0
votes
1answer
18 views

Markov Processes: How to show $\int P(X_t\in B\mid X_0)dPX_0^{-1}=P(X_t\in B)$?

Let $\left\{X_t \right\}_{t\in T}$ be a time homogeneous Markov process with state space $S$. How do I formally demonstrate$$P(X_t\in B)=\int_S P(X_t\in B\mid X_0)dPX_0^{-1}$$(here $PX_0^{-1}$ is the ...
2
votes
1answer
68 views

transition kernel

I've got some trouble with transition kernels. We look at Markov process with statspace $(S,\mathcal{S})$ and initial distribution $\mu^0$. We have a transition kernel $P:S\times ...
0
votes
1answer
56 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, ...
1
vote
0answers
158 views

Is there monotone class theorem used in one of these steps?

IN Rogers & Williams "Diffusions, Markov Process and Martingales" they introduce the resolvent as: $$R_\lambda f(x):=\int_{[0,\infty)}e^{-\lambda t}P_tf(x)dt=\int_ER_\lambda(x,dy)f(y)$$ where ...
0
votes
1answer
41 views

Characterizing the Dependence Structure of a Rewards for a Finite State Homogenous Markov Chain

Let $\{X_n, n\geq 1\}$ be a finite state homogenous Markov chain with states $i = 1, \ldots, N$ . Let $g$ denote a function which returns out a reward for any given state of the Markov chain. Let ...
3
votes
1answer
143 views

Applying equation to Markov process

This seems as an easy question, but however I can't handle it. In the following I need this fact: If $X=(X_t)$ is a Markov process with transition semigroup $(K_t)$ and initial distribution $\mu$ ...
12
votes
2answers
419 views

Difference in probability distributions from two different kernels

I wonder if the probability kernels of Markov processes on the same state space are close enough, does it also hold for the probabilities of the event that depend only on first $n$ values of the ...