Reports of the Academy of Sciences of the USSR

1962, Volume 147, No. 2

MATHEMATICS

M. G. Shur

ON A CLASS OF MARKOV PROCESSES WHOSE EXIT PROBABILITIES ARE MAJORIZED BY THE EXIT PROBABILITIES OF A WIENER PROCESS

(Presented by Academician P. S. Aleksandrov, 14 V 1962)

In the \(l\)-dimensional Euclidean space \(R^l\) let us consider the Wiener process \(\hat X=(\hat x_t,\hat{\mathcal M}_t,\hat P_x)\) and a strictly Markov process
\(X=(x_t,\xi,\mathcal M_t,P_x)\), obtained by a random change of time in some subprocess of the process \(\hat X\). Denote by \(\tau_U\) (respectively, \(\hat\tau_U\)) the moment of first exit of the process \(X\) (the process \(\hat X\)) from the domain \(U\)*, and put

\[ \pi_U(x,\Gamma)=P_x\{x(\tau_U)\in\Gamma\},\qquad \hat\pi_U(x,\Gamma)=\hat P_x\{\hat x(\hat\tau_U)\in\Gamma\} \]

for any Borel set \(\Gamma\) in \(R^l\). The system of measures \(\pi_U(x,\Gamma)\), evidently, satisfies the condition:

A. \(\pi_U(x,\Gamma)\leq \hat\pi_U(x,\Gamma)\) for all Borel \(\Gamma\).

If, moreover, \(P_x\{\xi>0\}=1\) for all \(x\in R^l\), then the following condition also holds:

B. \(\pi_{U_n}(x,R^l)\to 1\) as \(n\to\infty\) for any sequence of domains \(U_n\), each of which contains the point \(x\) and whose diameters tend to zero as \(n\to\infty\).

This assertion admits a converse. More precisely, the following theorem is true (by \(B^l\) is denoted the \(\sigma\)-algebra of Borel sets in \(R^l\)).

Theorem. Whatever standard process
\[ X=(x_t,\xi,\mathcal M_t,P_x), \]
given in the measurable space \((R^l,B^l)\) and satisfying conditions A and B, there exists an equivalent process \(X\), obtained by means of a random change of time in some subprocess of the Wiener process**.

In the case when \(\xi\equiv\infty\) and in condition A the equality sign stands, this theorem was proved by Mackeene and Tanaka \((^6)\). The supposition of its validity in the general case was expressed by E. B. Dynkin \((^4)\).

1. Denote by
   \[ \bar X=(\bar x_t,\bar\xi,\bar{\mathcal M}_t,\bar P_x) \]
   some subprocess of the Wiener process, and let \(\bar\tau_U\) be the moment of first exit of the process \(\bar X\) from the domain \(U\), while
   \[ \bar\pi_U(x,\Gamma)=\bar P_x\{\bar x(\bar\tau_U)\in\Gamma\}. \]
   Our immediate aim will be the construction of such a process \(\bar X\) for which the measures \(\bar\pi_U(x,\Gamma)\) coincide with \(\pi_U(x,\Gamma)\) for all domains \(U\) and all \(x\in R^l\). Everywhere, unless the contrary is stated, it is assumed that \(l\geq 2\).

* For the terminology, see \((^3,^4)\). The class of processes considered here was investigated in \((^4)\).

** Recall that \(\tau_U=\xi\), if \(x_t\in U\) for all \(t\) \((0\leq t<\xi)\), and
\[ \tau_U=\inf(t:t>0,\ x_t\notin U) \]
otherwise.

*** Two Markov processes given on one and the same measurable space are called equivalent if their transition functions coincide. A homogeneous, right-continuous, strictly Markov process
\[ X=(x_t,\xi,\mathcal M_t,P_x), \]
given in \((R^l,B^l)\), is called standard if \(\mathcal M_t\supset \mathcal N_{t+0}\) and if, for any \(x\in R^l\) and any nondecreasing sequence of random variables \(\tau_n\), not depending on the future, \(x(\tau_n)\) as \(n\to\infty\) tends to \(x(\tau)\), where
\[ \tau=\lim_{n\to\infty}\tau_n, \]
\(P_x\)-almost surely on the set
\[ \Omega_1=\{\tau<\xi\}. \]

