G. I. Kalmykov

CORRELATION FUNCTIONS OF A GAUSSIAN MARKOV PROCESS

(Presented by Academician S. N. Bernstein on January 3, 1964)

1. In this article a two-dimensional stationary Gaussian Markov process \(\{\xi(t)\}=\{(\xi_1(t),\xi_2(t))\}\) is considered. It is assumed that \(M\xi_1(t)=M\xi_2(t)=0,\; M\xi_1^2(t)=M\xi_2^2(t)=1\). By \(R(t)=\|r_{ij}(t)\|\) we denote the correlation matrix function, where \(r_{ij}(t)\) is defined as follows:
   \[ r_{ij}(t)=M\xi_i(s)\xi_j(s+t),\quad i,j=1,2. \]
   The matrix function \(R(t)\) satisfies the equation \(r_{ij}(t)=r_{ji}(-t)\). Therefore it is sufficient to investigate \(R(t)\) for \(t\ge 0\). Below, \(t\) is everywhere considered nonnegative. The process is assumed to be nondegenerate \((r=r_{12}(0)=r_{21}(0)\ne \pm 1)\) and nonsingular. The correlation functions \(r_{ij}(t)\) must satisfy the conditions:

I. \(|r_{ij}(t)|\le 1\), since the process is normalized.

II. The principal minors of the matrix
\[ S(t)= \begin{vmatrix} 1 & r & r_{11}(t) & r_{12}(t)\\ r & 1 & r_{21}(t) & r_{22}(t)\\ r_{11}(t) & r_{21}(t) & 1 & r\\ r_{12}(t) & r_{22}(t) & r & 1 \end{vmatrix} \]
must be positive.

The purpose of this article is to determine which matrix functions can be correlation matrix functions of a Gaussian Markov process. In doing so, we shall start from conditions I and II and from the system of functional equations for the correlation functions \(r_{ij}(t)\) of a Gaussian Markov process*:
\[ (1-r^2)r_{11}(t_1+t_2)=r_{11}(t_1)r_{11}(t_2)+r_{12}(t_1)r_{21}(t_2)- \]
\[ {}-r\,[r_{12}(t_1)r_{11}(t_2)+r_{11}(t_1)r_{21}(t_2)]; \tag{1} \]
\[ (1-r^2)r_{22}(t_1+t_2)=r_{22}(t_1)r_{22}(t_2)+r_{21}(t_1)r_{12}(t_2)- \]
\[ {}-r\,[r_{21}(t_1)r_{22}(t_2)+r_{22}(t_1)r_{12}(t_2)]; \tag{2} \]
\[ (1-r^2)r_{12}(t_1+t_2)=r_{12}(t_1)r_{22}(t_2)+r_{11}(t_1)r_{12}(t_2)- \]
\[ {}-r\,[r_{12}(t_1)r_{12}(t_2)+r_{11}(t_1)r_{22}(t_2)]; \tag{3} \]
\[ (1-r^2)r_{21}(t_1+t_2)=r_{21}(t_1)r_{11}(t_2)+r_{22}(t_1)r_{21}(t_2)- \]
\[ {}-r\,[r_{21}(t_1)r_{21}(t_2)+r_{22}(t_1)r_{11}(t_2)]. \tag{4} \]

1. From the system of functional equations (1)—(4) one can pass to a system of differential equations with constant coefficients, which agrees with the remark made in work \((^1)\):
   \[ (1-r^2)r'_{11}(t)=[r'_{11}(0)-rr'_{21}(0)]\,r_{11}(t)+[r'_{21}(0)-rr'_{11}(0)]\,r_{12}(t); \tag{5} \]
   \[ (1-r^2)r'_{12}(t)=[r'_{12}(0)-rr'_{22}(0)]\,r_{11}(t)+[r'_{22}(0)-rr'_{12}(0)]\,r_{12}(t); \tag{6} \]
   \[ (1-r^2)r'_{21}(t)=[r'_{11}(0)-rr'_{21}(0)]\,r_{21}(t)+[r'_{21}(0)-rr'_{11}(0)]\,r_{22}(t); \tag{7} \]
   \[ (1-r^2)r'_{22}(t)=[r'_{12}(0)-rr'_{22}(0)]\,r_{21}(t)+[r'_{22}(0)-rr'_{12}(0)]\,r_{22}(t). \tag{8} \]
   The system of equations (5)—(8) splits into two systems of equations: the system (5)—(6) and the system (7)—(8). Both of these systems of equations have one and the same characteristic equation
   \[ \begin{vmatrix} r'_{11}(0)-rr'_{21}(0)-k & r'_{21}(0)-rr'_{11}(0)\\ r'_{12}(0)-rr'_{22}(0) & r'_{22}(0)-rr'_{12}(0)-k \end{vmatrix} =0. \tag{9} \]

