A. V. SKOROKHOD

ON MULTIPLICATIVE UNIFORMLY CONTINUOUS FAMILIES OF RANDOM PROBABILITY OPERATORS

(Presented by Academician A. N. Kolmogorov on 27 III 1964)

1. Consider a set \(Y\) with a distinguished \(\sigma\)-algebra \(\mathfrak{B}\) of subsets on it, containing all one-point subsets of \(Y\). Denote by \(B\) the linear normed space of numerical functions \(f(y)\), defined on \(Y\) and \(\mathfrak{B}\)-measurable, with norm
   \[ \|f\|=\sup_{y\in Y}|f(y)|. \]
   A linear operator \(\Pi\), defined on \(B\), is called a probability operator if it takes nonnegative functions into nonnegative functions and \(\Pi 1=1\).

Let \(\{\Omega,\mathfrak{F}\}\) be a measurable space (\(\mathfrak{F}\) is a \(\sigma\)-algebra of subsets of \(\Omega\)). Consider a family of \(\sigma\)-algebras \(\mathfrak{F}_t^s\), defined for \(0\leq s\leq t\) and satisfying the condition: if \([s_1,t_1]\subset [s,t]\), then
\[ \mathfrak{F}_{t_1}^{s_1}\subset \mathfrak{F}_{t}^{s}\subset \mathfrak{F}. \]

Suppose that for all \(0\leq s<t\) and \(\omega\in\Omega\) probability operators \(\Pi_t^s(\omega)=\Pi_t^s\) on \(B\) are defined, satisfying the conditions:

M-1. For all \(f\in B\) and \(s<t\), the quantity \(\Pi_t^s f(y)\), as a function of \(\omega\) and \(y\), is measurable with respect to the \(\sigma\)-algebra \(\mathfrak{F}_t^s\times\mathfrak{B}\).

M-2. For \(s<t<u\) the relation
\[ \Pi_t^s\Pi_u^t=\Pi_u^s \]
holds.

Such a family of operators will be called a multiplicative family of probability operators.

Suppose also that condition M-3 holds. For all \(\omega\in\Omega\) and \(T>0\),
\[ \lim_{h\to 0}\sup_{0\leq s\leq T}\sup_{0<t-s<h}\|\Pi_t^s-I\|=0, \]
where \(I\) is the identity operator.

Then the family is called uniformly continuous. A family of operators satisfying conditions M-1, M-2, M-3 will be called an \(M\)-family.

In addition to multiplicative families of operators, certain additive families of operators, called \(A\)-families, will be used; their definition is given below.

A family of linear operators \(A_t^s(\omega)=A_t^s\) on \(B\), defined for all \(0\leq s<t\) and \(\omega\in\Omega\), is called an \(A\)-family if the following conditions are satisfied:

A-1. Whatever \(0\leq s<t\) and \(\omega\in\Omega\) may be, the operator \(I+\varepsilon A_t^s\) is a probability operator for \(\varepsilon<\|A_t^s\|^{-1}\).

A-2. The function \(A_t^s(\omega)f(y)\) is measurable with respect to \(\mathfrak{F}_t^s\times\mathfrak{B}\), for fixed \(s\) and \(t\), for all \(f\) from \(B\).

A-3. For all \(\omega\in\Omega\) and \(T>0\),
\[ \lim_{h\to 0}\sup_{0\leq s\leq T}\sup_{0<t-s<h}\|A_t^s\|=0. \]

There exists a one-to-one correspondence between \(M\)-families and \(A\)-families.

Theorem. To every \(M\)-family \(\Pi_t^s\) there corresponds an \(A\)-family \(A_t^s\), connected with \(\Pi_t^s\) by the equation: for \(s \in [0,t]\)

\[ \Pi_t^s = I+\int_s^t (d_u A_t^u)\Pi_t^u . \tag{1} \]

To every \(A\)-family \(A_t^s\), equation (1) assigns a unique \(M\)-family \(\Pi_t^s\), which is expressed in terms of \(A_t^s\) by the formula

\[ \Pi_t^s = I+\int_s^t d_u A_t^u+\cdots+ \int_{s \le u_1<\cdots<u_k \le t} d_{u_1}A_t^{u_1}\cdots d_{u_k}A_t^{u_k}+\cdots . \tag{2} \]

Thus, the description of \(M\)-families can be reduced to the description of the corresponding \(A\)-families, which in some cases is much simpler.

