TWO TYPES OF TERMINATION OF TRAJECTORIES OF A MARKOV PROCESS

L. V. SEREGIN

(Presented by Academician A. N. Kolmogorov on 14 I 1963)

In the modern theory of Markov processes one considers processes that are terminated at a random time \(\zeta\). In this note a class of \(\sigma\)-processes is singled out which, roughly speaking, are characterized by the fact that for them one may speak of a definite point of termination \(x_{\zeta-0}\). In a certain sense the opposite class is the class of \(\delta\)-processes. Under certain restrictions a general process is constructed from uniquely determined \(\sigma\)- and \(\delta\)-processes. In accordance with \((^1)\) we introduce the notation: \((E, \mathscr{B})\) is the phase space (measurable space); \(\Omega\) is the space of elementary events; \(x_t=x_t(\omega)\) is the trajectory of the process; \(\mathscr{N}\) is the \(\sigma\)-algebra in \(\Omega\) generated by the sets \(\{x_t\in \Gamma\}\) \((t\ge 0,\ \Gamma\in\mathscr{B})\); \(P(t,x,dy)\) is the transition function; \(P_x\) are probability measures on \(\mathscr{N}\). We also introduce the notation:
\[ \xi(\Gamma)=\inf\{t:\ x_t\in\Gamma\}\quad(\Gamma\in\mathscr{B}); \]
\[ f(x_{\zeta-0})=\lim_{t\to\zeta-0} f(x_t) \]
(if the limit exists); \(\psi_s(\omega)\) is the realization coinciding with \(x_t(\omega)\) for \(t<s\) and terminating at the time \(s\). Unspecified notation corresponds to \((^1)\). All processes under consideration are assumed to be standard (see \((^2)\)). Processes are considered up to equivalence.

1. Put
   \[ \Gamma_n=\left\{x:\ P\left({1\over n},x,E\right)>{1\over n}\right\}. \]
   Then \(\xi(\Gamma_n)\uparrow \zeta\) (a.s. \(P_x\)). Terminating trajectories that are bounded by one of the sets \(\Gamma_n\), i.e. \(\xi(\Gamma_n)=\zeta\), will be called \(\sigma\)-trajectories. Nonterminating trajectories and trajectories for which \(\xi(\Gamma_n)<\zeta\) \((n=1,2,\ldots)\), \(\xi(\Gamma_n)\uparrow\zeta\), will be called \(\delta\)-trajectories. If \(\{\tau_n<t<\zeta\}\in\mathscr{N}_t\) \((n=1,2,\ldots)\) and \(\tau_n\uparrow\zeta\), then on almost all (a.s.) \(\sigma\)-trajectories, starting with some \(n\), \(\tau_n=\zeta\). On a.s. \(\delta\)-trajectories the quantities \(\tau_n=\min\{n,\xi(\Gamma_n)\}\) do not coincide with \(\zeta\) for any \(n\). If \(E\) is a complete metric space and \(X\) has no discontinuities of the second kind, then for a.s. \(\sigma\)-trajectories there exists \(x_{\zeta-0}\). If the semigroup \(T_t\) maps into itself the space of continuous functions tending to \(0\) at infinity, then conversely: a.s. trajectories for which \(x_{\zeta-0}\) exists are \(\sigma\)-trajectories. Introduce the notation: \(\Omega_X^\sigma\) is the set of \(\sigma\)-trajectories of the process \(X\); \(\Omega_X^\delta\) is the set of \(\delta\)-trajectories;
   \[ p_\sigma(x)=P_x(\Omega_X^\sigma),\qquad p_\delta(x)=P_x(\Omega_X^\delta). \]
   Let
   \[ e(x)=M_x\min\{\zeta,1\}. \]
   Then
   \[ \Omega_X^\sigma=\left\{0<\inf_{t<\zeta} e(x_t)\le \sup_{t<\zeta} e(x_t)<1\right\}; \]
   \[ \Omega_X^\delta\cap\{\zeta<\infty\}=\{e(x_{\zeta-0})=0\}; \]
   \[ \{\zeta=\infty\}=\{e(x_{\zeta-0})=1\}\quad\text{(a.s. }P_x\text{)}. \]

