UDC 519.217

MATHEMATICS

A. V. SKOROKHOD

ON THE LOCAL STRUCTURE OF CONTINUOUS MARKOV PROCESSES

(Presented by Academician Yu. V. Linnik on 21 X 1965)

The first continuous Markov processes considered by A. N. Kolmogorov \((^{1})\) were diffusion Markov processes. Subsequently, W. Feller \((^{2})\) found the general form of continuous Markov processes on the line. Among these processes there turned out to be also some that were not diffusions. E. B. Dynkin, however, showed (see, for example, \((^{3})\)) that they can be obtained from diffusions by rather simple transformations; in particular, nonterminating processes can be obtained from diffusions by changing the spatial variable and by a random change of time. In \((^{4})\) it was shown that multidimensional Markov processes that are martingales can be obtained by a random change of time from diffusion ones. In the present note we continue to study the connection between continuous processes and diffusion Markov processes. In what follows we use the terminology and notation of the book by E. B. Dynkin \((^{5})\).

1. Let \(X\) be a locally compact space, and let \(U\) be some neighborhood of \(X\) having compact closure. We shall consider a continuous strictly Markov process \(\{x_t,\zeta,\mathfrak M_t,P_x\}\) in the space \(X\). Since we shall be interested only in the local properties of the process, and a process with terminating trajectories can be obtained from a nonterminating process by shortening its lifetime, we shall assume that the killing time \(\zeta\) coincides with the time of first exit from \(U\). We shall also assume that the family of \(\sigma\)-algebras \(\mathfrak M_t\) coincides with the intersection, over all measures \(P_x\), of the completions of the \(\sigma\)-algebra generated by the quantities \(x_s\) for \(s \leq t\). One of the principal tools in the investigation of the process will be the \(M\)-functionals of the process, defined below.

Definition. An additive almost homogeneous functional \(a_t\) is called an \(M\)-functional if it satisfies the following conditions:
1) \(M_x a_t = 0\) for \(x \in X\);
2) \(M_x a_t^2\) is a measurable bounded function;
3) \(a_t\) is a continuous function of \(t\) with \(P_x\)-probability 1, whatever \(x\) may be;
4) there exists a positive additive almost homogeneous functional \(\langle a,a\rangle_t\) for which \(M_x a_t^2 = M_x\langle a,a\rangle_t\) for \(x \in X\).

The most important example of an \(M\)-functional is the functional

\[ \hat\varphi_t=\varphi(x_t)-\varphi(x_0)-\int_0^t A\varphi(x_s)\,ds,\qquad \varphi\in D_A, \]

where \(A\) is the infinitesimal operator of the process, and \(D_A\) is its domain of definition.

If \(\alpha_t\) and \(\beta_t\) are \(M\)-functionals, then so is \(\alpha_t+\beta_t\). With each pair of \(M\)-functionals \(\alpha_t\) and \(\beta_t\) we associate the additive functional \(\langle\alpha,\beta\rangle_t\):

\[ \langle \alpha,\beta\rangle_t = {}^{1}/_{2}\bigl[\langle \alpha+\beta,\alpha+\beta\rangle_t-\langle \alpha,\alpha\rangle_t-\langle \beta,\beta\rangle_t\bigr]. \]

Theorem 1. Let a sequence of partitions of the interval \([0,t]\):
\(0=t_{n0}<\cdots<t_{nn}=t\) be such that \(\max_k [t_{nk+1}-t_{nk}]\to 0\). Then

\[ \sum_{k=0}^{n-1}(\alpha_{t_{nk+1}}-\alpha_{t_{nk}})(\beta_{t_{nk+1}}-\beta_{t_{nk}}) \to \langle \alpha,\beta\rangle_t \]

in probability \(\mathbf P_x\), whatever \(x\in X\).

Theorem 2. For any two \(M\)-functionals \(\alpha_t\) and \(\beta_t\), the function \(\langle \alpha,\beta\rangle_t\) is absolutely continuous with respect to \(\langle \alpha,\alpha\rangle_t\) with probability \(\mathbf P_x\), equal to 1 for all \(x\in X\), i.e., there exists a function \(\psi(s)\) such that, with probability \(\mathbf P_x\) equal to 1, the relation

\[ \langle \alpha,\beta\rangle_t=\int_0^t \psi(s)\,d\langle \alpha,\alpha\rangle_s \]

holds.

Remark. The function \(\psi(s)\) has the form \(g(x_s)\), where \(g(x)\) is some measurable function.

