On the Nonexplosion and Explosion for Nonhomogeneous Markov Pure Jump Processes

Abstract

In this paper, we obtain new drift-type conditions for nonexplosion and explosion for nonhomogeneous Markov pure jump processes in Borel state spaces. The conditions are sharp; e.g., the one for nonexplosion is necessary if the state space is in addition locally compact and the Q-function satisfies weak Feller-type and local boundedness conditions. We comment on the relations of our conditions with the existing ones in the literature and demonstrate some possible applications.

Appendix

In this appendix, we present some auxiliary results and the proofs of some lemmas stated in the previous sections.

Proposition 6.1

Let $\mu _n$ and $\mu$ be $[0,\infty ]$-valued measures on the Borel space X,  and $f_n$ (resp., f) be $(-\infty ,\infty )$-valued $\mu _n$-integrable (resp., measurable) functions thereon, and $g_n$ be nonnegative measurable functions on X. Suppose $\mathop {\underline{\lim }}_{n\rightarrow \infty }\mu _n(\Gamma )\ge \mu (\Gamma )$ for each $\Gamma \in \mathcal{B}(X)$, and $f_n(x)\rightarrow f(x)$ for each $x\in X.$ If $|f_n(x)|\le g_n(x)$ for each $x\in X$ and $n\ge 1,$ and

$$\begin{aligned} \mathop {\overline{\lim }}_{n\rightarrow \infty } \int _X g_n(y)\mu _n(\mathrm{d}y)\le \int _X \mathop {\underline{\lim }}_{n\rightarrow \infty }g_n(y)\mu (\mathrm{d}y)<\infty , \end{aligned}$$

then f is $\mu$-integrable, and

$$\begin{aligned}&\lim _{n\rightarrow \infty } \int _X f_n(y)\mu _n(\mathrm{d}y)=\int _X f(y)\mu (\mathrm{d}y)\in (-\infty ,\infty ). \end{aligned}$$

Proof

See Theorem 2.4 of [24] $\square$

Proof of Lemma 2.1

Let us firstly justify that (b), (c) and (d) are equivalent. Clearly, (d) implies (b), which in turn implies (c). That (c) implies (b) is true because clearly $t\rightarrow P(0,x,t,S)$ is monotone nonincreasing in $t\ge 0$. Now suppose (b) is true. By the Chapman–Kolmogorov equation, we see that for each $x\in S$ and $0\le s\le t$,

$$\begin{aligned} P_q(s,y,t,S)=1 \end{aligned}$$

(40)

for almost all y with respect to $P_q(0,x,s,\hbox {d}y)$. Now suppose for contradiction that (d) does not hold, so that there exist some $0\le s<t$ and $x\in S$ such that

$$\begin{aligned} P_q(s,x,t,S)<1. \end{aligned}$$

(41)

Then

$$\begin{aligned} 1= & {} P_q(0,x,t,S)=P_q(0,x,s,\{x\})P(s,x,t,S)+\int _{S\setminus \{x\}} P_q(0,x,s,\hbox {d}y)P_q(s,y,t,S)\\= & {} P_q(0,x,s,\{x\})P(s,x,t,S)+ P_q(0,x,s,S\setminus \{x\})\\&\quad <P_q(0,x,s,\{x\})+ P_q(0,x,s,S\setminus \{x\})=1, \end{aligned}$$

where the first equality and the last equality are by assumption that (b) holds, the second equality is by the Chapman–Kolmogorov equation, the third equality is by (40), and the inequality is by (41). This is a desired contradiction, and thus, assertions (b), (c) and (d) are equivalent.