2. We shall call a process a \(\sigma\)-process if \(p_\sigma(x)\equiv 1\), and a \(\delta\)-process if \(p_\delta(x)\equiv 1\). Introduce the notation:
   \[ \overline{a}A=\{\psi_s\omega:\ \omega\in A,\ s\le \zeta(\omega)\}, \]
   where \(A\subseteq\Omega\). A property of a \(\sigma\)-process: if \(A\subseteq\Omega\), \(\overline{P}_x(A)=1\), then \(\overline{P}_x(\overline{a}A)=1\). For a \(\delta\)-process, on the contrary, if \(\overline{P}_x(A)=1\), then \(\overline{P}_x(\overline{a}A)=0\). The class of \(\delta\)-processes is characterized by the fact that there exists \(\{\Gamma_n\}\) \((\Gamma_n\in\mathscr{B})\) such that for a.s. \(\omega\in\{\zeta<\infty\}\)
   \[ \xi(\Gamma_n)<\zeta,\qquad \xi(\Gamma_n)\uparrow\zeta. \]
   If for a \(\delta\)-process \(P(t,x,E)<1\) \((t>0,\ x\in E)\), then for
   \[ \Gamma_n=\left\{x:\ {1\over n}<P\left({1\over n},x,E\right)<1-{1\over n}\right\} \]
   we have \(\xi(\Gamma_n)<\zeta\),

\(\xi(\Gamma_n)\uparrow \zeta\) (a.s. \(P_x\)). The \(\sigma\)-processes include, for example, Feller processes in a compact space \(E\) with \(P_x\{\zeta<\infty\}\equiv1\). For a \(\sigma\)-process without ruptures of the second kind in a complete metric space, a.s. \(P_x\) there exists \(x_{\zeta-0}\).

1. For what follows we shall need one type of transformation of a process. Let \(f(x)\) be an excessive function (e.f.) of the process \(X\) with transition function \(P(t,x,dy)\) (see \((2)\)). Denote by \(X^f\) the process in the phase space \(\{x:f(x)>0\}\), corresponding to the transition function
   \[ P'(t,x,dy)=P(t,x,dy)[f(x)]^{-1}f(y). \]
   We denote the corresponding measures on \(\mathcal N\) by \(P_x^f\). For \(A\in\mathcal N^t\),
   \[ P_x^f(A)=[f(x)]^{-1}M_x[\chi_A(\omega)f(x_t)]. \]
   If \(\{\tau<t<\zeta\}\in\mathcal N^t\), \(\tau=\zeta\) (a.s. \(P_x\)), then \(\tau=\zeta\) (a.s. \(P_x^f\)). Hence it follows that if \(X\), a.s., has one of the following properties: a) it is continuous, b) it has no ruptures of the second kind, c) it is \(F\)-bounded, where \(F=\{\Gamma_n\}\), \(\Gamma_n\in\mathcal B\) \((n=1,2,\ldots)\), then, a.s., this property holds for \(X^f\).

2. Let \(X_1,X_2\) be two processes. We put \(X\in S\{X_1,X_2\}\) if there exist e.f.’s \(f_1(x),f_2(x)\) for \(X\) such that \(X_1=X^{f_1}\), \(X_2=X^{f_2}\), \(f_1(x)+f_2(x)\equiv1\). Let \(X_1,X_2\) be processes in phase spaces \(E_1,E_2\) with transition functions \(P_1(t,x,dy),P_2(t,x,dy)\). Then, in order that \(S(X_1,X_2)\) be nonempty, it is necessary and sufficient that: a) the set \(E_1\setminus E_2\) be absorbing for \(X_1\), and \(E_2\setminus E_1\) for \(X_2\); b) for \(x\in E_1\cap E_2\), \(dy\subset E_1\cap E_2\),
   \[ P_2(t,x,dy)=P_1(t,x,dy)[f(x)]^{-1}f(y), \]
   where \(f(x)\) is an e.f. for \(X_1\), \(f(x)>0\) for \(x\in E_1\cap E_2\). If \(X_0\in S(X_1,X_2)\), then \(S(X_1,X_2)\) coincides with the set of processes \(X_0^{f_1+c f_2}\), where \(c>0\). If \(X_0\in S(X_1,X_2)\), \(P_{0x},P_{1x},P_{2x}\) are the corresponding measures on \(\mathcal N\), and \(f_1(x),f_2(x)\) are e.f.’s for \(X_0\) such that \(X_0^{f_1}=X_1\), \(X_0^{f_2}=X_2\), then for \(A\in\mathcal N\)
   \[ P_{0x}(A)=f_1(x)P_{1x}(A)+f_2(x)P_{2x}(A). \]
   It follows from this that if \(\overline P_{1x}(A)=1\), \(\overline P_{2x}(B)=1\), then
   \[ \overline P_{0x}(A\cup B)=1. \]