Remark 1. Every linear operator \(C\) on \(B\) has the form

\[ Cf(y)=\int f(y')\,c(y,dy'), \]

where \(c(y,A)\), for fixed \(y\), is a countably additive function of bounded variation on \(\mathfrak B\), and for fixed \(A \in \mathfrak B\) the function \(c(y,A)\) belongs to \(B\). Let

\[ \Pi_t^s f(y)=\int \pi_t^s(y,dy')\,f(y'),\qquad A_t^s f(y)=\int \alpha_t^s(y,dy')\,f(y'), \]

and let \(\delta(y,A)=1\) for \(y\in A\), \(\delta(y,A)=0\) for \(y\notin A\). Then relations (1) and (2) may be rewritten in the form

\[ \pi_t^s(y,A) = \delta(y,A)+ \int_s^t\int_Y \pi_t^u(y',A)\,d_u\alpha_t^u(y,dy'), \tag{3} \]

\[ \pi_t^s(y,A)=\delta(y,A)+\cdots \]

\[ \cdots+ \int_{s\le u_1<\cdots<u_k\le t} \int_Y\cdots\int_Y d_{u_1}\alpha_t^{u_1}(y,dy_1)\cdots d_{u_k}\alpha_t^{u_k}(y_{k-1},A)+\cdots . \tag{4} \]

In the proof of the theorem it is established that \(d_s\pi_t^s(y,dy')\) and \(d_s\alpha_t^s(y,dy')\) have bounded variation on \([0,T]\times Y\), so that the integrals in (3) and (4) have the usual meaning.

Remark 2. If a probability measure \(P\) is given on \(\Omega\) and the relations A-2, A-3, M-2, M-3 hold with probability 1, then the theorem remains valid if one requires (1) and (2) also to hold only with probability 1.

Remark 3. If instead of conditions M-3 and A-3 one requires the fulfillment of the conditions:

M-4. For every \(\varepsilon>0\), with probability 1,

\[ \lim_{h\to 0}\sup_{0\le s\le T}\sup_{0<t-s<h} P\{\|\Pi_t^s-Y\|>\varepsilon/\mathfrak F_s^0\}=0. \]

A-4. For every \(\varepsilon>0\), with probability 1,

\[ \lim_{h\to 0}\sup_{0\le s\le T}\sup_{0<t-s<h} P\{\|A_t^s\|>\varepsilon/\mathfrak F_s^0\}=0, \]

then the theorem remains in force if the equalities (1) and (2) hold with probability 1, and the integrals are understood as the corresponding limits in the sense of convergence in probability.

1. Let now a Markov process \(x_t\) be defined in some space \(X\), and let \(\mathfrak F_t^s\) denote the minimal \(\sigma\)-algebra with respect to which the \(x_u\), \(u\in[s,t]\), are measurable. With every multiplicative family of ope-

with the operators \(\Pi_t^s\) one can associate a conditionally Markov process \(y_t\) with values in \(Y\) so that the pair \((x_t,y_t)\) is also a Markov process and

\[ \mathbf M\{f(y(t))/\mathfrak F_t^s,\, y(s)\}=\Pi_t^s f(y(s)) \]

(the question of adjoining a conditionally Markov process to a Markov process was considered by A. D. Venttsel’ in \((^1)\)).

In the case when the theorem is applicable, the description of all multiplicative families reduces to the description of additive families \(A_t^s\). Since such a family is determined by a function \(\alpha_t^s(y,A)\), \(y\in Y\), \(A\in\mathfrak B\), it is sufficient to specify this function. From the conditions imposed on \(A_t^s\) it follows that, for \(y\in \bar A\), the quantity \(\alpha_t^s(y,A)\) is a nonnegative additive functional of the process \(x_t\); the same functional will be \(-\alpha_t^s(y,\{y\})=\alpha_t^s(y,Y-\{y\})\); \(\{y\}\) is the set consisting of the point \(y\). In the case of homogeneous processes, nonnegative additive homogeneous functionals are completely described (see \((^2)\), Ch. 6). This makes it possible to describe all homogeneous \(M\)-families of operators by means of additive functionals (instead of condition M-3 one may impose condition M-4).

In the simplest case, when \(Y\) is a finite set \(\{y_1,\ldots,y_n\}\), the probability operator \(\Pi_t^s\) is determined by a stochastic matrix \(\|p_t^s(i,j)\|\), \(i,j=1,\ldots,n\). If the functions \(p_t^s(i,j)\) satisfy the condition: for every \(\varepsilon>0\), with probability 1,

\[ \lim_{h\to0}\sup_{s\le T}\sup_{0<t-s<h}\mathbf P\{|p_t^s(i,j)-\delta_{ij}|>\varepsilon/x_s\}=0, \]

where \(\delta_{ij}\) are the elements of the identity matrix, then there exist nonnegative additive functionals \(\alpha_t^s(i,j)\), \(i\ne j\), \(i,j=1,\ldots,n\), such that

\[ d_s p_t^s(i,j)=\sum_{k\ne i}[p_t^s(k,j)-p_t^s(i,j)]\,d_s\alpha_t^s(i,k). \]

Kiev State University

Received
25 III 1964

REFERENCES

\(^1\) A. D. Venttsel’. Abstracts of reports of the VII Conference on Probability Theory and Statistics, Markov Processes, 1–7, Tbilisi, 1963. \(^2\) E. B. Dynkin, Markov Processes, Moscow, 1963.