It is clear that (a) implies (b), c.f. Proposition 2.3 and (6). That (b) implies (a) follows from the relation $\{t_\infty =\infty \}=\bigcap _{n=1}^\infty \{X_n\in S\},$ c.f. (6). The proof is completed. $\square$

Proof of Lemma 4.1

It is clear that $\int _S g(y)q(\hbox {d}y|x,s)$ is well defined. For the second part of the statement, note that for each $x\in S$ and $s\ge 0$,

$$\begin{aligned} \int _S g(y)\tilde{q}(\hbox {d}y|x,s)-g(x)q_x(s)\ge & {} -||g||_f \int _S f(y)\tilde{q}(\hbox {d}y|x,s)-||g||_f f(x)q_x(s)\\= & {} -||g||_f \int _S f(y)q(\hbox {d}y|x,s)- 2||g||_ff(x)q_x(s) \\\ge & {} -||g||_f cf(x)- 2||g||_ff(x)q_x(s)\\= & {} -||g||_f f(x)(c+2q_x(s)), \end{aligned}$$

where the second inequality is by the fact that f is a c-drift function. Thus,

$$\begin{aligned} \int _S g(y)q(\hbox {d}y|x,s)\ge -||g||_f f(x)(c+2q_x(s)), \end{aligned}$$

the integral on the left-hand side being well defined. Similarly, one can check that

$$\begin{aligned} \int _S g(y)q(\hbox {d}y|x,s)\le ||g||_f f(x)(c+2q_x(s)), \end{aligned}$$

as required. $\square$

Proof of Lemma 4.2

The first assertion follows from the proof of the relevant assertion in Theorem 3.1 of [15]. The second assertion follows from the first assertion because for each $x\in S$ such that $\int _{S}f(y)q(\hbox {d}y|x,s)\ge 0,$

$$\begin{aligned} \left( \int _S f(y)q(\hbox {d}y|x,s)\right) ^+=\int _{S}f(y)q(\hbox {d}y|x,s) \le cf(x),\quad \forall ~s\ge 0, \end{aligned}$$

whereas for each $x\in S$ such that $\int _{S}f(y)q(\hbox {d}y|x,s)< 0,$

$$\begin{aligned} \left( \int _S f(y)q(\hbox {d}y|x,s)\right) ^+=0 \le cf(x),\quad \forall ~s\ge 0. \end{aligned}$$

The statement is proved. $\square$

Lemma 6.1

Let f be a c-drift function with respect to the Q-function q on S. Suppose

$$\begin{aligned} \sup _{x\in S,~s\ge 0}\{q_x(s)\}<\infty . \end{aligned}$$

Then for each $x\in S,$ $0\le s\le t$, and f-bounded (measurable) function g on S,

$$\begin{aligned} \int _S g(z)P_q(s,x,t,\mathrm{d}z)=g(x)+\int _s^t \left( \int _Sg(z) q(\mathrm{d}z|y,u)\right) P_q(s,x,u,\mathrm{d}y)\mathrm{d}u. \end{aligned}$$

In particular,

$$\begin{aligned} P_q(s,x,t,S)=1,\quad \forall ~x\in S, ~s,t\in [0,\infty ),~s\le t. \end{aligned}$$

(42)

Proof

The first assertion immediately follows from Proposition 2.2 and Lemma 4.2. The last assertion follows because $f\equiv 1$ is a c-drift function, and one can put $g\equiv 1$ in the first assertion. $\square$

Proof of Lemma 4.3

We only need justify that (22) implies (23) as follows. Let some $x \in S$ and $t\in [0,\infty )$ be arbitrarily fixed. Since (22) holds by assumption,

$$\begin{aligned} \int _0^t \mathbb {E}^q_x\left[ \left( \int _S f(y)q(\hbox {d}y|X_s,s)\right) ^-\right] \hbox {d}s= & {} \int _0^t \mathbb {E}^q_x\left[ \left( \int _S f(y)q(\hbox {d}y|X_s,s)\right) ^+\right] \hbox {d}s\\&-\mathbb {E}^q_x[f(X_t)]+f(x)<\infty \end{aligned}$$

by Lemma 4.2. Consequently, (24) holds. According to the Fubini theorem,

$$\begin{aligned} \int _0^t \mathbb {E}^q_x\left[ \int _S f(y)q(\hbox {d}y|X_s,s)\right] \hbox {d}s= \mathbb {E}^q_x\left[ \int _0^t \left( \int _S f(y)q(\hbox {d}y|X_s,s)\right) \hbox {d}s\right] \in (-\infty ,\infty ). \end{aligned}$$