Denote by \(\rho(x,y)\) the distance between points \(x\) and \(y\) in \(R^l\), and consider the sequence \(\tau_k^{(n)}\), defined as follows. We set \(\tau_0^{(n)} \equiv 0\). If \(\tau_k^{(n)}\) has already been defined, then \(\tau_{k+1}^{(n)}\) is set equal to the lower bound of the times \(t\) such that \(t>\tau_k^{(n)}\) and
\[ \rho\bigl(x(t),x(\tau_k^{(n)})\bigr)\geq \sqrt{l/2^{n+1}}, \]
provided that the set of such \(t\) is nonempty; otherwise \(\tau_{k+1}^{(n)}=\zeta\). The random variables \(x(\tau_k^{(n)})\), for any fixed \(n\), form a Markov chain.

Next, put \(x_n(t)=x(\tau_k^{(n)})\), if \(\tau_k^{(n)}<\zeta\) and \(k\cdot 2^{-n}\leq t<(k+1)2^{-n}\) (when \(\tau_k^{(n)}\geq \zeta\) and \(t\geq k\cdot 2^{-n}\), the quantity \(x_n(t)\) is not defined). The random functions \(x_n(t)\) are trajectories of a certain nonhomogeneous Markov process
\[ X_n=(x_n(t),\xi_n,\mathcal M_t^s(n),P_{s,x}^{(n)}), \]
defined on the same set of elementary events as \(X\), and having transition function
\[ P_n(s,x,t,\Gamma)=P_x\{x(\tau_v^{(n)})\in\Gamma\}, \]
where \(v\) is the difference of the integer parts of the numbers \(2^n t\) and \(2^n s\). In an analogous way we construct the chains \(\hat x(\hat\tau_k^{(n)})\) and the process \(\hat X_n\) with transition function \(\hat P_n(s,x,t,\Gamma)\), starting from the Wiener process. From consideration of the sequences \(x(\tau_k^{(n)})\) it is not difficult to derive that, for any \(x\in R^l\), the trajectory \(x_t(\omega)\) is continuous in \(t\) \((0\leq t<\zeta)\) \(P_x\)-almost surely.

The transition function of the desired process \(\overline X\) will subsequently be constructed as the limit \(P_n(0,x,t,\Gamma)\).

It is important to note that
\[ \hat P_n(0,x,t,\Gamma)\geq P_n(0,x,t,\Gamma), \]
and that the study of the distributions of the quantities \(\hat x(\hat\tau_k^{(n)})\) is in an obvious way reduced to the study of the distributions of normalized sums of independent random vectors \(\xi_i\), each of which is uniformly distributed on the circle of unit radius with center at the origin of the coordinates in \(R^l\). Using the theorem of paper \((^1)\) and the known estimates for the maximum of sums of independent quantities (see \((^2)\), Ch. 3, Theorem 2.2), we easily obtain\(^*\):

a) whatever \(T>0\) and \(\varepsilon>0\) may be, there exist numbers \(N\) and \(K\) such that for \(n>N\) the inequality
\[ \hat P_n(s,x,t,\Gamma)\leq K\lambda(\Gamma)+\varepsilon \]
holds for all \(x\in R^l\) and all numbers \(s\geq 0,\ t\geq 0\) for which \(t-s>T\);

b) for fixed \(t\in\Lambda\) and \(n\to\infty\), the function \(\hat P_n(0,x,t,\Gamma)\) tends to the transition function \(\hat P(t,x,\Gamma)\) of the Wiener process, uniformly in \(x\in R^l\) and \(\Gamma\in B^l\);

c) \(\delta>0\) can be chosen so that, for all sufficiently large \(n\), the quantity \(\alpha_n^\varepsilon(\delta)/\delta\) does not exceed any preassigned number.

We shall say that the Borel measures \(\mu_n\) converge weakly as \(n\to\infty\) to the Borel measure \(\mu\), if
\[ \int f\,d\mu_n\to \int f\,d\mu \]
for every continuous function \(f(x)\) tending to zero at infinity. Denote by \(\tau_n'\) the first exit time of \(X_n\) from \(U\) \((U\in C)\), and put
\[ \mu_n(x,\Gamma)=P_{0,x}^{(n)}\{x_n(\tau_n')\in\Gamma\}. \]

Lemma 1. Whatever \(x\in U\) \((U\in C)\) may be, the measures \(\mu_n(x,\Gamma)\) converge weakly to \(\pi_U(x,\Gamma)\) as \(n\to\infty\).

