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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81 views

Reality check: $\mathbb E \{ B_s B_t ^2\}=0 $

I desire to calculate $\mathbb E \{ B_s B_t ^2\} $, where $B$ is a standard brownian motion starting from zero. I want to be sure I am not making any mistake on both reasoning and result, even if I ...
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1answer
89 views

Why is there “markov property” in proving the renewal equation for a renewal process?

When proving the renewal equation for a renewal process in Wikipedia The renewal function satisfies $$ m(t) = F_S(t) + \int_0^t m(t-s) f_S(s)\, ds $$ where $F_S$ is the cumulative ...
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1answer
554 views

Differences between a Markov jump process and a continuous-time discrete-state Markov process?

What are the difference and relation between a Markov jump process and a continuous-time discrete-state Markov process? By "a continuous-time discrete-state Markov process", I understand it same as a ...
2
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1answer
390 views

Two different ways of constructing a continuous time Markov chain from discrete time one

Consider a homogeneous continuous time Markov chain (CTMC) with Markov transition function $p(t)$ and infintesimal generator $G$. Its embeded discrete time Markov chain (DTMC) has its transition ...
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2answers
31 views

A state $i$ is recurrent, if and only there exists $n \geq 1$ s.t. $p_{ii}^{(n)} =1$?

For a homogeneous discrete time Markov chain with transition matrix $p$, a state $i$ is recurrent, if and only there exists $n \geq 1$ s.t. $p_{ii}^{(n)} =1$? I have it copied from somewhere in my ...
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2answers
83 views

Covariance for periodic weakly stationary process

Let $X(n),n\in \mathbb N_0$ be a weakly stationary process with $X(n) = X(n+N)$ for some $N \in \mathbb N_0$. What is the covariance function $b(k):=\operatorname{Cov}[X(n+k),X(n)]$?
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1answer
237 views

Fail of optional sampling theorem

Could anyone help me see why the optional sampling theorem ($E(M_{\tau}\mid\mathcal{F}_{\sigma})=M_{\sigma}$ a.s.) fails for certain stopping times $\sigma\leq\tau$ for the not uniformly integrable ...
0
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1answer
58 views

Process with independent increments: relation of increments to process value at later time

Let $X_t,t\geq 0$ be a process with independent increments, $X_{t+s}-X_t$ is independent of $X_r,r\leq t$. Can something similar be said about a later value and and an earlier increment, for example ...
0
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1answer
288 views

Exchanging limit and expectation for $L^2$ random variables

Let $X_n$ be a sequence of random variables in $L^2$, i.e. $\mathbb E[\vert X_n \vert^2]<\infty$. Since the expectation value can be interpreted as a scalar product on $L^2$, can one exchange limit ...
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1answer
1k views

Doob's supermartingale inequality

I'm trying to prove that For a non-negative supermartingale $M$ it holds that for all $\lambda>0$ we have $$\lambda P\{\sup_{n}M_{n}\geq\lambda\}\leq E(M_{0})$$ My idea was to use Markov's ...
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1answer
45 views

Meyer's Theorem in Williams & Rogers

In Diffusions, Markov Processes and Martingales Volume 2 by Rogers and Williams they state the following theorem due to Meyer: $\mathbf{Theorem }$ Le $M\in\mathcal{M}^2_0$. Then there exists a ...
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40 views

Concepts: time homogenous and independent increments

Can someone give me an illustrative example for a time homogenous process without independent increments and for a process that is not time homogenous, but has independent increments?
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75 views

How verification argument really works?

Let $C(u,s)$ be cost functional for an admissible control $u$ with initial state of the system being $s$. Our aim is the solution of the following problem: $$\inf_u E(C(u,s))$$ We defined the value ...
2
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1answer
233 views

Construction of positive recurrent Markov chain

Let $\{X_i\}_{i\geq 1}$ be i.i.d. with values in $\mathbb N_0$. Define a Markov chain via the following transition matrix: $$p(0,n) = \mathbb P(X_1 = n-1) \qquad p(m,n) = \mathbb P\left(\sum_{k=1}^m ...
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1answer
33 views

Is $\{ r \mapsto X_{r} \text{ is continuous for all } s < t \} \in \sigma(X_s : s \leq t)$?

If $(X_t)_{t \geq 0}$ is a stochastic process, is $\{ r \mapsto X_{r} \text{ is continuous for all } $s < t$ \} \in \sigma(X_s : s \leq t)$? I'm particularly interested in the case where $X_t$ is ...
7
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2answers
244 views

Construction of Brownian Motion

In Wiener's construction of Brownian Motion, it is assumed that there exists a probability space $(\Omega,\mathcal F,\mathbb P)$ and random variables $X_n:\Omega\rightarrow\mathbb R$ for $n\in\mathbb ...
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0answers
508 views

Use of Martingale Representation Theorem

I am working on the following problem, and struggling with it. Can anyone help? Let $$H=e^{\int_0^T B_s\,ds}$$ where $T>0$. Show first $E[H^2]<\infty$. Then find an adapted process ...
2
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2answers
459 views

First jump time of Poisson process (and general right-continuous processes).

