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I've just mentioned that I forgot to write that the sequence $A_n$ is non-increasing. Since the limit is understood as a set which belongs to all $A_n$.
Maybe - unfortunately I haven't found it there. At least at the moment we have the hypothesis is true for the finite-state Markov chain (as I understand also for infinite if there $u = \min\limits_y Q(y,y)>-\infty$). For the general piecewise-deterministic homogeneous Markov process it's not true (I've constructed a counterexample). For the case of diffusions it seems that it should be enough to claim that $$ E_x\int\limits_0^\infty I[X_t\in D] = 0 $$ where $D = x\in E|\sigma(x) = 0$ and $\sigma$ is a diffusion matrix. For example it will not be the case for PDP's and non-homogeneous MPs.
You mean that we can make an equivalent change of measure only on a finite horizon? Ok, it will be enough for me. Unfortunately as I understand the change of the diffusion term is impossible even on the finite horizon because it will change a quadratic variation which is a local characteristic of a process.
But I like your logic - this is prove based on the generator which I'm missing for the general homogeneous Markov process. If you would explain, I could try to make an analogue for the general case.
As I understand you denote $(e^{tQ})(y,z)$ as an element of the matrix on row $y$ and column $z$. Then you are mixing $A$ and $y$ easily - this is a little bit confusing, could you explain what is $(e^{tQ})(y,z)$ and $(e^{tQ})(y,A)$ in your notation?
Do I understand correct, roughly speaking calculating expectations with the Law of Large Numbers will give us different results P-a.s.? But what about diffusion with drift? Can't we use the same argument to prove that on the interval $[0,+\infty)$ there is no equivalent cahnge of measure?
Yes, it's clear for me now about diffusions. But I'm not sure I completely understand what do you mean about the Poisson process. By the construction in Shreve, we use $Z(T)$ to define a new measure $\tilde{P}$. On the other hand we use the same to change the drift of diffusion - and we tell that this is the change on the whole real line.
Sorry, I'm missing a lot in your answer. 1. For the diffusion case you suppose that my hypothesis is true if the coefficient of diffusion term is always non-zero? (It seems to me too) 2. For the continuous Markov chain what did you prove, that I'm right or wrong?
See e.g. Shreve, Stochastic Calculus for Finance II, chapter 11 - change of measure for the Poisson process. There he provide an equivalent change of measure which changes an intensity of the Poisson process.
As I understand, it is a non-homogeneous Markov process since it is written as $$ X_t = X_0 + \int\limits_0^t u(s,X_s)dB_s. $$ In the topic I mentioned that I am looking for the homogeneous one.