* The equations were communicated to me by O. V. Sarmanov.

If equation (9) has two real distinct roots \(k_1=\lambda\), \(k_2=\mu\), then the solution of system (5)—(8) consists of linear combinations of the two exponential functions \(e^{\lambda t}\) and \(e^{\mu t}\). If the root of equation (9) is multiple, \(k_{1,2}=\lambda\), then the solution of system (5)—(8) consists of linear combinations of the two functions \(e^{\lambda t}\) and \(t e^{\lambda t}\). If, however, the roots of (9) are complex: \(k_{1,2}=\lambda \pm i\nu\), then the solution of system (5)—(8) consists of linear combinations of the functions \(e^{\lambda t}\cos \nu t\) and \(e^{\lambda t}\sin \nu t\). Since \(r_{ij}(t)\) are bounded, \(\operatorname{Re} k_{1,2}\leqslant 0\).

Below it will be clarified under what additional restrictions the solution of system (9)—(12) satisfies conditions I and II.

1. Consider the case when \(r_{ij}(t)\) have the form

\[ r_{ij}(t)=a_{ij}e^{\lambda t}+b_{ij}e^{\mu t}, \tag{10} \]

where \(a_{ij}+b_{ij}=r_{ij}(0)\).

Lemma 1. The coefficients \(a_{ij}\) satisfy the equations

\[ (1-r^2)a_{11}=a_{11}^2+a_{12}a_{21}-r a_{11}(a_{12}+a_{21}); \tag{11} \]

\[ (1-r^2)a_{22}=a_{22}^2+a_{12}a_{21}-r a_{22}(a_{12}+a_{21}); \tag{12} \]

\[ (1-r^2)a_{12}=a_{12}(a_{11}+a_{22})-r(a_{11}a_{22}+a_{12}^2); \tag{13} \]

\[ (1-r^2)a_{21}=a_{21}(a_{11}+a_{22})-r(a_{11}a_{22}+a_{21}^2). \tag{14} \]

Lemma 2. If at least one of the inequalities \(a_{11}\ne a_{22}\), \(a_{12}\ne a_{21}\) holds, then the relations

\[ 1-r^2=a_{11}+a_{22}-r(a_{12}+a_{21}),\qquad a_{11}a_{22}=a_{12}a_{21}. \tag{15} \]

hold.

Lemma 3. If \(a_{11}=a_{22}\) and \(a_{12}=a_{21}\), then either \(a_{11}=a_{22}=1\) and \(a_{12}=a_{21}=r\), or \(b_{11}=b_{22}=1\), \(b_{12}=b_{21}=r\).

Theorem 1. Let \(\{\xi(t)\}=\{(\xi_1(t),\xi_2(t))\}\) be a Gaussian Markov process with correlation functions (10). Then, by a nondegenerate linear transformation

\[ \widetilde{\xi}_1(t)=\alpha \xi_1(t)+\beta \xi_2(t),\qquad \widetilde{\xi}_2(t)=\gamma \xi_1(t)+\delta \xi_2(t) \tag{16} \]

of the random variables \(\xi_1(t), \xi_2(t)\), one can obtain a normalized Gaussian Markov process \(\{\widetilde{\xi}(t)\}=\{(\widetilde{\xi}_1(t),\widetilde{\xi}_2(t))\}\), whose correlation functions have the form

\[ \widetilde r_{11}(t)=e^{\lambda t},\qquad \widetilde r_{12}(t)=\widetilde r e^{\lambda t},\qquad \widetilde r_{21}(t)=\widetilde r e^{\mu t},\qquad \widetilde r_{22}(t)=e^{\mu t}; \tag{17} \]

the correlation coefficient \(\widetilde r\) is determined by the formula

\[ \widetilde r=\alpha\gamma+r(\alpha\delta+\beta\gamma)+\beta\delta. \tag{18} \]

