Standard transition semigroups and regularity of jump processes

Let $X = \left\{ X(t), t \geqslant 0\right\}$ be a homogeneous standard (i.e., $p_{i j}(t) \rightarrow \delta_{i j}$ as $t \rightarrow 0+$) Markov chain with a finite or countable state space $\mathrm{S}$ and transition probabilities $p_{i j}(t)$, considered in its separable modification, and let $Q = \left(q_{i j}\right)_{i, j \in \mathrm{S}}$ be its infinitesimal matrix:

Let $q_{i} = \infty$ for a state $i$. Then the probability that the process $X$ stays in this state for a positive time is zero, i.e., having entered this state, the process leaves it instantly.

Let $X = \left\{ X(t), t \geqslant 0\right\}$ be a homogeneous standard (i.e., $p_{i j}(t) \rightarrow \delta_{i j}$ as $t \rightarrow 0+$) Markov chain with a finite or countable state space $\mathrm{S}$ and transition probabilities $p_{i j}(t)$, and let $Q = \left(q_{i j}\right)_{i, j \in \mathrm{S}}$ be its infinitesimal matrix:

Prove that $\sup_{i} q_{i} < \infty$ if and only if $p_{i i}(t) \rightarrow 1$ as $t \rightarrow 0+$ uniformly in $i \in \mathrm{S}$ (or, equivalently, $p_{i j}(t) \rightarrow \delta_{i j}$ as $t \rightarrow 0+$ uniformly in $i, j \in \mathrm{S}$ ).

Give an example of transition probabilities for which the standardness condition (49) does not hold.

Definition 9. A homogeneous Markov chain $X = \left\{ X_{t}, t \geqslant 0\right\}$ (or the semigroup $P(t) = \left(p_{i j}(t)\right)$ generated by it) is called standard if $P(t) \rightarrow I$ as $t \rightarrow 0+$, i.e., for all $i, j \in \mathrm{S}$

where, as usual, $\delta_{i j}$ is the Kronecker symbol.

Give an example where property (49) holds but (89) does not.

Condition (89), where $q_{i j} = \left.\frac{d^{+} p_{i j}(t)}{d t}\right|_{t = 0}$ are the entries of the infinitesimal matrix $Q$:

Definition 9. A homogeneous Markov chain $X = \left\{ X_{t}, t \geqslant 0\right\}$ (or the semigroup $P(t) = \left(p_{i j}(t)\right)$ generated by it) is called standard if $P(t) \rightarrow I$ as $t \rightarrow 0+$, i.e., for all $i, j \in \mathrm{S}$

where, as usual, $\delta_{i j}$ is the Kronecker symbol.

Let the state space $\mathrm{S}$ consist of a finite number $N$ of elements, and let the transition matrix $P(t)$ be standard. Prove that then this matrix admits the following representation:

where the $N \times N$ matrix $Q$ satisfies the condition

Conversely, if conditions (95) hold for a matrix $Q$, then formula (94) defines a standard matrix of transition probabilities.

The backward system of Kolmogorov equations for the transition probabilities $p_{i j}(t)$ of a homogeneous Markov chain with infinitesimal matrix $Q = \left(q_{i j}\right)_{i, j \in \mathrm{S}}$ has the form

and the forward system has the form

Equations (65) and (72) can be understood either in the narrow sense (i.e., that all $p_{i j}^{\prime }(t)$ are continuous for $t \geqslant 0$ and the equations under study hold for all $t \geqslant 0$ ), or in the broad sense, i.e., that $p_{i j}(t)$ are absolutely continuous and the equations under consideration hold for almost all (with respect to Lebesgue measure) $t \geqslant 0$.

Prove that for the systems of Kolmogorov equations both interpretations given above (in the narrow and in the broad sense) are equivalent. For the forward equations this exercise should be classed among the very difficult ones.

Let $Q = \left(q_{i j}\right)_{i, j \in \mathrm{S}}$ be a matrix with real entries, where $\mathrm{S}$ is countable, such that

The backward and forward systems of Kolmogorov equations for the matrices $P(t) = \left(p_{i j}(t)\right)_{i, j \in \mathrm{S}}$, $t \geqslant 0$, have, respectively, the form

and

(see also Exercise 34). Prove that if $\sup_{i} q_{i} < \infty$, then both systems of Kolmogorov equations have a unique solution.

Let $X$ be a continuous-time Markov chain with countable state space $S$ and semigroup $\left\{ \mathbf{P}_{t}\right\}$. Show that $p_{i j}(t)$ is a continuous function of $t$. Let $g(t) = -\log p_{i i}(t)$; show that $g$ is a continuous function, $g(0) = 0$, and $g(s+t) \leq g(s)+g(t)$. We say that $g$ is 'subadditive', and a well known theorem gives the result that

Deduce that the limit $g_{i i} = \lim_{t \downarrow 0} t^{-1}\left\{ p_{i i}(t)-1\right\}$ exists.

Prove that the transition probabilities $p_{i j}(t)$ of a standard stochastic semigroup are uniformly continuous on $\mathbb {R}_{+}$.

Let $\xi_{t}, t \geqslant 0$, be a homogeneous Markov process with countably many states $(0,1, \ldots )$. Prove that if the transition functions $P_{ij}(t)$ are continuous at $t = 0$, then they are uniformly continuous for $t \geqslant 0$.

Let $\xi_{t}$ and $\eta_{t}, t \geqslant 0$, be two homogeneous Markov processes with countably many states, $P_{ij}(t)$ and $Q_{ij}(t)$ the corresponding transition functions. Prove that if for some $t_{0} > 0$ and all $i$ and $j$, $P_{ij}(t) = Q_{ij}(t)$ for $0 \leqslant t \leqslant t_{0}$, then $P_{ij}(t) = Q_{ij}(t)$ for all $t \geqslant 0$.

Let $\xi_{t}, t \geqslant 0$, be a homogeneous Markov process with finitely many states $\left\{ 0,1, \ldots , N\right\}$ and transition functions $P_{ij}(t)$. Prove that the determinant of the matrix $P(t)$ with entries $P_{ij}(t), 0 \leqslant i, j \leqslant N$, is positive for all $t > 0$.

Let $\xi_{t}, t \geqslant 0$, be a homogeneous Markov process with finitely many states $\left\{ 0,1, \ldots , N\right\}$ and transition functions $P_{ij}(t)$. Suppose that $P_{ij}(t)$ is continuous for all $t \geqslant 0$. Prove that the finite limits

exist, and $\sum_{j \neq i} a_{ij} = -a_{ii}$.

Give an example of a homogeneous Markov process with countably many states for which the finite limits

exist, but $\sum_{j \neq i} a_{ij} \neq -a_{ii}$.

Show that there is no Markov process $\xi_{t}$, $t \geqslant 0, \xi_{0} = 0$, with two states $\left\{ 0,1\right\}$ and trajectories almost surely continuous, whose transition probabilities equal $P_{00}(t) = e^{-t}, P_{01}(t) = 1-e^{-t}, P_{11}(t) = 1$.