UDC 519.21

Yu. I. GOLOSOV

ON ONE METHOD FOR COMPUTING THE RADON—NIKODYM DERIVATIVE OF TWO GAUSSIAN MEASURES

(Presented by Academician A. N. Kolmogorov, 22 XII 1965)

Let \(\xi(t,\omega)\), \(t \in [a,b]\), \(\omega \in \Omega\), be a family of real-valued functions; \(B\) the smallest \(\sigma\)-algebra of subsets of \(\Omega\) with respect to which the family \(\xi(t)\), \(t \in [a,b]\), is measurable; \(\mathcal P_i\), \(i=1,2\), measures on \((\Omega,B)\) with respect to which \(\xi(t)\) is a Gaussian process with \(M_i\xi(t)=0\), \(M_i\xi(t)\xi(s)=R_i(s,t)\), \(i=1,2\). In this note, on the basis of the general theory of Aronszajn \((^4)\) of Hilbert spaces reproducing the kernel \(R(s,t)\) (r.k.h.s. \(R(s,t)\)), a new formulation is given of necessary and sufficient conditions that the kernels \(R_i(s,t)\) must satisfy in order that the measures \(\mathcal P_1\) and \(\mathcal P_2\) be equivalent (this circumstance will be denoted by the symbol \(\mathcal P_1 \sim \mathcal P_2\)), and a formula is indicated for the Radon—Nikodym derivative \(d\mathcal P_2/d\mathcal P_1[\xi(t)]\) of the measures \(\mathcal P_2\) and \(\mathcal P_1\). The results obtained are related to a recent elegant result of Yu. Rozanov \((^3)\), concerning the case of Gaussian measures corresponding to stationary processes and formulated in spectral terms. They apply to the case when, with respect to both measures, the process \(\xi(t)\) is a continuous-in-mean-square Gaussian Markov process (g.m. process). The latter situation was studied in \((^{12})\) under additional restrictions concerning the regularity of the kernels \(R_i(s,t)\). Further, the case is considered of Gaussian measures \(\mathcal P_1\) and \(\mathcal P_2\) in the space of functions \(\xi(s,t)\) of two variables, where the measure \(\mathcal P_1\) corresponds to a Wiener field (in the sense of \((^{13})\)), while \(\mathcal P_2\) differs from \(\mathcal P_1\) in its mean value or correlation function.

1. Denote by \(H_i(\xi)\) the Hilbert spaces that are the linear closures of the random variables \(\xi(t)\), \(t \in [a,b]\), in the norm corresponding to the scalar product \(\langle \gamma,\eta\rangle_i=M_i\gamma\eta\). For equivalence of the measures \(\mathcal P_1\) and \(\mathcal P_2\) it is necessary and sufficient that the symmetric operator \(S\) in \(H_1(\xi)\), defined by the equality \(M_2\gamma\eta=M_1\gamma S\eta\), satisfy the following conditions, established in \((^{2,6,10})\).

Condition I. The operator \(S\) is bounded and has a bounded inverse.

Condition II. The operator \(I-S\) is a Hilbert—Schmidt operator.

Let \(H(R_i)\) be the r.k.h.s. \(R_i(s,t)\), and let \(\langle\cdot,\cdot\rangle_i\) be the scalar product in \(H(R_i)\). The equality \(M_1\xi(t)\gamma=\varphi(t)\) defines an isomorphic correspondence \(\gamma \leftrightarrow \varphi(t)\) between \(H_1(\xi)\) and \(H(R_1)\). The operator \(S\) in \(H_1(\xi)\) then corresponds to an operator \(\overline S\) in \(H(R_1)\) (defined by the relation \(M_1\xi(t)S\gamma=\overline S\varphi(t)\), where \(\gamma \leftrightarrow \varphi(t)\)) such that \(\overline S\varphi(t)=\langle R_2(s,t),\varphi(s)\rangle_1\); to the operator \(I-S\) in \(H_1(\xi)\) there corresponds the operator \(\overline I-\overline S\) in \(H(R_1)\) such that \((\overline I-\overline S)\varphi(t)=\langle R_1(s,t)-R_2(s,t),\varphi(s)\rangle_1\). Conditions I and II, obviously, may also be referred to the operator \(\overline S\). Let us find \(d\mathcal P_2/d\mathcal P_1[\xi(t)]\). If \(\mathcal P_1 \sim \mathcal P_2\), then the operator \(S\) has a discrete spectrum. Let \(x_k\), \(k=1,2,\ldots\), be a complete orthonormal system in \(H_1(\xi)\) of eigenvectors of the operator \(S\), corresponding to eigenvalues \(\mu_k\), \(k=1,2,\ldots\). Then almost everywhere both with respect to \(\mathcal P_1\) and with respect to \(\mathcal P_2\),

\[ \frac{d\mathcal P_2}{d\mathcal P_1}[\xi(t)] = \lim_{n\to\infty} \exp \frac{1}{2} \sum_1^n \left[ \left(1-\frac{1}{\mu_k}\right)-\ln\mu_k \right] \times \]