3. Let \(X\) be some process. Then
   \[ X\in S(X^{p_\sigma},X^{p_\delta}),\qquad P_x^{p_\sigma}(\Omega_x^\sigma)=1,\qquad P_x^{p_\delta}(\Omega_x^\delta)=1. \]
   We shall call \(X^{p_\sigma}\) the \(\sigma\)-component, \(X^{p_\delta}\) the \(\delta\)-component. If \(p_\sigma(x)>0\) \((x\in E)\), then the \(\sigma\)-component is a \(\sigma\)-process; the \(\delta\)-component is always a \(\delta\)-process. Thus, if \(p_\sigma(x)>0\) \((x\in E)\), then \(X\in S(X_1,X_2)\), where \(X_1\) is a \(\sigma\)-process and \(X_2\) is a \(\delta\)-process. If \(p_\delta(x)>0\) \((x\in E)\), then \(X_1,X_2\) are determined uniquely.

4. We shall call an e.f. \(f(x)\) regular if
   \[ P_x^f\{f(x_{\zeta-0})<\infty\}=1, \]
   and singular if
   \[ P_x^f\{f(x_{\zeta-0})=\infty\}=1. \]
   Introduce the notation
   \[ dA=\{\psi_s\omega:\omega_f\in A,\ s\le \zeta(\omega)\}\quad(A\subset\Omega). \]
   If \(f(x)\) is regular, \(A\in\mathcal N\), \(\overline P_x(A)=1\), then
   \[ \overline P_x(dA)=1. \]
   If \(f(x)\) is singular, \(A=\{f(x_{\zeta-0})<\infty\}\), then \(\overline P_x(A)=1\),
   \[ \overline P_x(dA)=0. \]
   An e.f. \(f(x)\) decomposes into the sum of a regular \(f_1(x)\) and a singular \(f_2(x)\), and in a unique way:
   \[ f_1(x)=f(x)P_x^f\{f(x_{\zeta-0})<\infty\}, \]
   \[ f_2(x)=f(x)P_x^f\{f(x_{\zeta-0})=\infty\}. \]
   When e.f.’s are added, their components are added. E. B. Dynkin proved that \(f(x)\) is regular if \(f(x)=M_x\xi\), where \(\xi\) is an excessive random variable (see \((3)\)).

5. Let \(f(x)\) be an e.f. for \(X\); \(f(x)>0\) \((x\in E)\), \(T_\infty f(x)\equiv0\); let \(f_1(x),f_2(x)\) be the regular and singular components of \(f(x)\); \(Y=X^f\), and \(Y_1,Y_2\) the \(\sigma\)-component and \(\delta\)-component of \(Y\). Then, if
   \[ p_\delta(x)\equiv P_x\{\zeta=\infty\}, \]
   then \(f_1(x)\) maps \(X\) to \(Y_1\), and \(f_2(x)\) maps \(X\) to \(Y_2\).

Remark. The condition
\[ p_\delta(x)\equiv P_x\{\zeta=\infty\} \]
means that, a.s., the \(\delta\)-trajectories are nonterminating, and is fulfilled for nonterminating processes and \(\sigma\)-processes.

