All Questions
37
questions
1
vote
2
answers
381
views
Sum of i.i.d. random variables is a Markov process
Let $\{Y_j\}_1^\infty$ be i.i.d. real random variables on a common probability space.
$\forall n\in\mathbb{N}$, define $X_n = x_0 + \sum_{j=1}^nY_j$, where $x_0$ is a constant. Also define $X_0=x_0$. ...
- 501
9
votes
2
answers
426
views
If $Y\sim\mu$ with probability $p$ and $Y\sim\kappa(X,\;\cdot\;)$ otherwise, what's the conditional distribution of $Y$ given $X$?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(E,\mathcal E)$ be a measurale space
$\mu$ be a probability measure on $(E,\mathcal E)$
$X$ be an $(E,\mathcal E)$-valued random ...
- 13.9k
0
votes
0
answers
90
views
If $X$ is a Feller process, then $\sup_{x\in E}\text E\left[d(X_s,X_t)\wedge1\mid X_0=x\right]\xrightarrow{s-t\to0}\to0$
Let $(E,d)$ be a compact metric space, $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on $C(E)$, $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(X_t)_{t\ge0}$ be an $E$...
- 13.9k
0
votes
1
answer
222
views
Definition of Factorization of Conditional Expectation
I believe this is a very silly question or I am overlooking something fairly simple but I cannot make sense of the factorization of the conditional expectation in a very concrete application:
I am ...
- 919
1
vote
1
answer
940
views
Equivalent Definitions of the Markov Property
Assume we have a stochastic process $\{X_n\}_\mathbb{N}$ defined on some underlying probability space that takes values in another measurable space. One of the many definitions that I have seen of ...
- 313
1
vote
1
answer
193
views
If $(κ_t)_{t≥0}$ is the transition semigroup of a continuous Markov process, is $t↦(κ_tf)(x)$ continuous for all bounded continuous $f$ and fixed $x$?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$
$E$ be a metric space
$(X_t)_{t\ge0}$ be an $E$-valued right-...
- 13.9k
0
votes
1
answer
104
views
Equivalence of discrete definition of Markov property in the coutinuous case
In the book, Lectures from Markov Processes to Brownian Motion, it is stated that the oldest definition of Markov property is, for every integer $n\ge1$ and $0\le t_1<t_2<\cdots<t<u,$ and $...
- 183
1
vote
0
answers
47
views
Markov property for unbounded function
Let $(X_t)$ be a Markov process with respect to a filtration $\mathcal{F}_t$. Assume that $P(X_t>0 \, \forall t\geq 0) = 1 $.
Denote $E_x$ the expectation under the measure where $X_0=x$.
Is it ...
- 1,774
1
vote
1
answer
79
views
Finite-dimensional conditional distributions of a Markov process
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$I\subseteq\mathbb R$
$(\mathcal F_t)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$
$(E,\mathcal E)$ be a measurable space
$X$ be ...
- 13.9k
1
vote
1
answer
255
views
Show some property of a Markov process
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$I\subseteq\mathbb R$
$(\mathcal F_t)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$
$(E,\mathcal E)$ be a measurable space
$X$ be ...
- 13.9k
1
vote
2
answers
122
views
Markov Property and FDDs
Let $X,Y$ be two discrete time $\mathbb{R}^n$-valued stochastic processes with the same finite dimensional distributions. It may be that $X,Y$ are defined on two different probability spaces. Now, if $...
- 2,031
0
votes
1
answer
77
views
Show that $ (e^{\alpha X_t} \int^ t_ 0 e ^{-\alpha X_u}du, t \geq 0) $ is a Markov process
I want to show that $ (e^{\alpha X_t} \int^ t_
0 e ^{-\alpha X_u}du, t \geq 0) $ is a Markov process
whereas $(\int^ t_
0 e ^{-\alpha X_u}du, t\geq 0)$ is not. Here $X_t$ is a Levy Process and ...
- 630
1
vote
1
answer
497
views
ARCH(1) process is a Markov process
I have a question about the ARCH(1) process. Let $(\Omega, \mathcal F, P)$ be a probability space, let $(Z_t)_{t \in \mathbb Z}$ be a sequence of i.i.d. real-valued random variables with mean zero and ...
- 683
0
votes
1
answer
477
views
Conditional expectation of a Brownian motion
Let $B$ be a Brownian motion. Fix times $0 < r < t$.
Write $\mathcal{D}$ for the space of paths traced out by continuous maps from $[0,t]$ to $\mathbb{R}$ with a suitable (e.g. Skorokhod) ...
- 3,884
1
vote
0
answers
395
views
Markov property - equivalent notions
Why are these different notions of the markov property equivalent:
$$\forall A\in\mathcal{S}\qquad \mathbb{P}(X_t\in
A|\mathcal{F}_s)=\mathbb{P}(X_t\in A|X_s)$$
$$\forall f:S\to\mathbb{R} \text{ ...
- 402