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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110 views

does a simple random walk eventually hit every point?

Let $M_n=\sum_{k=0}^{n}X_k$ be a simple random walk starting at $0$, where $P(X_n=1)=P(X_n=-1)=\frac{1}{2}$. What is the probability the random walk hits the point $z\in\mathbb{Z}$? I have a feeling ...
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1answer
151 views

On the quadratic variation

I understand that the Quadratic Variation of Brownian Motion $B_t$ is $[B_t,B_t]=t$ and I know that the equality is under the meaning of $\mathcal{L}^2$ convergence. Yet I saw in some book saying that ...
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1answer
47 views

Not using stochasstic integral how to prove $E\int_0^T W^2(t)dt<+\infty$?

Can anyone help me to prove this? Suppose $W_t$ ~ $N(0,t)$, then not using stochasstic integral (or anything related with Ito) how to prove $E\int_0^T W^2(t)dt<+\infty$? Thanks.
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1answer
188 views

How is a simple birth process is time-homogeneous?

Why is it that a simple birth process is time-homogeneous? The incidence of a birth in a small time interval depends on the population size at the time of start of the interval. Doesn't this ...
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1answer
223 views

Ito vs Stratonovich SDE with irregular time-dependence in coefficients

Suppose I am interested in the Stratonovich SDE $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) \circ dB_t $$ If the coefficients are smooth enough in time and space, I can show this is equivalent to the Ito ...
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1answer
317 views

Joint density of Markov process as product of conditional densities.

Let $(X(t))_{t\geq0}$ be a Markov proces such that s $X(0)=x_0$. Consider the random vector $(X(t_1),\dots,X(t_n))$ with corresponding joint density $g(x_1,\dots ,x_n)$. Is it then true that ...
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2answers
821 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
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1answer
741 views

Is a Bernoulli process a Markov chain?

For a Bernoulli process, the outcome of a future trial is independent of the outcome of past trials. I.e., the future behaviour of a Bernoulli process is independent of its past, i.e. a Bernoulli ...
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1answer
70 views

Does $\sup_{t \leq T} |M_{n_k}(t)-M_{m_k}(t)|\to 0$ imply $\lim_k M_{n_k}(t)$ exists and is continuous?

This came up in proving that $\mathcal{M}^2_c$ is a complete metric space using the invariant metric induced by $$ ||M|| = \sum_k \frac{||M(k)||_2\wedge 1}{2^n}. $$ Suppose $M_n(t)$ is a sequence of ...
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1answer
738 views

Why is this infinite-state-space Markov chain positive recurrent?

Given the following transition matrix for a Markov chain, how can I see that the chain is positive recurrent? I want to convince myself that the chain has a limiting distribution, and the chain is ...
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0answers
82 views

an exetension of Doob's inequality

Doob's inequality gives an estimation of $$\mathbb{P}(\sup_{0\leq t\leq 1}|X_t|\geq\varepsilon)$$ where $X$ is a martingale. Now I wonder how to estimate $$\mathbb{P}(\sup_{0\leq t,s\leq 1, ...
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1answer
84 views

an issue with expectation

in book's Bernt.Øks SDE i read that book and i have some serious issues :( page 21 Example 7.4.2 ) Consider n-dimensional Brownian motion $W=(W_1, \ldots ,W_n)$ starting at $a=(a_1,\ldots,a_n) \in ...
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1answer
65 views

the Expectation is equal the probability of the first exit time of openset

let $X_{t}$ itô diffusion and $1_{A}(x):=\begin{cases} 1 & \text{if } x \in A, \\ 0 & otherwise. \end{cases}$ and $\tau$ be The first exit time defined by : $$ \tau=\inf\{t>0:x_{t} ...
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1answer
82 views

Positive conditional expectation

We have a log-normal variable $x = e^{\mu +\sigma w}$, where $w$ is standard normal. We want to compute $E[(x - K)^{+}\mid z]$. I'm not sure if I can write it as $E[(x ̃ - K) 1_{\{x ̃- K>0\} }]$, ...
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1answer
99 views

Autocovariance of a given stochastic process?

I need to find the autocovariance $C_{YY}(t,s)$ of the stochastic process $Y(t) = t^2 X(t) -2X'(t)$ where $C_{XX}(t,s) = e^{-t^2 -s^2}$ is given. Using known properties I can calculate the ...
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0answers
382 views

Hitting times and stopping times for cadlag processes

Let $X$ be a cadlag stochastic process. If $X$ is continuous, then I already know that $\inf\{t\geq 0: X_t \in C\}$ is a stopping time whenever $C$ is closed in $\mathbb{R}$. What if $X$ is only ...
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2answers
239 views

Counter-examples of right-continuous filtrations

A filtration $(\mathcal{F}_t)$ is said to be right continuous if $\mathcal{F}_t = \bigcap\limits_{h > 0} \mathcal{F}_{t + h}$. (A filtration $( \mathcal{F}_t)$ is a collection such that each ...
2
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1answer
119 views

Characteristics of stochastic integral?

