated at times defined by Brownian motion?

Let $X_t$ and $Z_t$ be independent, $\mathbb{R}$-valued Brownian motions. For each $t$, the process $X_{|Z_t|}$ defined as $$\omega\mapsto X_{|Z_t(\omega)|}(\omega)$$ is measurable (with respect to ...
1
vote
0answers
36 views

Formal examples of Poisson point processes

I am self-studying probability theory and currently the Poisson point process (PPP) gives me hell, firstly because the definition of a point process in general and PPP in particular seems rather ...
3
votes
1answer
44 views

Confusion regarding almost sure events. If given infinite time, will a discrete-time gaussian process cover the entire real line?

This question really pertains to any discrete time continuous-valued, stationary stochastic process on the real line, but the Gaussian process will be adequate for this question. I have this ...
0
votes
1answer
54 views

Stochastic process gambler's ruin [closed]

This is a gambler's ruin problem I would appreciate if anyone can give me a hint about how to solve it. So A, B play this game by tossing a coin. If H shows then B gets 1 dollar from A and if T shows ...
1
vote
1answer
30 views

Bound for the variance of a stochastic process

Given a random variable $X$ and $N$ realizations of the stochastic process associated to $X$, a theorem gives a bound for the $\sigma^2[X]$: $$\sigma^2[X]\le\frac{1}{4}(A-a)$$ where $A$ and $a$ are ...
1
vote
2answers
63 views

conditional expected value - Poisson process plus random variable

I've struggled with this actuary excercise for a while and I don't know how to do it: Each claim can be characterized by two random variables $(T,D)$, where $T$ is the moment of reporting the claim ...
0
votes
1answer
15 views

Recurrence of states in a function of a Markov chain

Suppose $X$ is a Markov chain (or process, for that matter) and suppose further $f(X)$ is also a Markov chain. Let $s$ be a recurrent state in $X$. Is there a general way to determine the recurrence ...
0
votes
0answers
30 views

Independent Brownian motions question

Let $B$ and $W$ be independent Brownian Motions and let $\tau$ be a stopping time of $W$. Is it true that $E[\int_0^{\tau} B_s \, dW_s] = 0\text{ ?}$ So far I have tried the following: The integral ...
1
vote
1answer
23 views

mean hitting time of a level and growth rate of maximum process

Let $X_t$ be the absolute value of Brownian motion starting at $0$, let $\tau_x$ be it's first hitting time of the level $x>0$, and let $M_t$ be it's running maximum up to time $t$. Suppose we knew ...
0
votes
0answers
22 views

Optimal stopping problem

Consider the OU process: $dX_t = -X_tdt + dW_t$, $X_0 = 0$. Compute the optimal stopping time for the following problem: $$v = \sup_{\tau} E[|X_{\tau}| - \tau]$$ So far I have set $L\phi = 0$, ...
1
vote
0answers
29 views

Correlation and First Order Stochastic Dominance

Suppose we have a random variable $X \sim [0,1]$ with a continuous distribution $F_X(x)$. Suppose $I \in \left\{0,1\right\}$ is a discrete random variable with $\text{Prob}(I=1 \ | \ X=x)$ strictly ...
0
votes
0answers
11 views

Entropy of non-ergodic process

Two coins have been kept in a box, One is fair while the other is biased. One coin is picked. The probability of either coin being picked is equal. The picked coin is then tossed again and again to ...
0
votes
1answer
17 views

$n$-step transition probability of a Markov chain

Let $(X_t)_{t\in\mathbb{N}_0}$ be a time-homogenous Markov chain over a finite state space $\left\{1,\dots,m\right\}$, so that we've got $$\Pr(X_{t+1}=j\mid X_t=i_t,\dots,X_0=i_0)=\Pr(X_{t+1}=j\mid ...
3
votes
1answer
54 views

My understanding of “$\sigma-$algebra represents information”.

