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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What is the distribution of a stochastic process?

Let $(\Omega,\mathcal{A})$ be a measurable space $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$ $I\subseteq\mathbb{R}$ $X_t$ be a random variable on ...
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Scaling Gaussian process

If $X\sim N(0,I)$, then $VX \sim N(0,VV^\top)$. Consider the Gaussian process $Y\sim \mathcal{G}(0,k(t,t')),t,t'\in[0,1]$, that is any finite-dimensional distribution of $Y$ is a multivariate normal. ...
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22 views

Why is the expectation of essential supremum equal the supremum of expectations

Let $\{X_i\}$ be a sequence sequence of nonnegative r.v. which has the lattice property. This implies that there is a sequence of indices $\{i_n\}$ so that $\{X_{i_n}\}$ is nondecreasing and ...
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19 views

Can an indicator function be a valid Radon Nikodym derivative?

Take a process $X_t$ defined on a canonical space with probability $\mathbb{P}$. Can the indicator function $\mathbb{1}_{X_t< U}$ be a Radon Nikodym derivative? That is can we have a measure ...
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53 views

Regarding “Two Singular Diffusion Problems” by William Feller

I'm currently reading the research paper, Two Singular Diffusion Problems, by William Feller (1950). However, I don't understand how Feller derived the solution $(3.5)$ given equation $(3.4)$ in his ...
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35 views

how to prove this function is a probability measure in $U_B$

Let $(\Omega, U, P)$ be a probability space. and $B\in U$, $P(B)\gt 0$ $U_B =\{A: A=B\cap C, C\in U\}$ its class in $\Omega$ is a $\sigma$-algebra and $P_B : U_B \to \Bbb R$ $A \to ...
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2answers
23 views

Transition probability matrix

In the article here it had this question. A walker moves on two positions a and b. She begins at a at time 0, and is at a next time as well. Subsequently, if she is at the same position for two ...
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1answer
55 views

Long term probability in Markov Chains

I was practicing some questions on transition probability matrices and I came up with this question. You have 3 coins: A (Heads probability 0.2),B (Heads probability 0.4), C (Heads probability ...
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13 views

Ross Intro to Probability Models--Example 4.4

Can someone explain to me please how we derived the Transition Matrix? Why we decided to put $P_{00} =0.7$ and $1 - P_{00} = P_{02}$. I just don't see it the way Ross defined the different states. ...
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1answer
54 views

Separable generating function of a pair of dependent discrete random variables

Independence is sufficient but not necessary for the generating function of the sum of two random variables to be the product of their individual generating functions. I am trying to come up with an ...
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1answer
29 views

Prove that a right-continuous stochastic process is product measurable

Let $X=(X_t,t\ge 0$ be a real-valued stochastic process on a measurable space $(\Omega,\mathcal{A})$ with almost surely right-continuous paths $\mathbb{F}:=(\mathcal{F}_t,t\ge 0)$ be a filtraiton on ...
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1answer
38 views

Continuity of $x \mapsto E_{x}[F]$, Brownian motion

I have a question about Brownian motion. Let $(\Omega,\mathcal{F},P)$ be a Probability space and $(B_{t})_{t \in [0,\infty[}$ be a standard $1$-dimensional Brownian motion defined on ...
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1answer
61 views

Strong solutions SDE inequality with an application of Gronwall's inequality

Suppose that we have a general SDE on a probability space $(\Omega,\mathcal{F},P)$ defined by: $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) d W_t, $$ where $W$ is a Brownian motion and $b$ and $\sigma$ are ...
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12 views

Cholesky decomposition of positive semi-definite covariance matrix

I am trying to simulate a column vector of random complex variables, $\boldsymbol{x}$, which has a has a given covariance matrix: $$ \boldsymbol{C}=E\left[\boldsymbol{x}\boldsymbol{x}^{*}\right] $$ ...
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27 views

what is determinantal process?

