Tagged Questions

0
votes
1answer
35 views

Order of convergence of a sum

Let $(X_t)_{t\geq 0},\;X_0=0$, be a positive stochastic process such that \begin{align*} \mathbb{E}\left[\sum_{n=1}^{\infty}X_t^n\right]=\sum_{n=1}^{\infty}\mathbb{E}[X_t^n]<\infty. \end{align*} ...
0
votes
0answers
30 views

Help me solve the invariant measure of $Q$

My $Q$ matrix is given by: \begin{bmatrix} -\lambda &0 &\lambda &0 &0 &... \\ \mu&-(\lambda+\mu) &0 &\lambda &0 &... \\ 0&\mu ...
1
vote
1answer
45 views

Stochastic process, Gaussian, with zero mean is a Wiener process

Let $(\Omega, \mathcal F , \mathbb P)$ be a probability space and let $\mathcal F = \{\mathcal F_t\}_{t\ge} $ a filtration. Let $W=\{W_t;t ≥ 0\}$ be a stochastic process adapted to $\mathcal F$. ...
0
votes
0answers
14 views

Levy process absolute moment

For a Levy process $(X_t)_{t\geq 0}$, we have $\mathbb{E}[X_t]=t\mathbb{E}[X_t^1]$ and $\text{Var}(X_t)=t\text{Var}(X_t^1)$. Does the same hold for the first absolute moment, i.e. does ...
0
votes
0answers
10 views

Analytic tools in the theory of Galton-Watson processes

The questions basically aims at discussing the relative power of using probability generating functions, moment generating functions and characteristic functions as an example for ...
0
votes
1answer
36 views

Continuous Non negative martingale converging to 0

Is there any (non trivial) continuous non negative martingale which converges to 0?
1
vote
2answers
27 views

Generalization of Doob Dynkin for Stochastic processes

Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
1
vote
0answers
41 views

Exponential Levy process

We assume that the stochastic process L is a Levy process with the predictable characteristics triplet $(b,c,\nu)$. Which integrability conditions we should assume for the new stochastic process ...
3
votes
1answer
64 views

Optimal probability measure

Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
1
vote
2answers
53 views

Moment generating function of a stochastic integral

Let $(B_t)_{t\geq 0}$ be a Brownian motion and $f(t)$ a square integrable deterministic function. Then: $$ \mathbb{E}\left[e^{\int_0^tf(s) \, dB_s}\right] = \mathbb{E}\left[e^{\frac{1}{2}\int_0^t ...
1
vote
1answer
35 views

The weighted distribution function for combination of two variables

For example, we have two random variables $a$ and $b$. And they have cumulative distribution function $F(x)$ and $H(x)$. We have number $0 < p < 1$. Suppose, some machine get this random ...
1
vote
0answers
57 views

$dX_t=1_{X_t\not=0} dW_t$

Given The SDE : $dX_t=1_{X_t\not=0} dW_t$ with $ X_{0}=\xi $ how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss Indeed Consider the stopping time $$ ...
2
votes
0answers
52 views

Progressive measurability of stopped process

Let $(\mathcal{F}_t)_{t\in \mathbb{R}_+}$ be a filtration and let $X$ be a stochastic process progressively measurable with respect to $(\mathcal{F}_t)_{t\in \mathbb{R}_+}$. Let $T$ be a stopping time ...
0
votes
0answers
14 views

Submartingale bounds

Let $X_1,X_2,\ldots$ be a submartingale with respect to the filtration generated by it. Is it possible to get any bounds for the probability $\mathbb{P}(X_2 < 0\mid X_1 >0)$ ?
5
votes
1answer
150 views

Interchange supremum and expectation

Let $B_n:=\{f\in L^\infty_+\mid f\le n \}$, where we consider $L^\infty$ with the weak$^*$ topology. I have the following sets $$D(z):=\{h\in L^0_+(\mathcal{F}_T)\mid h\le Z_T \mbox{ for a }Z\in ...
1
vote
1answer
69 views

Reverse Hölder Continuity and Hausdorff dimension

Let $f$ be a function on $[0,1]$. Say that $f$ is reverse Hölder continuous of exponent $\beta > 0$ if there is a $C >0$ such that for any $s<t\in [0,1]$, there exists $s',t'\in [s,t]$ such ...
1
vote
0answers
19 views

Estimate on Galton-Watson process distribution

Let $(Zn)_{n\in \mathbb N_0}$ be a Galton-Watson process, i.e. $$ Z_{n+1} = \sum_{k=1}^{Z_n}\xi_{n,k},\qquad (\xi_{n,k})_{n\in \mathbb N_0,k \in \mathbb N} \quad \text{i.i.d } \mathbb N_0 \text{ ...
1
vote
0answers
83 views

Change of probability measure and a continuous-time Markov chain

Let $(\Omega,\mathcal{F},\mathbb{P},\mathbb{F})$ be a complete filtered probability space, with $W$ a Wiener process and $\alpha$ a continuous-time Markov chain (taking values in $\{1,...,M\}$). We ...
0
votes
0answers
17 views

How to calculate auto-correlation of a bpsk modulated signal or how to calculate expectation value of complex exponential function [migrated]

How to calculate auto-correlation of a bpsk modulated signal or how to calculate expectation value of complex exponential function manually not by using matlab or any other software? for example,if ...
1
vote
0answers
30 views

Cylindrical sigma algebra answers countable questions only.