I've read that the first jump time of the Poisson process is totally inaccessible (definition at the bottom for anyone interested). This made me wonder if the first jump time is a stopping time. I ...
0
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1answer
553 views

Distribution of stochastic integral w.r.t. to centered Poisson process

Let $N(t)$ be a Poisson process with intensity $\lambda$ and define the centered process as $N_0(t):= N(t)-\lambda t$. A stochastic integral can be properly defined w.r.t. to $N_0(t)$ (but not w.r.t. ...
0
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1answer
52 views

Conditional expectations of a stochastic process

Let $(X_t)_{t\geq0}$ be a stochastic process such that $X_t>0$, $X_t\to X_0>0$ pathwise, $\mathbb{P}(X_t>M)=o(\sqrt{t})$ for all $M>X_0$, and $\displaystyle\lim_{t\to ...
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3answers
1k views

The variance of a simple random walk/process

I've been trying to wrap my head around this for the past day. Please help! Let $\epsilon_i = \pm 1$ with equal probabilities independently for $i=1,...,N$. Then $Z_i = \epsilon_1 + ... + \epsilon_i$ ...
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2answers
3k views

Prove Markov Chain by definition

I came across this problem in homework: $X_n$ are i.i.d random variables with $\mathbb{P}[X_n=1]=\mathbb{P}[X_n=-1]=\frac{1}{2}$ and we we also have $S_n=X_1+...+X_n.$ Show that $S_n'=|S_n|$ is a ...
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1answer
118 views

The random walk $S_n=a+\sum_{i=1}^nX_i$

Consider a variant of random walk defined as $$S_n=a+\sum_{i=1}^nX_i,$$ where $X_i$ takes either value $2$ with prob= $p$ or value $-1$ with prob =$1-p$. What is $P(S_n=b)$?
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2answers
149 views

Conditional expectation of a functional of an Itô's semimartingale under its equivalent martingale measure

Consider a probability filtered space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfying the habitual conditions and is generated by $1 d $- ...
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1answer
181 views

Difference of two Poisson processes with same parameter

If I have two Poisson processes, $X$ and $Y$, each with rate $\lambda$, then what is the rate of $Z$ where $Z=X-Y$. Is it $2 \lambda$? and would this differ if $X$ and $Y$ had different rates? ...
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1answer
165 views

Multivariate extension of the Glivenko-Cantelli Theorem

Is there a multivariate extension of the Glivenko-Cantelli theorem for empiric process. If not, are there some weaker statements or other results about the probability limits of empirical ...
2
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1answer
69 views

A problem related with sum of uniform variables

This problem appeared in my mid-term exam. Let $U_n$, $n\ge1$, be i.i.d. random variables which are uniform in $(0, 1)$. Given a constant $t>0$, let $N$ denote the value of $n$ such that ...
2
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1answer
237 views

Approximation of stochastic integral

Let $f \in C^2_C(\mathbb{R})$ and $$X_t = X_0 + \int_0^t \sigma(s) \, dB_s + \int_0^t b(s) \, ds$$ be an (one-dimensional) Itô process where $\sigma,b: [0,\infty) \times \Omega \to \mathbb{R}$ ...
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87 views

Showing a certain process has $\limsup X_t$ bounded almost surely.

This question has been solved. I'm working on a problem where I need to show $$\limsup_{t \rightarrow \infty} X_t \leq \sqrt{c}\quad \text{a.s.}$$ where $X_t$, $t \geq 0$ is a stochastic process ...
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1answer
84 views

Can someone explain to me Feyman Kac and walk through an example?

I kind of understand what needs to be done to convert an SDE to a PDE but I don't understand why we're allowed to do it. What is the generator? ie: given $dS(t) = rS(t)dt + \sigma S(t)dB(t)$ we get ...
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1answer
58 views

On discrete-time stochastic attractivity

Let $m$ be a probability measure on $Y \subseteq \mathbb{R}^p$, so that $m(Y)=1$. Consider a function $f: \mathbb{R}^n \times Y \rightarrow \mathbb{R}^n$, continuous on the first arguments, ...
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1answer
2k views

Poisson process expected value

Let {${N(t), t \geq0}$} be a Poisson process with rate $\lambda$. Let $S_n$ denote the time of the $nth$ event. What is $E[N(4)-N(2)|N(1)=3]$? (Note: $E[X]$ is the expected value or mean). I know ...
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0answers
69 views

test for fractional brownian motion

Given a time series (real data), how can I check if this time series is a fractional brownian motion? I mean, I would start to check for stationary increments. Is it enough to do exploratory plots? Is ...
3
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1answer
156 views