The transformation matrix \(A=\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix}\) is determined from the system of equations

\[ \alpha^2 b_{11}+\alpha\beta(b_{12}+b_{21})+\beta^2 b_{22}=0, \tag{19} \]

\[ \alpha\gamma b_{11}+\alpha\delta b_{12}+\beta\gamma b_{21}+\beta\delta b_{22}=0, \tag{20} \]

\[ \alpha\gamma a_{11}+\alpha\delta a_{21}+\beta\gamma a_{12}+\beta\delta a_{22}=0, \tag{21} \]

\[ \gamma^2 a_{11}+\gamma\delta(a_{12}+a_{21})+\delta^2 a_{22}=0, \tag{22} \]

\[ \alpha^2+2r\alpha\beta+\beta^2=1,\qquad \gamma^2+2r\gamma\delta+\delta^2=1. \tag{23} \]

Relying on Lemma 2, one can prove that a solution of system (19)—(23) exists and is uniquely determined.

Remark. If \(\lambda=\mu\), then the process \(\{\xi(t)\}=\{(\xi_1(t),\xi_2(t))\}\) is a process whose correlation functions already have the form (17).

Theorem 1 makes it possible to reduce the investigation of correlation functions of the form (10) to the investigation of correlation functions \(\widetilde r_{ij}(t)\) of the form (17). Let a Gaussian Markov process have functions (17) as its correlation functions. It turns out that condition II holds if and only if the inequality

\[ \frac{\widetilde r^2}{(1+\sqrt{1-\widetilde r^2})^2} \leqslant \frac{\lambda}{\mu} \leqslant \frac{(1-\sqrt{1-\widetilde r^2})^2}{\widetilde r^2} \tag{24} \]

holds.

Therefore the following holds.

Theorem 2. Let \(R(t)=\|r_{ij}(t)\|\) be a matrix function, where \(r_{ij}(t)\) have the form (10), with \(|r|<1\). Then \(R(t)\) is the correlation matrix function of a Gaussian Markov process if and only if: 1) the coefficients \(a_{ij}\) satisfy equations (11)—(14); 2) \(\lambda\) and \(\mu\) satisfy inequality (24), where \(\tilde r\) is defined according to (18), and the coefficients \(\alpha,\beta,\gamma,\delta\) are determined from the system (19)—(23).

Remark. If \(\tilde r=0\), then inequality (24) imposes no restrictions on \(\lambda\) and \(\mu\).

1. Let the correlation functions \(r_{ij}(t)\) have the form:

\[ r_{ij}(t)=e^{\lambda t}\,[r_{ij}(0)+a_{ij}t]. \tag{25} \]

The case in which all \(a_{ij}\) vanish was considered in Sec. 3. In Sec. 4 we shall consider the case in which at least one of the coefficients \(a_{ij}\) is nonzero.

Lemma 4. The coefficients \(a_{ij}\) satisfy the equations

\[ a_{11}^{2}+a_{12}a_{21}-ra_{11}(a_{12}+a_{21})=0; \tag{26} \]

\[ a_{22}^{2}+a_{12}a_{21}-ra_{22}(a_{12}+a_{21})=0; \tag{27} \]

\[ r(a_{11}a_{22}+a_{12}^{2})-a_{12}(a_{11}+a_{22})=0; \tag{28} \]

\[ r(a_{11}a_{22}+a_{21}^{2})-a_{21}(a_{11}+a_{22})=0. \tag{29} \]

Lemma 5. The coefficients \(a_{12}\) and \(a_{21}\) are not equal to each other.

