All Questions
36
questions
2
votes
1
answer
33
views
A cadlag Feller process for $\mathcal F$ is Markov w.r.t $\mathcal F_+$ (Th. 46, Chap. 1, Stochastic Integration - Protter)
In page 35 of the book Stochastic Integration by P. Protter, he defines a Feller process as follows:
Then he states the following theorem.
In the proof, he used the following strategy:
Next, he ...
- 581
1
vote
1
answer
46
views
Decomposing a general stopping time into stopping components
Let $(X_n)_{n \geq 0}$ be a discrete-time Markov chain taking values in a finite state space $S$, with transition matrix $P$. Let $(\mathcal F_n)_{n\geq 0}$ be the natural filtration and let $\tau \...
- 581
5
votes
1
answer
84
views
Sum of conditional expectations of a bounded stochastic process
Is there a proof for the following statement or is there a counter-example?
Let $\{X_t\}$ be a stochastic process
adapted to the filtration $\{\mathcal{F}_t\}$.
Assuming $0 \leq X_t \leq 1$,
and $\...
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
1
vote
1
answer
47
views
Interpretation of condition probability of $X_0 = x$ for a Markov process and semigroup $(P_t)$ [closed]
When studying Markov processes, I have seen a lot of authors define the semigroup as $P_tf(x) = \mathbb E_x(f(X_t))$ (with the assumption that $X_t$ is homogeneous) and the call $\mathbb E_x$ as the &...
- 581
1
vote
1
answer
88
views
How can we rigorously show conditional independence here?
Let
$(E,\mathcal E,\lambda)$ be a measure space;
$p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$c:=\int p\:{\rm d}\lambda\in(0,\infty)$$ and $\mu$ denote the measure with density $\frac pc$ ...
- 13.9k
0
votes
1
answer
51
views
Equivalence of defintion for Markov Property
I've seen definitions of Markov Property for a process $X$ indexed with the positive integers with values in $S$, that are supposedly equavalent. I consider the canonical filtration $\mathcal F=(\...
- 1,032
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
2
votes
1
answer
238
views
Showing an equivalence between the martingale property and a markov property.
I really am not sure how to get a rigorous answer to the following, any help would be greatly appreciated.
Let $(X_n)_{n\geq0}$ be an integrable process, taking values in a countable set $E ⊆ \mathbb{...
- 1,023
3
votes
2
answers
194
views
Markov transition kernels of process with independent increments
Suppose that $\{X_t : Ω → S := \mathbb{R}^d, t\in T\}$ is a stochastic
process with independent increments and let $\mathcal{B}_t :=\mathcal{B}_t^X$ (natural filtration) for all $t\in T$. Show, for ...
- 1,387
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
91
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
2
answers
377
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