Running maximum absolute value of Wiener process

In Wikipedia a formula is given for the distribution of $$M_t = \max_{0\leq s \leq t} W_s$$ even conditioned on $W_t$. I wonder if there is also a simple expression for (note the absolute value) ...
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1answer
247 views

Proving a Probability Generating Function satisfies a partial differential Equation

We have N animals grazing in a field. The animals graze independently, and periods of grazing and resting alternate for the animals. If an animal is resting at time t, the probability it begins ...
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2answers
687 views

Probability of Extinction in a simple Birth and Death Process

We are asked to show that the probability of extinction $\zeta=\lim_{t\to \infty} P\left(X(t)=0\right)$ given by: $$\zeta=\begin{cases}1&\text{if }\lambda\le \mu,\\ \left(\frac \mu\lambda ...
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1answer
63 views

what does $(\Omega^T,\mathcal{A}^T)$ mean?

Let $(\Omega_t,\mathcal{A}_t), t\in T$ be a collection of measurable spaces. What does the notation mean? $(\Omega^T,\mathcal{A}^T)$
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1answer
36 views

$\Pr\{Z_S=\epsilon\}$ where S is a stopping time.

I have a process $Z_t$ that satisfies $\mathrm{d} Z_t = \dfrac{a}{Z_t}\mathrm{d}t +\mathrm{d}W_t$, Then I am given that $S=\min\{s>0: Z_s=\epsilon \text{ or } Z_s=\alpha\}$, then I need to find ...
4
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1answer
344 views

Why is this process a certain density process?

We are given a stochastic process $X$ and denote by $\mathbb{P}$ the set of all equivalent local martingale measure, that is the set of all equivalent measures $Q\approx P$, such that $X$ is a local ...
2
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1answer
44 views

What is the norm on the functional space used in defining the generator of a homogeneous Markov process?

From Wikipedia: Given a strongly continuous semigroup $T : \mathbb{R}_+ \to L(B)$ on a Banach space $B$, its infinitesimal generator $A$ of a strongly continuous semigroup $T$ is defined as a ...
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1answer
60 views

Inequality between two Random Walks

Let's consider two Random Walks, $$x^{(1)}_t = x_0 + \sum_{i=1}^{t}\xi^{(1)}_i,$$ $$x^{(2)}_t = x_0 + \sum_{i=1}^{t}\xi^{(2)}_i.$$ The random variables $\xi^{(1)}_i$ are i. i. d. They take values on ...
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0answers
660 views

Generator of a Markov process

For a (index-)homogeneous Markov process $X_t$, its infinitesimal generator A is defined to act on suitable functions $f : \mathbb R^n → \mathbb R$ by $$ A f (x) = \lim_{t \downarrow 0} ...
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0answers
68 views

Rate of increase of maximum process of Brownian Motion

Suppose $M_t=\sup_{0\leq s\leq t}\{B_s\}$, where $\{B_t\}_0^{\infty}$ is a standard Brownian Motion. I would like to know if it is true that $M_t e^{-t}$ converges to 0 almost surely? Thanks!
3
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1answer
154 views

A Coupled Random Walk on the xy-Plane

Consider a point on the $xy$-plane whose position is updated in iterations. In each iteration, the point undergoes, with equal probability, either an $A$- or a $B$-update, defined as follows: ...
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1answer
846 views

Expected time, exponential distribution

Suppose that you arrive at a single-teller bank to find five other customers in the bank, one being served and the other four waiting in line. You join the end of the line. If the service times are ...
3
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1answer
173 views

Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $ is a random variable? The kronecker product ...
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1answer
138 views

Martingale inequality

Let $f: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}$ be a deterministic function, as nice as you want, $W$ a Brownian motion and define $$ Y^r_t := \int_0^t f(r,s) dW_s $$ For each fixed $r$, ...
0
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1answer
514 views

Why can all adapted left-continuous stochastic processes be adapted processes?

The definition of an adapted process $X$ is that $X_i$ be $(\mathcal{F_i}, \Sigma)$-measuriable where $\mathcal{F.} = (\mathcal{F_i})_{i \in S}$ is a filtration of the sigma algebra $\mathcal{F}$ ...
2
votes
1answer
410 views

Condition Expectation of Difference between Two Poisson processes

$P_t$ and $Q_t$ are poisson processes with rates $a$ and $b$. How do I calculate $E[(P_t-Q_t)]^2|Q_t=m-P_t]$?
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1answer
167 views

Conditional variance of arrival times

Given a poisson process $P_t$ with rate $r$, with arrival times $S_n$ How do I calculate the Variance of $S_2-S_1|P_t=2$?