Lemma 6. The coefficients \(a_{ij}\) satisfy the equations

\[ a_{11}+a_{22}=r(a_{12}+a_{21}), \qquad a_{11}a_{22}=a_{12}a_{21}. \tag{30} \]

Theorem 3. Let \(\{\xi(t)\}=\{(\xi_1(t),\xi_2(t))\}\) be a Gaussian Markov process with correlation functions (25). Then there exists a linear transformation (16) of the process \(\{\xi(t)\}\) such that the process \(\{\tilde \xi(t)\}=\{(\tilde \xi_1(t),\tilde \xi_2(t))\}\) is a Gaussian Markov process with correlation functions

\[ \tilde r_{11}(t)=\tilde r_{22}(t)=e^{\lambda t}, \qquad \tilde r_{21}(t)=0, \qquad \tilde r_{12}(t)=\tilde a_{12}te^{\lambda t}. \tag{31} \]

The coefficient \(\tilde a_{12}\) is expressed by the formula

\[ \tilde a_{12}=\alpha\gamma a_{11}+\alpha\delta a_{12}+\beta\gamma a_{21}+\beta\delta a_{22}. \tag{32} \]

Let us note that if \(a_{11}=a_{22}\), then, according to Lemma 6, \(a_{12}a_{21}=0\). Without loss of generality one may assume that \(a_{21}=0\). Then from equation (28) it follows that \(ra_{12}^{2}=0\). Since not all \(a_{ij}\) vanish in the case under consideration, \(r=0\). The process \(\{\xi(t)\}\) is the desired one. In what follows, without loss of generality, one may assume that \(a_{11}\ne0\). Transformation (16) is determined as follows:

\[ \alpha=-\frac{a_{12}}{a_{11}}\beta;\qquad \gamma=-\frac{a_{21}}{a_{11}}\delta;\qquad \beta=\frac{a_{11}}{\sqrt{a_{12}^{2}-2ra_{11}a_{12}+a_{11}^{2}}}; \]

\[ \delta=\frac{a_{11}}{\sqrt{a_{21}^{2}-2ra_{11}a_{21}+a_{11}^{2}}}. \tag{33} \]

Let \(R(t)=\|r_{ij}(t)\|\), where \(r_{ij}(t)\) have the form (25), be the correlation matrix function of the process \(\{\xi(t)\}=\{(\xi_1(t),\xi_2(t))\}\). Then \(\tilde R(t)=\|\tilde r_{ij}(t)\|\), where \(\tilde r_{ij}(t)\) is defined according to (31), is the correlation matrix function of the Gaussian process \(\{\tilde \xi(t)\}=\{(\tilde \xi_1(t),\tilde \xi_2(t))\}\). The process \(\{\tilde \xi(t)\}\) is centered and normalized. The function \(\tilde R(t)\) must satisfy conditions I and II of Sec. 3. However, \(\tilde R(t)\) satisfies conditions I and II if and only if

\[ |\tilde a_{12}|\le -2\lambda. \tag{34} \]

Theorem 4. Let \(R(t)=\|r_{ij}(t)\|\) be a matrix function, where \(r_{ij}(t)\) have the form (25). The matrix function \(R(t)\) is a correlation матриц-

function of a Gaussian Markov process if and only if (34) holds and the coefficients \(a_{ij}\) satisfy (26)—(29).

1. Let the correlation functions \(r_{ij}(t)\) have the form:

\[ r_{ij}(t)=e^{\lambda t}\bigl[r_{ij}(0)\cos \nu t+a_{ij}\sin \nu t\bigr],\qquad i,j=1,2. \tag{35} \]

Lemma 7. The coefficients \(a_{ij}\) satisfy the equations

\[ 1-r^2=-a_{11}^2-a_{12}a_{21}+ra_{11}(a_{12}+a_{21}); \tag{36} \]

\[ 1-r^2=-a_{22}^2-a_{12}a_{21}+ra_{22}(a_{12}+a_{21}); \tag{37} \]

\[ r(1-r^2)=ra_{11}a_{22}+ra_{12}^2-a_{12}(a_{11}+a_{22}); \tag{38} \]

\[ r(1-r^2)=ra_{11}a_{22}+ra_{21}^2-a_{21}(a_{11}+a_{22}). \tag{39} \]