I need to describe a couple of integrals which are supposed to be evaluated in terms of Ito calculus. $$ I_1 = \int_0^t e^{-2\tau}dW(\tau); \\ I_2 = \int_0^t e^{-3 W(\tau)} dW(\tau); $$ Here ...
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2answers
178 views

How to understand Markov property?

I'm learning stochastic process in college. How to understand Markov property?I'm curious about what is the power and validity of Markov property ? A stochastic process has the Markov property if ...
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0answers
117 views

question about the sequential continuity of the set of probability measures

I have a question about the sequential continuity of the set of probability measures. Let $\Omega$ be the space of continuous functions defined in $[0,1]$ taking values in $\mathbb{R}$. Let ...
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1answer
64 views

Given a function and how do you determine the pdf of the left side given the pdf of the right side variables?

Given a function and how do you determine the pdf of the left side given the pdf of the right side variables? Specifically what is the pdf of W, given the equation $$ W = I^2 R$$ with $I$ and $R$ are ...
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1answer
269 views

inequality with sup and Expectations

let $X_{t}$ Itô diffusion how we can get : $$ \sup_{x\in \mathbb{R}^{n}}|E^{x}[f(X_{t})]| \leq \sup_{y \in \mathbb{R}^{n}}|f(y)|\sup_{x\in\mathbb{R}}E^{x}[1]=\sup_{y\in\mathbb{R}^{n}}|f(y)|.$$ Here ...
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3answers
9k views

How to generate a Cauchy random variable

How do I calculate a Cauchy random variable and how do I calculate the probability mass function to show it is "heavy tailed"
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1answer
2k views

Joint pdf of independent randomly uniform variables

Given two random uniform variables, $U$ and $V$, that are uniformly distributed over [0,1], how do you calculate the joint pdf of $X$, $Y$ where $X = F(U,V)$ and $Y = G(U,V)$ and where is the joint ...
3
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1answer
100 views

Long-memory process and convergence of finite dimensional distributions

We assume that $(X_t)_{t\in\mathbb{N}}$ is a stationary sequence of standard normal random variables such that for the autocovariance function holds $\gamma _X (k)\sim Ck^{2d-1}$ with $d \in (0,1/2)$ ...
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1answer
560 views

Hilbert's Barber Shop

Hilbert opens a barber shop with an infinite number of chairs and an infinite number of barbers. Customers arrive via a Poisson random process with an expected 1 person every 10 minutes. Upon arrival, ...
2
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2answers
317 views

Progressive measurability of a specific set related to Brownian motion

Let $\{W_t: t \in R_+\} $ be a standard Brownian motion process on a given probability space. I am interested in assessing the progressive measurability of the following set: $Z(\omega) := \{t: ...
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0answers
127 views

Conditional expectation with three random variables

We have $N_1, N_2, N_3$ normally distributed random variables with $µ_i =E[N_i]$, $σ_{ij}=Cov(N_i,N_j)$. We also have $\tilde{µ}_i=E[N_i|N_2 = x] $, $\tilde{σ}_{ij}=Cov(N_i,N_j|N_2 = x)$ and $v^2 ...
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1answer
199 views

What is the expected value of the product of 3 Ito Integrals?

How can I calculate the expected value $\mathbb{E}(I_{110}^2 * I_{10})$, where $I_{110}$=$\int_{t_0}^T \int_{t_0}^{s_3} \int_{t_0}^{s_2} 1\, \, dW(s_1) dW(s_2) ds_3$ and $I_{10}=\int_{t_0}^T ...
4
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1answer
198 views

Poisson process and an independent Wiener process

Let $\{W_t\}_{t\geq 0}$ be a standard one-dimensional Wiener process and $\{N_t\}_{t\geq 0}$ an independent rate-1 Poisson process. Define $T$ to be the first time (if ever) $t \geq 0$ such that $W_t ...
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1answer
77 views

Relation between three normally distributed random variables

We have $N_1, N_2, N_3$ normally distributed random variables with $µ_i =E[N_i]$, $ σ_{ij}=Cov(N_i,N_j)$. How can I write a relation between these three random variables to prove that: $E[N_i|N_2 = ...
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1answer
53 views

Uniform convergence in probability and its connection to the convergence of distribution functions

I have the following question: Let $(X_i)_{i\in\mathbb{N}}$ and $(Y_i)_{i\in\mathbb{N}}$ be real stationary sequences with finite variances and further let $Z$ be a real valued random variable with ...
2
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1answer
227 views

Quadratic variation of a purely discontinuous martingale

Definition A cadlag $L^2$-martingale $M$ is called purely discontinuous, if $M \cdot N$ is a martingale for any continuous $L^2$-martingale $N$. (If not otherwise mentioned, I always assume that the ...
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1answer
109 views