1. If \(f(x)\) is a positive e.f. for \(X\), then the corresponding transformation is invertible: if \(Y=X^f\), then \(X=Y^{1/f}\). Consider positive e.f.’s that transform \(X\) into a \(\sigma\)-process or a \(\delta\)-process. Let \(P_x\{\zeta<\infty\}\equiv1\). Put
   \[ e(x)=M_x\min\{\zeta,1\}. \]
   Then \(e(x)\) is an e.f., \(e(x)>0\) \((x\in E)\);

\(Y = X^\varepsilon\) is a \(\sigma\)-process; \(X = Y^f\), where \(f(x)=\dfrac{1}{e(x)}\). Consider processes for which one of the functions \(p_\sigma(x)\), \(p_\delta(x)\), \(r(x)=P_x\{\zeta=\infty\}\) is positive.

1) If \(p_\sigma(x)>0\) \((x\in E)\), then \(X=Y^{1+f}\), where \(Y\) is a \(\sigma\)-process, \(f(x)\) is a singular function for \(Y\), and this representation is unique,

\[ Y=X^{p_\sigma},\qquad f(x)=\frac{1-p_\sigma(x)}{p_\sigma(x)}. \]

2) If \(r(x)>0\) \((x\in E)\), then \(X=Y^{1+f}\), where \(Y\) is a nonterminating process, \(f(x)\) is an excessive function for \(Y\), \(T'_\infty f(x)\equiv 0\), where \(T'_t\) is the semigroup for \(Y\). The representation is unique,

\[ Y=X^r,\qquad f(x)=\frac{1-r(x)}{r(x)}. \]

3) If \(p_\delta(x)>0\) \((x\in E)\), then \(X=Y^f\), where \(Y\) is a \(\delta\)-process, for example,

\[ Y=X^{p_\delta},\qquad f(x)=\frac{1}{p_\delta(x)}. \]

1. Every continuous process \(X\) is a subprocess of a continuous \(\delta\)-process (a generalization of Theorem 1 \((^4)\)). Consider the relation between trajectories of the two types for the process \(X\) and the subprocess \(Y\) corresponding to the multiplicative functional \(\alpha_t>0\) \((t>\zeta)\) (see (1)). We have \(\Omega_Y^\delta=\Omega_X^\delta\cap\{\alpha_{\zeta-0}>0\}\). It follows that, if \(\alpha_{\zeta-0}=0\) (a.s. \(P_x\)) or \(X\) is a \(\sigma\)-process, then \(Y\) is a \(\sigma\)-process.

2. Some formulas.

\[ p_\sigma(x)=\lim_{n\to\infty}\lim_{h\to0}h^{-1}\int_0^\infty \{T_t[\chi_n(x)\,[1-P(h,x,E)]]\}\,dt, \]

where \(\chi_n(x)\) is the indicator of the set

\[ \left\{x:\ P\left(\frac1n,x,E\right)\geq\frac1n\right\}. \]

The regular component \(f_1(x)\) of the excessive function \(f(x)\) is given by the formula

\[ f_1(x)=\lim_{n\to\infty}\lim_{h\to0}h^{-1}\int_0^\infty \{T_t[\chi_{\{f(x)<n\}}(x)\,[f(x)-T_h f(x)]]\}\,dt, \]

if \(T_\infty f(x)\equiv0\), and by the formula

\[ f_1(x)=\lim_{n\to\infty}T_\infty[\chi_{\{f(x)<n\}}(x)\,f(x)]. \]

if \(T_\infty f(x)\equiv f(x)\). The proof of the formulas is based on Lemma 1.1 \((^5)\).

The author expresses his gratitude to E. B. Dynkin for assistance in preparing the note for publication.

Received
7 I 1963

REFERENCES

1. E. B. Dynkin, Foundations of the Theory of Markov Processes, Moscow, 1959.
2. E. B. Dynkin, DAN, 127, No. 1 (1959).
3. E. B. Dynkin, Proc. Fourth Berkeley Symposium on Math. Stat. and Probability, Berkeley, 1961.
4. V. A. Volkonskii, Teoriya Veroyatn. i ee Primen., 4, 2 (1959).
5. L. V. Seregin, Teoriya Veroyatn. i ee Primen., 6, 1 (1961).