Definition. Let \(\alpha_t\) and \(\beta_t\) be two \(M\)-functionals and

\[ \langle \alpha,\beta\rangle_t=\int_0^t g(x_s)\,d\langle \alpha,\alpha\rangle_s. \]

Then we shall call \(g(x)\) the derivative of the functional \(\beta_t\) with respect to the functional \(\alpha_t\), and denote

\[ g(x)=\frac{\partial \beta}{\partial \alpha}(x). \]

The existence of the derivative (possibly nonunique) follows from Theorem 2.

1. Let us now consider stochastic integrals with respect to \(M\)-functionals. If \(g(x)\) is such a function that

\[ \sup_x \mathbf M_x \int_0^t g(x_s)^2\,d\langle \alpha,\alpha\rangle_s<\infty, \]

then by the usual method one can construct the integral \(\int_0^t g(x_s)\,d\alpha_s\). However, this integral will be the limit, with respect to the measure \(\mathbf P_x\), of certain integral sums and therefore will depend on \(x\). It turns out that there exists an \(M\)-functional \(\gamma_t\) such that

\[ \gamma_t=\int_0^t g(x_s)\,d\alpha_s \tag{1} \]

with probability \(\mathbf P_x\) equal to 1 for all \(x\). In what follows, by a stochastic integral we shall always mean this \(M\)-functional. If the functional \(\gamma_t\) is defined by relation (1), then

\[ \langle \gamma,\gamma\rangle_t=\int_0^t g(x_s)^2\,d\langle \alpha,\alpha\rangle_s. \]

In the study of \(M\)-functionals the following generalization of Itô’s formula for stochastic integrals is useful.

Theorem 3. Let \(\alpha_t^{(1)},\ldots,\alpha_t^{(k)}\) be \(M\)-functionals, and let \(g(t,\xi_1,\ldots,\xi_k)\) be a function of \(k+1\) real variables for which there exist, are bounded, and are continuous the derivatives \(g_t'(t,\xi_1,\ldots,\xi_k)\), \(g_{\xi_i}'(t,\xi_1,\ldots,\xi_k)\),

\(\xi_{\xi_i\xi_j}(t,\xi_1,\ldots,\xi_k),\ i,j=1,\ldots,k\). Then

\[ \begin{aligned} g(t,\alpha_t^{(1)},\ldots,\alpha_t^{(k)}) &= g(0,0,\ldots,0)+\int_0^t g_s'(s,\alpha_s^{(1)},\ldots,\alpha_s^{(k)})\,ds \\ &\quad+\sum_{i=1}^k \int_0^t g_{\xi_i}'(s,\alpha_s^{(1)},\ldots,\alpha_s^{(k)})\,d\alpha_s^{(i)} +\frac12\sum_{i,j=1}^k\int_0^t g_{\xi_i\xi_j}''(s,\alpha_s^{(1)},\ldots,\alpha_s^{(k)})\,d\langle \alpha_i,\alpha_j\rangle_s . \end{aligned} \tag{2} \]

Corollary. Let \(\varphi_1(x),\ldots,\varphi_k(x)\) be functions from \(D_A\); let \(g(\xi_1,\ldots,\xi_k)\) be a twice continuously differentiable function. Then

\[ g(\varphi_1(x_t),\ldots,\varphi_k(x_t))- g(\varphi_1(x_0),\ldots,\varphi_k(x_0)) = \]

\[ = \sum_{i=1}^k \int_0^t g_{\xi_i}'(\varphi_1(x_s),\ldots,\varphi_k(x_s))\,d\hat{\varphi}_i(s)+ \]

\[ +\sum_{i=1}^k \int_0^t g_{\xi_i}'(\varphi_1(x_s),\ldots,\varphi_k(x_s))\,A\varphi_i(x_s)\,ds+ \]

\[ +\frac12\sum_{i,j=1}^k\int_0^t g_{\xi_i\xi_j}''(\varphi_1(x_s),\ldots,\varphi_k(x_s))\, d\langle \hat{\varphi}_i,\hat{\varphi}_j\rangle_s, \tag{3} \]

where

\[ \hat{\varphi}_i(s)=\varphi_i(x_s)-\varphi_i(x_0)-\int_0^s A\varphi_i(x_u)\,du . \]

1. Introduce in the set of \(M\)-functionals the relation of subordination: the functional \(\alpha_t\) is subordinate to \(\beta_t\) \((\alpha_t<\beta_t)\), if

\[ \alpha_t=\int_0^t g(x_s)\,d\beta_s, \]

where necessarily \(g(x)=\dfrac{\partial\alpha}{\partial\beta}(x)\). The functional \(\alpha_t\) is called maximal if the relation \(\alpha_t<\gamma_t\) implies \(\gamma_t<\alpha_t\).