In stochastic process $\{X_t\}_{t\ge0}$ adapted to $\{\mathcal F_t\}_{t\ge0}$ where $\mathcal F_s\subset\mathcal F_t,\forall s<t$. Many textbook say that $\{\mathcal F_t\}_{t\ge0}$ represents a ...
3
votes
2answers
34 views

Interpretation for the determinant of a stochastic matrix?

Is there a probabilistic interpretation for the determinant of a stochastic matrix (i.e. an $n \times n$ matrix whose columns sum to unity)?
0
votes
1answer
32 views

Prove of Stopping time

Let $(X_k)_{k\in\mathbb{N}}$ be iid random variables with $\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=\frac{1}{2}$. Let $Z_n=\prod_{k=1}^n(1+X_k)$, so $Z_n$ a martingale. Consider ...
1
vote
1answer
39 views

Markov property question

In every book I can find, the Markov property for ito diffusions, $E[f(X_{t+h})\mid F_s] = E^{X_t}f(X_h)$ is stated for $\textbf{bounded}$ Borel functions. However, I have the following statement ...
1
vote
0answers
10 views

Holder continuity, brwonian motion [duplicate]

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 ...
0
votes
0answers
17 views

integrability condition stochastic process

Consider the finite time interval $[0,T]$ and the stochastic process $(X(t); t\leq s)$ Can the integral \begin{align} \int_{0}^{T}X(s)ds \end{align} de defined if the stochastic process $X$ is not ...
1
vote
1answer
59 views

Stopped sigma-algebra for a counting process

let $(\Omega, \mathcal{A}, P)$ be a probability space and $(N_t)_{t \geq 0}$ a right-continuous counting process with jumps of size 1, $N_0 = 0$ and canonical filtration $\mathcal{F}_t := \sigma( N_u ...
0
votes
0answers
73 views

balls in bins — waiting time until $k$ bins are occupied

Consider the classic balls in bins problem: we throw balls one by one into $n$ bins independently and uniformly. Define $\tau(k)$ for $1 \le k \le n$ to be the number of balls we have thrown until $k$ ...
1
vote
0answers
47 views

Stochastically continuous but a.s. discontiuous process

This is a homework question so no answers please The problem is: Find a process $X_{t}$ s.t. $\forall t_{0}\geq 0$ and $\varepsilon>0$ we have $lim_{n\to ...
0
votes
0answers
18 views

Extension of martingale representation theorem.

It seems that the proof I am reading of the Martingale Representation Theorem, "A square integrable RCLL martingale which is adapted to the augmented filtration of a Brownian Motion must be an Ito ...
2
votes
1answer
40 views

Square integrable stochastic process

Suppose that for a stochastic process we have \begin{align} \mathbb{E}\left[\int_{0}^{T}X^{2}(t)dt \right]<\infty \end{align} where $T<\infty$. Does it holds that $|X(t)|<M$, where $M$ ...
1
vote
0answers
27 views

Locate proof of Second Fundamental Theorem of Asset Pricing

Where can I find a $\textbf{rigorous}$ proof of the Second Fundamental Theorem of Asset Pricing. That is, A market is complete if and only if it has a unique risk neutral measure. Please do not ...
0
votes
1answer
22 views

Is random walk on half-line a martingale?

Let $X_n$ denote a random walk on $\mathbb Z^+$ starting at $0$. Is it a martingale? In Probability with Martingales by David Williams on page 99 it is claimed that it is, but I cannot understand ...
1
vote
1answer
24 views

Joint Quadratic variation

Let $X,Y$ be square integrable Right continuous martingales. If $Z$ is the total variation of $\langle X,Y\rangle$, how can I show that $$Z \leq \frac{1}{2}[\langle X\rangle + \langle Y\rangle].$$ I ...
0
votes
0answers
16 views

Sum over stochastic processes on the same set of categories

I have a stochastic process consisting of multiple (stochastic) steps, for which I want to know if I can substitute (or at least approximate) it by summing over the deterministic and stochastic parts ...