Would anyone please explain what does this mean? A random point process $P$ on a discrete base set $Y = \{1,\ldots,N\}$ is a probability measure on the set $2^Y$ of all subsets of $Y$. Let $K$ ...
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40 views

2 User Queuing Model Probability Problem

Consider two users who arrive to a system with exponential arrival times with parameters $\lambda_a$ and $\lambda_b$. Once they arrive, the users stay in the system for an exponentially distributed ...
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83 views

Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$

I am currently working on an exercise that requires the knowledge of the distribution of $\int_0^t \frac{W_s}{s} \,ds$, where $W$ is a Brownian motion. I can compute the distribution of $\int_{0}^T ...
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1answer
34 views

Strong Markov Property and Product of Expectations

Let $(B_{t})_{t\geq0}$ be a Brownian motion and let $\tau=\inf\left\{ t\geq0:B_{t}\leq-4\right\} $ be a stopping time. Then the strong Markov property ensures that e.g. $A:=\left\{ ...
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55 views

What is the integral of a family of diffusion processes? [closed]

Let $S$ be an infinite subset of $[0,1]$. For all $s \in S$, let W_s(t) be a standard Wiener process. Definite P(s)_t = \mu(P,s,t) dt + \sigma(P,s,t) dW^s_t Can we characterize? $$F_t= \int_S P(s)_t ...
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10 views

Informal Introduction to Cox Processes

Can anyone recommend a good introduction (preferably not terribly formal) to Cox processes? (Inhomogenous Poisson processes with a parameter that is also stochastic.) This introduction to spatial ...
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1answer
26 views

Show a random walk is transient

I was going through some problems related to Markov chains and I got stuck on this bit: We are given a random walk on $Z$, defined by the transition matrix $p_{i,i+1}=p$ and $p_{i,i-1}=1-p$. How to ...
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1answer
21 views

Limiting distribution of a Markov chain?

I have the problem below. There are n identical machines. They are all operational at time 0. The lifetime of each one is an exponential random variable with rate L. There are r repairmen (1 ≤ r ≤ ...
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21 views

$2$-dimensional density of Brownian bridge?

I know that a $1$-dimensional Brownian bridge $B(t)$ just follows a normal distribution with mean $0$ and variance $t(1-t)$. But how do I compute the 2-dimensional density? I mean, $\{B(s), B(t)\}$ ...
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46 views

Ito Differential Equation example [closed]

Could someone explain Ito through an example as following? How to use Ito differential equation to find $dy$ , where $y = e^{w(t)}$
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28 views

If two Brownian motion starts and end at the same points, can we say something about there difference?

Let $X$ and $Y$ be two standard Brownian motions with mean $0$ and variance $1$, both started at zero. If we know that \begin{align} X_n &= Y_n, \end{align} for some $n>0$, can we say ...
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1answer
24 views

If $B=(B_t,t\ge 0)$ is a Brownian motion and $(\mathcal{F}_t,t\ge 0$ is its generated filtration, then $X_t-X_s$ are independent of $\mathcal{A}_s$

A real-valued stochastic process $B=(B_t,t\ge 0)$ is called Brownian motion $:\Leftrightarrow$ $B_0=0$ $B$ has independent and stationary increments, i.e. ...
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37 views

Insurance claims Poisson problem derving expected value and variance

If I have that claims arrive at an insurance company according to a Poisson process $\{N(t) : t \ge 0\}$ at a rate $\lambda > 0$ and $X_i$ denotes the claim size of the $ith$ claim. I assume that ...
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21 views

Examples of Non-Markov process with continuous time and finite set of states.

What is the best real world examples of non-Markov process with continuous time, but with finite set of states?
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40 views

Expectation of squared Ito integral

Let $\omega$ be a standard Brownian motion. How do you compute the expectation involving the square of an Ito integral: $ ...
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14 views

Prove that an operator from $L^2(\Omega;C(s,T;\mathbb R^n ))$ into itself is well defined

I need an help proving the following estimate. First, we fix the notation. Let $L^2(\Omega;C(s,T;\mathbb R^n ))$ be the set of continuous and adapted processes $\{X_t:t\in [s,T]\}$ (valued in ...
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Definition of rough path

There are many books and notes on the rough path theory. A rough path is defined as an ordered pair $(X, \mathbb X)$, where $X$ is a path mapping from $[0, T]$ to some Banach space $V$ and $\mathbb X: ...
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Proving that a process is class DL

Let $(X_{t})$ be a stochastic process with $X_{t}\sim\mathcal{N}(\xi_{t},\sigma_{t}^{2})$ where $\xi_{t}\downarrow0$ and $\sigma_{t}^{2}\uparrow1/2$. What would be the most straightforward way ...
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23 views

Is a linear random walk with jump recurrent?