From assertions a) and c), Lemma 1, and estimate (6.28) of book \((^3)\), one may conclude that

d) for any fixed \(x\in R^l\), \(t\geq 0\), and \(\varepsilon>0\), one can indicate so small an \(s>0\) that, for all sufficiently large \(n\), the following will be fulfilled—

\(^*\) In what follows, \(\lambda\) is the ordinary Lebesgue measure in \(R^l\); \(V_\varepsilon(x)\) is the exterior of the \(\varepsilon\)-neighborhood of the point \(x\); \(\alpha_n^\varepsilon(\delta)\) is the upper bound of the values of \(\hat P_n(s,x,t,V_\varepsilon(x))\) for \(x\in R^l\) and \(t-s\leq\delta\); \(C\) is the collection of domains whose boundaries are \((l-1)\)-dimensional smooth manifolds of class 2; \(\Lambda\) is the collection of nonnegative dyadic-rational numbers.

the inequalities hold

\[ P_n(0,x,s,V_\varepsilon(x))<\varepsilon,\qquad P^{(n)}_{0,x}\{\xi_n>s\}>1-\varepsilon,\qquad P^{(n)}_{0,x}\{t+\dot{s}>\xi_n>t\}<\varepsilon. \tag{1} \]

1. Let us proceed to the construction of the transition function of the desired process \(\overline{X}\). Consider the family of functions

\[ \Phi_n(s,x,t,\Gamma)=\int \widehat{P}(s,x,dy)\,P_n(s,y,t,\Gamma) \qquad (n\geq 0,\ t\geq s,\ \Gamma\in B^l). \]

For any fixed \(s>0\) this family of functions is equicontinuous in \(x\). Moreover, if \(\varepsilon>0\), \(s_0>0\), and \(x\in R^l\) are fixed, then one can choose \(s>0\) such that, for all \(\Gamma\in B^l\), \(t>s_0\), and all sufficiently large \(n\),

\[ \left|P_n(0,x,t,\Gamma)-\Phi_n(s,x,t,\Gamma)\right|<2\varepsilon. \tag{2} \]

Indeed, assuming \(s\) to satisfy the second of inequalities (1) and recalling the Chapman–Kolmogorov equality for \(X_n\), we obtain, for large values of \(n\), that

\[ \left|P_n(0,x,t,\Gamma)-\int \widehat{P}_n(0,x,s,dy)\,P_n(s,y,t,\Gamma)\right|<\varepsilon, \]

whence, according to b), our assertion follows.

Lemma 2. There exists a numerical sequence \(\{n_k\}\) such that, for any \(x\in R^l\) and \(t\in\Lambda\), the measures \(P_{n_k}(0,x,t,\Gamma)\) converge weakly as \(k\to\infty\) to some measure \(\overline{P}(t,x,\Gamma)\).

Lemma 2 is easily derived from the assertion: for any \(s_0>0\) there exists a sequence \(\{n_k\}\) such that, for all \(x\in R^l\) and all \(t>s_0\) \((t\in\Lambda)\), the measures \(P_{n_k}(0,x,t,\Gamma)\) converge weakly to some measure \(Q(s_0,t,x,\Gamma)\). To prove this assertion, fix \(s_0>0\), \(t>s_0\) \((t\in\Lambda)\), and \(n\geq 0\).

For each \(x\in R^l\) choose \(s=s(n,x)\) \((0<s(n,x)\leq 2^{-n})\) so as to satisfy (2) with \(\varepsilon=2^{-n-1}\). Then about each \(x\) describe an open ball \(V(x,n)\) such that, for all points \(y\) in it,

\[ \left|\Phi_m(s(n,x),x,t,\Gamma)-\Phi_m(s(n,x),y,t,\Gamma)\right|<2^{-n} \]

for any \(m\geq 0\), \(\Gamma\in B^l\). Keeping \(n\) fixed for the time being, from the system of balls \(V(x,n)\) extract a countable covering of the whole of \(R^l\). Denote the centers of the balls entering this covering by \(x_{n,r}\), and choose \(\{n_k\}\) so that our assertion is true for all \(x_{n,r}\) for any \(n\) and \(r\). The sequence \(\{n_k\}\) is the desired one. Indeed, if \(x_0\in R^l\), \(2^{-n_0}\leq s(n,x_0)\), and \(x_0\in V(x_{n_0,r_0},n_0)\), then

\[ \left|P_m(0,x_0,t,\Gamma)-P_m(0,x_{n_0,r_0},t,\Gamma)\right|<4\cdot 2^{-n} \]

for all sufficiently large \(m\).