Lemma 8. The coefficients \(a_{ij}\) satisfy the equation

\[ a_{11}+a_{22}=r(a_{12}+a_{21}). \tag{40} \]

Theorem 5. Let \(\{\xi(t)\}=\{(\xi_1(t),\xi_2(t))\}\) be a Gaussian Markov process with correlation functions (35). Then by a linear nonsingular transformation (16) of the random variables \(\xi_1(t), \xi_2(t)\) one can obtain a Gaussian Markov process \(\{\tilde{\xi}(t)\}=\{(\tilde{\xi}_1(t),\tilde{\xi}_2(t))\}\) whose correlation functions have the form

\[ \tilde r_{11}(t)=\tilde r_{22}(t)=e^{\lambda t}\cos \nu t,\qquad \tilde r_{12}(t)=\tilde a_{12}e^{\lambda t}\sin \nu t, \tag{41} \]

\[ \tilde r_{21}(t)=\tilde a_{21}e^{\lambda t}\sin \nu t. \]

In this case

\[ \tilde a_{12}=\alpha\gamma a_{11}+\alpha\delta a_{12}+\beta\gamma a_{21}+\beta\delta a_{22}; \tag{42} \]

\[ \tilde a_{21}=\alpha\gamma a_{11}+\beta\delta a_{21}+\alpha\delta a_{21}+\beta\delta a_{22}. \tag{43} \]

The matrix

\[ A=\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix} \]

of the nonsingular transformation (16) is determined by equations (23) and the equations

\[ a_{11}\alpha^2+(a_{12}+a_{21})\alpha\beta+a_{22}\beta^2=0, \tag{44} \]

\[ a_{11}\gamma^2+(a_{12}+a_{21})\gamma\delta+a_{22}\delta^2=0, \tag{45} \]

up to interchange of the rows of the matrix \(A\).

Theorem 5 makes it possible to reduce the study of the correlation functions of the Gaussian Markov process \(\{\xi(t)\}=\{(\xi_1(t),\xi_2(t))\}\), having the form (35), to the study of the correlation functions \(\tilde r_{ij}(t)\) of the corresponding Gaussian Markov process \(\{\tilde{\xi}(t)\}=\{(\tilde{\xi}_1(t),\tilde{\xi}_2(t))\}\), having the form (41). The correlation matrix function \(\tilde R(t)=\|\tilde r_{ij}(t)\|\) must satisfy conditions I and II. It turns out that condition I is fulfilled if and only if the inequalities

\[ \min\{|\tilde a_{12}|,|\tilde a_{21}|\}\ge \frac{\nu}{\sqrt{\lambda^2+\nu^2}} \exp\left\{\frac{\lambda}{\nu}\operatorname{arc\,tg}\left(-\frac{\nu}{\lambda}\right)\right\}; \tag{46} \]

\[ \max\{|\tilde a_{12}|,|\tilde a_{21}|\}\le \frac{\sqrt{\lambda^2+\nu^2}}{\nu} \exp\left\{-\frac{\lambda}{\nu}\operatorname{arc\,tg}\left(-\frac{\nu}{\lambda}\right)\right\}. \tag{47} \]

As for condition II, it is fulfilled if and only if

\[ \nu^2(\tilde a_{12}+\tilde a_{21})^2\le 4\lambda^2. \tag{48} \]

Thus the following holds.

Theorem 6. Let the functions \(r_{ij}(t)\) have the form (35). The functions \(r_{ij}(t)\) form the correlation matrix function of a normalized and centered Gaussian Markov process if and only if: 1) the coefficients \(a_{ij}\) satisfy equations (36)—(39); 2) inequalities (46)—(48) hold, where \(\tilde a_{12}\) and \(\tilde a_{21}\) are determined according to (42) and (43). The transformation matrix (16) is determined from equations (23), (44), and (45).

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
1 I 1964

CITED LITERATURE

1. J. L. Doob, Ann. Math. Stat., 15, No. 3 (1944).