Now for each $0\le u\le t,$ (22) implies

$$\begin{aligned} \mathbb {E}_x^q[f(X_t)]-f(x)= & {} \mathbb {E}_x^q\left[ \int _0^t\int _S f(y)q(\hbox {d}y|X_s,s)\hbox {d}s\right] \\= & {} \mathbb {E}_x^q\left[ \int _0^u\int _S f(y)q(\hbox {d}y|X_s,s)\hbox {d}s\right] +\mathbb {E}_x^q\left[ \int _u^t\int _S f(y)q(dy|X_s,s)\hbox {d}s\right] \\= & {} \mathbb {E}_x^q[f(X_u)]-f(x)+\int _u^t\mathbb {E}_x^q\left[ \int _S f(y)q(\hbox {d}y|X_s,s)\right] \hbox {d}s. \end{aligned}$$

All the expressions are finite. Now (23) follows. $\square$

Lemma 6.2

For a c-drift function f with respect to the Q-function on S that satisfies relation (22), it holds that for each $d\in (-\infty ,\infty )$, $x\in S$ and $t\ge 0$,

$$\begin{aligned} \hbox {e}^{\mathrm{d}t} \mathbb {E}^q_x[f(X_t)]=f(x)+\int _0^t \mathrm{e}^{\mathrm{d}u} \left( \mathbb {E}^q_x\left[ \int _S f(y)q(\mathrm{d}y|X_u,u)\right] +d \mathbb {E}^q_x[f(X_u)]\right) \mathrm{d}u. \end{aligned}$$

Proof

Keeping in mind Lemmas 4.2 and 4.3 (and especially (24)), direct calculations give

$$\begin{aligned}&\int _0^t \hbox {e}^{\mathrm{d}s}\mathbb {E}_x^q\left[ \int _S f(y)q(\hbox {d}y|X_s,s)\right] \hbox {d}s\\&\quad =\int _0^t \int _0^s \left( d \hbox {e}^{\mathrm{d}u}\hbox {d}u\right) \mathbb {E}^q_x\left[ \int _S f(y)q(\hbox {d}y|X_s,s)\right] \hbox {d}s +\mathbb {E}^q_x[f(X_t)]-f(x)\\&\quad = \int _0^t d \hbox {e}^{\mathrm{d}u} \left( \int _u^t \mathbb {E}^q_x\left[ \int _S f(y)q(\hbox {d}y|X_s,s)\right] \hbox {d}s\right) \hbox {d}u+\mathbb {E}^q_x[f(X_t)]-f(x)\\&\quad = \hbox {e}^{\mathrm{d}t} \mathbb {E}^q_x[f(X_t)](\hbox {e}^{\mathrm{d}t}-1)- \int _0^t d\hbox {e}^{\mathrm{d}u} \mathbb {E}^q_x[f(X_u)]\hbox {d}u+\mathbb {E}^q_x[f(X_t)]-f(x), \end{aligned}$$

where the first and the third equalities are by (23). All the expressions are finite. The statement now follows. $\square$

Lemma 6.3

Consider a c-drift function f with respect to the Q-function q on S. For each $x\in S,$ $s,t\ge 0$, $s\le t$ and $\Gamma \in \mathcal{B}(S)$, the following relation holds;

$$\begin{aligned} P_{q^f}(s,x,t,\Gamma )=\frac{\mathrm{e}^{-c(t-s)}}{f(x)}\int _\Gamma f(y) P_q(s,x,t,\mathrm{d}y). \end{aligned}$$