Lemma 2 and assertion a) allow us, from the Chapman–Kolmogorov equality for the process \(X_n\), to conclude that, for all \(x\in R^l\) and \(\Gamma\in C\) (and consequently also for all \(\Gamma\in B^l\)),

\[ \overline{P}(s+t,x,\Gamma)=\int \overline{P}(s,x,dy)\,\overline{P}(t,y,\Gamma), \tag{3} \]

where \(s,t\in\Lambda\). For \(t\notin\Lambda\) \((t>0)\), define the measure \(\overline{P}(t,x,\Gamma)\) as the weak limit of the measures \(\overline{P}(u,x,\Gamma)\) as \(u\downarrow t\) (the correctness of this definition follows from c)). Taking a) into account, one can verify that the measures \(\overline{P}(t,x,\Gamma)\) satisfy (3) for all \(s,t\geq 0\), and, thus, \(\overline{P}(t,x,\Gamma)\) is a transition function. In view of the fact that \(\overline{P}(t,x,\Gamma)\) is majorized by the transition function of the Wiener process, the transition function \(\overline{P}(t,x,\Gamma)\) corresponds to some subprocess of the Wiener process (7). We shall take this subprocess as \(\overline{X}\).

3. It is not difficult to verify that the finite-dimensional distributions of the processes \(X_{n_k}\) converge weakly, as \(k \to \infty\), to the finite-dimensional distributions of the process \(\overline{X}\).

By means of Theorem 3.1.2 from \((^5)\) we establish that the measures \(\mu_n(x,\Gamma)\), for every \(U \in C\), converge weakly to \(\overline{\pi}_U(x,\Gamma)\), and, consequently, in accordance with Lemma 1, \(\pi_U(x,\Gamma)=\overline{\pi}_U(x,\Gamma)\) if \(U \in C\). This equality is extended to all domains \(U\) with the aid of Lemma 3.

Lemma 3. Let \(Y=(y_t,\xi',\mathcal{M}'_t,Q_x)\) be a standard process satisfying conditions A and B. Let \(\xi_n\) be a nondecreasing sequence of random variables independent of the future, and let \(\xi=\lim_{n\to\infty}\xi_n\).

Let the event \(H\) consist in the fact that \(\sup_{0\le t<\xi}|y_t|<\infty\), where \(|y_t|\) denotes the distance of \(y_t\) from the origin in \(R^l\). Then the measures

\[ \mu_x^{(n)}(\Gamma)=Q_x\{H,\; y(\xi_n)\in\Gamma\} \]

converge weakly to the measures

\[ \mu_x(\Gamma)=Q_x\{H,\; y(\xi)\in\Gamma\} \]

as \(n\to\infty\).

Now, from a theorem of Blumenthal, Getoor, and McKean one may conclude that the process \(X\) is equivalent to some process obtained from \(\overline{X}\) by a random change of time. In the case \(l=1\), however, our theorem follows from the works \((^8,^9)\).

Received
11 V 1962

CITED LITERATURE

\(^1\) W. Richter, Teor. Veroyatn. i ee Primen., 3, No. 1 (1958).
\(^2\) J. L. Doob, Stochastic Processes, Moscow, 1956.
\(^3\) E. B. Dynkin, Foundations of the Theory of Markov Processes, Moscow, 1959.
\(^4\) E. B. Dynkin, DAN, 144, No. 3 (1962).
\(^5\) A. V. Skorokhod, Teor. Veroyatn. i ee Primen., 1, No. 3 (1956).
\(^6\) H. P. McKean Jr., H. Tanaka, Mem. Coll. Sci. Univ. Kyoto, ser. A, 33, No. 3 (1961).
\(^7\) P. A. Meyer, Fonctionnelles multiplicatives et additives de Markov, Thèses, Université de Paris, 1961.
\(^8\) V. A. Volkonskii, Teor. Veroyatn. i ee Primen., 3, No. 3 (1958).
\(^9\) V. A. Volkonskii, Teor. Veroyatn. i ee Primen., 4, No. 2 (1959).