All Questions
37
questions
10
votes
1
answer
234
views
Exploiting the Markov property
I've encountered the following problem when dealing with short-rate models in finance and applying the Feynman-Kac theorem to relate conditional expectations to PDEs.
Let $(\Omega,\mathcal{F},\{\...
- 1,524
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
6
votes
1
answer
1k
views
How to Prove that a (Centered) Gaussian Process is Markov if and only if this Equation Holds?
A centered Gaussian process is Markov if and only if its covariance
function $\Gamma: \mathbb{R}\times\mathbb{R} \to \mathbb{R}$ satisfies
the equality:
$$\Gamma(s,u)\Gamma(t,t)=\Gamma(s,t)\...
- 21.3k
5
votes
1
answer
85
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
2
answers
198
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 ...
- 1,397
3
votes
1
answer
640
views
Stationary Markov process properties
Let $X$ be a right-continuous process with values in $(E,\mathcal{E})$, defined on $(\Omega, \mathcal{F}_t,P)$. Suppose that $X$ has stationary, independent increments.
I now want to show the ...
user155670
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
1
answer
48
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 ...
- 603
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
2
votes
1
answer
380
views
Markov property for a stochastic process with discrete state space.
Consider a stochastic process $\{X_s\}_{s\in\mathcal S\subseteq\mathbb R}$ with value in $(\mathbb R,\mathcal B(\mathbb R))$ adapted to a filtration $\{\mathcal F_s\}$ (we can suppose that $\{\...
- 13.5k
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
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
1
vote
1
answer
47
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 \...
- 603
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
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