Theorem 4. Every \(M\)-functional is subordinate to some maximal functional.

Theorem 5. If \(\alpha_t\) and \(\beta_t\) are two maximal functionals, then the functions \(\langle\alpha,\alpha\rangle_t\) and \(\langle\beta,\beta\rangle_t\) are absolutely continuous with respect to one another.

Corollary. If \(\alpha_t\) is an arbitrary \(M\)-functional, and \(\bar{\alpha}_t\) is maximal, then there exists a function \(g(x)\) for which

\[ \langle\alpha,\alpha\rangle_t=\int_0^t g(x_s)\,d\langle\bar{\alpha},\bar{\alpha}\rangle_s . \]

Take an arbitrary maximal functional \(\bar{\alpha}_t\) and define \(\tau_t\) from the relation \(\langle\bar{\alpha},\bar{\alpha}\rangle_{\tau_t}=t\). Put further \(y_t=x_{\tau_t}\) (the process \(y_t\) is obtained from the process \(x_t\) by means of the random change of time generated by the functional \(\langle\bar{\alpha},\bar{\alpha}\rangle_t\)). As follows from § 5, Ch. 10 of [5], \(y_t\) is also a continuous strong Markov process. It is easy to see that under such a change of time \(M\)-functionals pass into \(M\)-functionals, and maximal functionals into maximal functionals. Since for the maximal functional \(\tilde{\alpha}_t=\bar{\alpha}_{\tau_t}\) we have \(\langle\tilde{\alpha},\tilde{\alpha}\rangle_t=t\), it follows that for any two \(M\)-functionals

of the process \(y_t\), \(\beta_t\) and \(\gamma_t\), the expression \(\langle \beta,\gamma\rangle_t\) will be absolutely continuous with respect to \(t\), and hence

\[ \langle \beta,\gamma\rangle_t=\int_0^t g(y_s)\,ds. \]

Let \(\varphi_1,\varphi_2,\ldots,\varphi_n,\ldots\) be an arbitrary sequence of functions from the domain of definition of the infinitesimal operator \(\tilde A\) of the process \(y_t\), and let \(g(\xi_1,\ldots,\xi_N)\) be a twice continuously differentiable function of its arguments. Denote by \(\tilde A[\varphi_i\varphi_j](x)\) the function for which

\[ \langle \hat\varphi_i,\hat\varphi_j\rangle_t = \int_0^t \tilde A[\varphi_i\varphi_j](y_s)\,ds. \]

Then from formula (3) there follows the relation

\[ \mathbf{M}_y g(\varphi_1(y_t),\ldots,\varphi_N(y_t)) - g(\varphi_1(y),\ldots,\varphi_N(y_t)) = \]

\[ = \int_0^t \left[ \sum_{i=1}^{N} \frac{\partial g}{\partial \xi_i} (\varphi_1(y_s),\ldots,\varphi_N(y_s))\,A\varphi_i(y_s) + \right. \]

\[ \left. + \frac{1}{2} \sum_{i,j=1}^{k} \frac{\partial^2 g}{\partial \xi_i \partial \xi_j} (\varphi_1(y_s),\ldots,\varphi_N(y_s))\, \tilde A[\varphi_i\varphi_j](y_s) \right]\,ds. \tag{4} \]

If we now choose the sequence \(\varphi_n\) so that it can be regarded as local coordinates in \(U\), then it will follow from (4) that the quasicharacteristic operator of the process is a second-order differential operator, and this means precisely that the process \(y_t\) is quasidiffusive (see \({}^{5}\), p. 31).

Institute of Mathematics
Academy of Sciences of the Ukrainian SSR

Received
14 X 1965

REFERENCES

\({}^{1}\) A. N. Kolmogorov, UMN, vol. 5, 5 (1938).
\({}^{2}\) W. Feller, Ann. Math., 60, 447 (1954).
\({}^{3}\) E. B. Dynkin, DAN, 144, 483 (1962).
\({}^{4}\) A. V. Skorokhod, Theory of Probability and Its Applications, 8, 379 (1963).
\({}^{5}\) E. B. Dynkin, Markov Processes, Moscow, 1963.
\({}^{6}\) K. Ito, Collected Translations. Mathematics, 3, 5, 131 (1959).