Proof

By Feller’s construction, c.f., (2) and (3), it suffices to show

$$\begin{aligned} \sum _{m=0}^n P_{q^f}^{(m)}(s,x,t,\Gamma )= & {} \frac{\mathrm{e}^{-c(t-s)}}{f(x)}\int _\Gamma f(y) \left( \sum _{m=0}^n P^{(m)}_q(s,x,t,\hbox {d}y)\right) ,~x\in S,\nonumber \\&\qquad ~s,t\ge 0,~ s\le t,~\Gamma \in \mathcal{B}(S), \end{aligned}$$

(43)

as follows. Let $x\in S,~s,t\ge 0,~ s\le t,~\Gamma \in \mathcal{B}(S)$ be arbitrarily fixed. Consider $n=0.$ Then

$$\begin{aligned} \frac{\hbox {e}^{-c(t-s)}}{f(x)}\int _\Gamma f(y) P^{(0)}_q(s,x,t,\hbox {d}y)= & {} \frac{\hbox {e}^{-c(t-s)}}{f(x)}\int _\Gamma f(y) \hbox {e}^{-\int _s^t q_x(v)\hbox {d}v} \delta _x(\hbox {d}y) \nonumber \\= & {} \delta _x(\Gamma ) \hbox {e}^{-\int _s^t q_x(v)\mathrm{d}v-c(t-s)}\nonumber \\= & {} P_{q^f}^{(0)}(s,x,t,\Gamma ). \end{aligned}$$

(44)

Now assume that (43) holds for $n=k$. Then

$$\begin{aligned}&\frac{\hbox {e}^{-c(t-s)}}{f(x)}\int _\Gamma f(y) \left( \sum _{m=0}^{k+1} P^{(m)}_q(s,x,t,\hbox {d}y)\right) \\&\quad =P_{q^f}^{(0)}(s,x,t,\Gamma )+\frac{\hbox {e}^{-c(t-s)}}{f(x)}\int _\Gamma f(y) \left( \sum _{m=1}^{k+1} P^{(m)}_q(s,x,t,\hbox {d}y)\right) \\&\quad = P_{q^f}^{(0)}(s,x,t,\Gamma )+\frac{\hbox {e}^{-c(t-s)}}{f(x)}\int _\Gamma f(y) \left( \sum _{m=0}^{k} P^{(m+1)}_q(s,x,t,\hbox {d}y)\right) \\&\quad = P_{q^f}^{(0)}(s,x,t,\Gamma )+\frac{\hbox {e}^{-c(t-s)}}{f(x)}\int _\Gamma f(y) \\&\qquad \quad \left( \sum _{m=0}^{k}\int _s^t \hbox {e}^{-\int _s^u q_x(v)\hbox {d}v} \left( \int _S \tilde{q}(dz|x,u) P^{(m)}_q(u,z,t,\hbox {d}y)\right) \hbox {d}u\right) \\&\quad = P_{q^f}^{(0)}(s,x,t,\Gamma )+\int _{s}^t \hbox {e}^{-\int _s^u (q_x(v)+c)\hbox {d}v} \int _S \frac{f(z)}{f(x)} \tilde{q}(dz|x,u) \\&\quad \qquad \left( \int _\Gamma \frac{f(y)}{f(z)} \hbox {e}^{-c(t-u)}\sum _{m=0}^k P^{(m)}_q(u,z,t,\hbox {d}y) \right) \hbox {d}u\\&\quad = P_{q^f}^{(0)}(s,x,t,\Gamma )+\sum _{m=0}^k P_{q^f}^{(m+1)}(s,x,t,\Gamma ), \end{aligned}$$

where the first equality is by (44), the third equality is by (2), and the last equality is by (2), the inductive supposition and (7). Now (43) holds for $n=k+1$, and thus for all $n=0,1,\ldots$ by induction. $\square$