Covariance combined with normal distribution

We have $N_1$ and $N_2$, normal distributed random variables with averages $µ_i=E[N_i]$ and variances $σ_i^2=Var[N_i]$ and $c = Cov(N_1, N_2)$. We want to compute $E[e^{N_1} I(N_2>0)]$, where I is ...
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1answer
75 views

Inequality brownian motion

I have the probability $P(e^{σB_t+ αt}>Kf_t)$, where σ, α, K constants, f is a function of t and $B_t$ brownian motion. This probability must be independent of t. So why is $f_t$ chosen such as ...
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1answer
272 views

Conditional expectation with brownian motion

I want to find $E[e^{σB_t}|∫_0^1B_s ds]$. I make the notation $∫_0^1B_s ds = z$, and I know that: $E[B_t|Z]= 3t(1 - t/2)z $, $Var [B_t|Z] = t - 3t^2(1 - t/2)^2$ and $Var(∫_0^1B_s ds) = 1/3$. The ...
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0answers
223 views

Blackwell–Girshick equation

I came across Blackwell–Girshick equation days before,but I have found nothing about it. What is Blackwell–Girshick equation? Can you give some explaination or links?
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0answers
58 views

Contradiction on equality with stochastic integrals

I want to compute $E[∫_0^tB_u \, du ∫_0^sB_u \, du]$ and I know from another source that should be equal to $ts^2/2$. But when I try to compute it like: $$\begin{align} & E\left[(tB_t- \int_0^tu ...
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1answer
114 views

Computing expectation of a stochastic integral

I need to compute the expectation $$E\left[\int_0^tu \, dB_u \int_0^s u \, dB_u \right].$$ Being that is my first question, how can I initialize MathJax if I have it on my hard drive.
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1answer
64 views

equivalent conditions of u.a.n

Suppose that $X_{1},X_{2},\cdots$ are independent random variables with mean $E[X_{k}]=m_{k}$ and variance $Var(X_{k})=\sigma_{k}^2$, and define $D_{n}^2=Var(X_{1}+\cdots+ X_{n})$.Show that ...
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1answer
40 views

Second order stochastic dominance characteristic

If x is Second Order stochastic dominant(SSD) to y, then it is equivalent to state that $r_x + \epsilon = r_y $ with $E[e|r_x] \le 0 $ Proof: $E(U(1+r_y) = E[U(1+r_x+e)] = E[E[U(1+r_x +e)|r_x]] ...
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1answer
694 views

Levy-Ottaviani's inequality

The question how to show convergence in probability imply convergence a.s. in this case? uses a result called Ottaviani's inequality. Where can I learn about the original Ottaviani's inequality, and ...
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1answer
113 views

Issue with a Poisson process and its jump times

Let $(N_t)_{t\geq 0}$ be a Poisson process and $$T_n = \inf\{t\geq 0, \ N_t \geq n\}$$ Now given $t \ge 0$ how to compute $$ \mathbb{E} \left[ \sum_{n=1}^{N_t} X_{T_n}\right] $$ ? where $(X_t)_{t\ge ...
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0answers
195 views

Variance of the first return time of a simple random walk on an hypercube graph

I am trying to solve this problem.... I have a simple random walk on a $d$-cube (finite graph). At each vertex of the graph, the particle chooses one of $d$ edges equally likely. I need to calculate ...
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1answer
224 views

If quadratic variation of a local martingale is zero then it is itself zero

Let $M$ be a local martingale, if we need it, we can assume that $M$ is continuous. We know that $\langle M\rangle =0$. This implies that $M$ and $M^2$ are local martingale. Can we conclude that ...
4
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1answer
148 views

How to compute $E\left[W_t \int_0^t s \, dW_s\right]$?

I want to compute $E\left[W_t \int_0^t s \, dW_s\right]$ where $W_t$ is a Brownian motion. My attempt below is based on some very shaky mathematics; in particular I have no justification of the 4th ...
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2answers
142 views

expected life absorbing Markov Chain

No idea on how to start this question. Any help would be much appreciated. A flea lives on a polyhedron with N vertices, labelled $1, . . . , N$. It hops from vertex to vertex in the following manner: ...
2
votes
1answer
290 views

Average number of bins occupied when throwing $n$ balls into $N$ bins

There are $n$ balls and $N$ bins. At each time, a ball is thrown in one bin of $N$ bins at random. This repeats n times. So that in total $n$ balls are thrown into bins. The question is, on average, ...
2
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0answers
141 views

Intuitive meaning of spectral radius of a Markov chain transition matrix?

What is the intuitive meaning of the eigenvalues and in particular of the spectral radius of the transition matrix corresponding to a Markov chain?
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1answer
57 views

Regarding change of measure

Suppose a Gaussian process $\{B_t\}$, apparently $2^{B_t}$ is not a martingale. Can someone teach me how to change the measure so that $2^{B_t}$ is a martingale? Thanks.