Let $\lambda_0=10^5$ or any other large integer. Define the recursive "process": $\lambda_t=\text{sample from a Poisson distribution with mean }\lambda_{t-1}$. Is this process recurrent? I mean, after ...
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1answer
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Function of mean square continuous process

I have been asked to prove that, if $\{X_t\}$ is a ($n$-dimensional) mean square continuous process and $f:\mathbb{R}^n \rightarrow \mathbb{R}^d$ is a Lipschitz function, the process $\{f(X_t)\}$ is ...
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1answer
31 views

Poisson process Probabilities

If I assume that $\{N(t)=: t \ge 0\}$ is a Poisson process with intensity $\lambda$. For $0<s<t$, how would I find the $\Pr\{N(t)>N(s)\}$?
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Markov Chain States

Given a MC with five states $\{1,2,3,4,5\}$ and transition matrix \begin{bmatrix} 0.5& 0.5 & 0 & 0 & 0 \\ 0.75 & 0.25 & 0 & 0 & 0 \\ 0 & 0.25 & ...
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dependent “time change” of a.s. convergent random variables

Let $(X_n)$ be a sequence of random variables, s.t. $\frac{X_n}{n^p}\to X$ a.s. for some $p>0$. Now let $(Y_t)$ be a discrete stochastic process, s.t. $\frac{(Y_t)^p}{t}\to Y>0$ a.s. We only ...
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1answer
31 views

Show that $((N_t-t)^2-t)_{t \geq 0}$ is a martingale for a Poisson process $(N_t)_{t \geq 0}$

I am asked to show that if $N$ is a poisson process of intensity $1$, then: $X_t=N_t-t$ is a martingale. $X_t^2-t$ is a martingale. I have done the first part easily, using independence of ...
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Application of Doob's Optional Stopping Time Theorem on new stopping time

Consider a random walk on a line starting at 0. On each step the probability of moving in either direction (right or left) is 1/2. There are two particular points on the line -a, and b. If $\tau$ is ...
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1answer
64 views

Exponential of a uniform integrable martingale is a submartingale

For reference I want to prove this Lemma: Let $M$ be a uniformly integrable martingale with the additional property that $\mathbb{E}[ \exp(M_\infty)] < 1$. Then $\exp(M)$ is a uniformly ...
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1answer
73 views

Compute Var(x=X1+X2+…+Xn)

Compute $Var(X_1+X_2+...+X_n)$ given $X_1,X_2...$ are iid.,$EX=\mu,Var(X)=\sigma ^2$,and $Var(N)=\sigma [n]^2$, N is a random variable of nonnegative integers independent with X, and my solution ...
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Pathwise solution of a stochastic integral equation, without stochastic calculus

Let $f$ be a Lipschitz continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $W$ be a standard Brownian motion. I don't know any stochastic calculus (nothing about stochastic integrals, nothing ...
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1answer
22 views

gaussian process convergence

if I have a series of gaussian processes : ($W_{t}^{n}$ is gaussian process for every n) and I know that for every t there exist $W_t $ s.t $ E|W_t^n-W_t|^2\to0 $as $n\to \infty$. how can I show that ...
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11 views

reference for regime shifting models

I'm looking for a good introduction to regime shifting models. It would be nice to see things like simple example of regime shifting models, ways to detect a regime shift in data, fitting regime ...
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1answer
19 views

Expectation of a Wiener process at a Stopping Time - 2

I am working through an answer to the following question and I do not understand a statement given towards the end of the solution, specifically why $\tilde{W}(\sigma) = 1$. (This question is related ...
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37 views

What is the resulting stochastic process of divided Geometric Brownian motions

Let $W_{1,t},W_{2,t},...,W_{n,t}$ be $n$ independent geometric Brownian motions. Now let's say I construct the following processes: $$ X_1 = \frac{W_1}{\sum_i^n W_{i,t}} $$ $$ X_2 = ...
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Renewal process past exam question

Consider a renewal process ($N_t$, t ≥ 0) with independent inter-occurrence times $X_n$, n ∈ N, all having the same cumulative distribution function: $P(X_1 ≤ x) = w_1*F_1(x) + w_2*F_2(x)$, $w_1, w_2 ...
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35 views

Markov factorization of the density of an AR(1) process

Suppose we have a causal, stationary AR$(1)$ process with i.i.d. innovations $Z_t$. Then we know that it is a Markov as future value $X_{t+1} = \phi X_t + Z_{t+1}$ given the past $X_1,\ldots X_t$ ...
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21 views

Random Variable being $F$-measurable

It is said the Random variable is $F$-measurable if $\{X\leq x\}$ is an element of $F$. Is $X$ not $F$measurable once it is not less than or equal to $1$ $x$ or only for all?
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A variation of Lévy's characterization of Brownian motion

It is shown here, without using stochastic calculus, that if $W_t$ is a standard Brownian motion, then $$ f(W_t)-\frac{1}{2}\int_0^t f''(W_s)ds $$ is a martingale, where $f\in C^2$ and compactly ...