I got a missing link in some in the following (standard) textbook question: Show that the cylindrical sigma algebra $\mathcal{F}_T$ on $\mathbb{R}^T$ (equals $\bigotimes_{t\in ...
2
votes
1answer
57 views

Weak convergence and generating function

Let $X_n$ be a sequence of $\mathbb N_0$ valued random variables and denote by $g_{X_n}$ their generating function, i.e. $g_{X_n}(s) = \mathbb E[s^{X_n}] = \sum_{k=0}^{\infty} s^k \mathbb P(X_n=k)$. ...
2
votes
1answer
33 views

Martingale equality

The question is to prove $$P\{\sup_{t\geq 0}M_{t}>x\mid \mathcal{F}_{0}\}=\min\left\{1,\frac{M_{0}}{x}\right\},$$ where $M$ is a positive continuous martingale which converges to 0 almost surely ...
1
vote
0answers
15 views

Analytic random function

Let $t\in[0,1]\mapsto X_t\in[0,1]$ an analytic random function. I'd like to say that the deterministic function $t\mapsto \mathbb{E}[X_t]$ is still analytic. What are the minimal conditions needed? ...
7
votes
2answers
56 views

Two martingales whose distributions agree for each time have the same overall distribution

Let $\{X_n\}$ and $\{Y_n\}$ be two martingales. Suppose that for each fixed $n \in \mathbb Z_+$, $X_n$ and $Y_n$ have the same distribution. Must it hold that the random sequences $\{X_n\}$ and ...
2
votes
1answer
62 views

$\sigma$-algebras and independent stochastic processes

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space. We consider a Wiener process $W$ with respect to his standard filtration $(\mathcal{F}_t^W)_{t \geq 0}$ and a process $X$ with ...
1
vote
0answers
24 views

Is open-ball-weak convergence of borel-measurable random elements the same as borel-weak-convergence?

The definitions are from David Pollard, Convergence of Stochastic Processes, IV.1.1 p.65 Let $(\Omega,\mathcal{A},P)$ be any probability space. Let $(S,\mathcal{S})$ be a metric space with any ...
0
votes
0answers
22 views

On discrete-time stochastic attractivity of linear systems

Let $m$ be a probability measure on $Y \subseteq \mathbb{R}^p$, so that $m(Y)=1$. Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^p$. Assume that $f(0) = 0$, and that there ...
2
votes
1answer
50 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 ...
0
votes
1answer
58 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
votes
1answer
32 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
votes
1answer
32 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 ...
1
vote
1answer
90 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 ...
1
vote
1answer
28 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 ...
5
votes
2answers
75 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 ...
2
votes
2answers
84 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
votes
1answer
32 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 ...
1
vote
2answers
87 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 $- ...
1
vote
0answers
39 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 ...
0
votes
1answer
46 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, ...
3
votes
2answers
78 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 ...
-2
votes
1answer
44 views

Please help me with the proof on conditional variance

Let $X$ be a square integrable random variable on $(\Omega,\mathcal{F},P)$. Let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. Define the conditional variance of $X$ given $\mathcal{G}$ by ...
6
votes
1answer
92 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 ...
0
votes
0answers
27 views

Non-arbitrage theory and existence of a risk premium

Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and isgenerated by $1 d $- ...
2
votes
1answer
98 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 $E[(P_t-Q_t)]^2|Q_t=m-P_t]$?
1
vote
1answer
105 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 $S_2-S_1|P_t=2$?
0
votes
1answer
97 views

Conditional CDF of Poisson process

$X_t$ and $Y_t$ are poisson processes with rates $a$ and $b$ (independent processes) $n = 1,2,3...$ Find the conditional CDF $F_X{}_t{}_|{}_X{}_t{}_+{}_Y{}_t{}_={}_n(x)$ I get an answer of ...
1
vote
1answer
75 views

Stopping time proof

Let $\{X_t, t \ge 0\}$ be a continuous stochastic process and adapted to the filtration $\{\mathcal{F}_t,t\ge 0 \}$ and consider $$ \alpha = \inf\{t, |X_t|>1\}, $$ the first time the the process ...
2
votes
0answers
69 views

Does weak convergence of two independent sequences of random processes imply weak convergence of the couple sequence?

I consider sequences of random processes in the Skorohod space. Let $F_n$ and $G_n$ converge weakly to F and G, respectively, in $D[-\infty,\infty]$ endowed with the supnorm. I suppose that this ...
0
votes
0answers
45 views

Continuous-time stochastic process that is left-continuous predictable process - why? [duplicate]

Predictable processes are basically deterministic processes - and I am wondering why continuous-time processes that are left-continuous are automatically predictable processes. To my eyes, ...
0
votes
1answer
30 views

Continue laplace transform of Levy-subordinator analytically to complex numbers with positive real part.

The following is from here part 1.2. If $\phi$ is the Laplace exponent of a subordinator, then there exist a unique pair $(k, d)$ of nonnegative real numbers and a unique measure $\Pi$ on ...

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