All Questions
12
questions with no upvoted or accepted answers
3
votes
0
answers
48
views
Is my proof of Markov Property for Reflected BM correct?
I want to show that $|B_{t}|$ is a Markov Process where, $B_{t}$ is a Standard Brownian Motion. I have seen the proof here and here. But I don't understand why the method below might fail (or if it's ...
- 1,285
2
votes
0
answers
91
views
Conditional distribution of Gaussian process completely determined by conditional expectation
I am reading the book Stochastic Processes by J. L. Doob and trying to understand the argument that, for any (real-valued) Gaussian process $X = (X_t)_{t\ge0}$, the Markov property is characterized by ...
- 344
1
vote
0
answers
22
views
Markov property of $X_t + \sigma Y_t$ when $\sigma \rightarrow 0$
Let $(X_t)$ and $(Y_t)$ be sample continuous stochastic processes on $[0,1]$ such that $Z_t (\sigma):= X_t + \sigma Y_t$, where $\sigma >0$, is Markov with regard to the filtration generated by $...
- 2,294
1
vote
0
answers
66
views
Is the Markov property under $\mathbb{P} $ preserved under change to a measure $\mathbb{Q} $ absolutely continuous to $\mathbb{P} $ .
Let $(\Omega,\mathcal{F}, \{\mathcal{F}_n \}, \mathbb{P})$, be a filtered probability space, $\mathcal{F}= \sigma\{F_n,n\in \mathbb{N}\} $, $M_n$ a nonnegative martingale, and $\mathbb{E} \mathrm{M}_n=...
- 73
1
vote
1
answer
92
views
Relation between the strong Markov property of a process and the strong Markov property of the associated canonical process on the path space
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(E,\mathcal E)$ be a measurable space;
$\pi_I$ denote the projection from $E^{[0,\:\infty)}$ onto $I\subseteq[0,\infty)$ and $\pi_t:=...
- 13.9k
1
vote
0
answers
27
views
If $(Y_n)$ is iid, then $Z_n:=\sum_{i=1}^nY_i$ is Markov
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_n)_{n\in\mathbb N_0}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$, $E$ be a $\mathbb R$-Banach space, $(Y_n)...
- 13.9k
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
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
0
answers
395
views
Markov property - equivalent notions
Why are these different notions of the markov property equivalent:
$$\forall A\in\mathcal{S}\qquad \mathbb{P}(X_t\in
A|\mathcal{F}_s)=\mathbb{P}(X_t\in A|X_s)$$
$$\forall f:S\to\mathbb{R} \text{ ...
- 402
1
vote
0
answers
33
views
Equation with the expectation of a assessed Markov process
In my book about Markov processes there is following equation in a proof and I don't see why it's right, I already ask some people in the university, but I had no success, can somebody help me?
$$E(\...
- 33
0
votes
0
answers
15
views
Does is hold that $E[f(X_t)1_{\{ s \geq T_1\}}| \mathcal F_{T_1}] = P_{t-T_1}(X_{T_1})1_{s \geq T_1}$ for $s\leq t$ and X is a strong Markov process
Let $X=(X_t)_{t\geq 0}$ be a homogeneous cadlag Markov process taking values in a finite state space $S$. Let $T_1$ be its first jump time and $f$ be a bounded measurable function. I would like to ...
- 41
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