\[ \times \exp \frac{1}{2} \sum_1^n \left[ x_k^2 \left(1-\frac{1}{\mu_k}\right) - \left(1-\frac{1}{\mu_k}\right) \right], \tag{1} \]

whereas \(\mathscr P_1 \sim \mathscr P_2\) if and only if the limit on the right-hand side of (1) exists almost everywhere with respect to both measures (from this conditions I and II are obtained). It is easy to verify that \(\mathscr P_1 \sim \mathscr P_2\) if and only if, both with respect to the measure \(\mathscr P_1\) and with respect to the measure \(\mathscr P_2\), there exists

\[ \underset{n\to\infty}{\operatorname{l.i.m.}}\ \frac12 \sum_1^n \left[ x_k^2\left(1-\frac1{\mu_k}\right)-\left(1-\frac1{\mu_k}\right) \right] =F[\xi(t)] . \]

Since, moreover, the series

\[ \sum_1^\infty \left[ \left(1-\frac1{\mu_k}\right)-\ln\mu_k \right] =D \]

converges, it follows that

\[ d\mathscr P_2/d\mathscr P_1[\xi(t)]=C\exp F[\xi(t)],\qquad C=\exp{}^{1/2}D . \]

This circumstance suggests the following construction.

Consider the random function \(\zeta(s,t)=\xi(s)\xi(t)-R_1(s,t)\), \(s,t\in[a,b]\times[a,b]\), and construct the Hilbert space \(H_1(\zeta)\) corresponding to it. Denote
\(\Phi_1(s,t;u,v)=M_1\zeta(s,t)\zeta(u,v)=R_1(s,u)R_1(t,v)+R_1(s,v)R_1(t,u)\), and let \(H(\Phi_1)\) be the r.k.H.s. of \(\Phi_1(s,t;u,v)\), the scalar product in which we denote by \([\cdot,\cdot]_1\). Since \(R_1(s,u)R_1(t,v)\) is the reproducing kernel of the direct product of the spaces \(H(R_1)\otimes H(R_1)\) (see \((^4)\)), we have
\[ H(\Phi_1)\equiv[H(R_1)\otimes H(R_1)], \]
where \([H(R_1)\otimes H(R_1)]\) denotes the space of symmetric functions \(f(s,t)=f(t,s)\) from \(H(R_1)\otimes H(R_1)\) with scalar product
\[ [\cdot,\cdot]_1=\frac12[\cdot,\cdot]. \]
The correspondence \(\gamma \leftrightarrow \varphi(s,t)\), where \(M_1\zeta(s,t)\gamma=\varphi(s,t)\), defines an isomorphism between \(H_1(\zeta)\) and \(H(\Phi_1)\). Under conditions I and II the operator \(\bar I-\bar S^{-1}\) is a symmetric Hilbert–Schmidt operator in \(H(R_1)\) and, consequently, is given by a kernel \(A(s,t)\in H(\Phi_1)\); moreover, since under condition I the operator \(\bar I-\bar S^{-1}\) is a Hilbert–Schmidt operator if and only if the operator \(\bar I-\bar S\) is such, the condition \(A(s,t)\in H(\Phi_1)\) is equivalent to the condition
\[ R_1(s,t)-R_2(s,t)\in H(R_1)\otimes H(R_1). \]
It is not difficult to check that \(F[\xi(t)]\in H_1(\zeta)\) and that \(\gamma \leftrightarrow A(s,t)\) under the isomorphism of \(H_1(\zeta)\) and \([H(R_1)\otimes H(R_1)]\). Thus the following is true.

Theorem 1. \(\mathscr P_1\sim\mathscr P_2\) if and only if:

1) the equation \(\langle R_2(s,t),\varphi(t)\rangle_1=\mu\varphi(s)\) in \(H(R_1)\) has no eigenvalue \(\mu=0\);

2) \(R_1(s,t)-R_2(s,t)\in H(R_1)\otimes H(R_1)\).

In this case

\[ d\mathscr P_2/d\mathscr P_1[\xi(t)] = C\exp\left\{\,{}^{1/2}[A(s,t),\zeta(s,t)]_1\right\}, \tag{2} \]

where \(A(s,t)\) is the kernel of the operator \(\bar A=\bar I-\bar S^{-1}\) in \(H(R_1)\), defined by the identity
\[ \langle\varphi,\psi\rangle_1-\langle\varphi,\psi\rangle_2 = \langle\varphi,\bar A\psi\rangle_1; \]
\(C\) is the constant defined above through the eigenvalues of the operator \(\bar S\).

Remark. In the case when \(M_1\xi(t)=0\), \(M_2\xi(t)=m(t)\),
\[ M_i[\xi(t)-M_i\xi(t)][\xi(s)-M_i\xi(s)]=R(s,t), \]
the conditions under which \(\mathscr P_1\sim\mathscr P_2\) and the corresponding formula for
\(d\mathscr P_2/d\mathscr P_1[\xi(t)]\) in terms of the r.k.H.s. \(R(s,t)\), assertions related to Theorem 1, were obtained by Parzen \((^8)\) and by Gack \((^5)\). The scalar product \([A(s,t),\zeta(s,t)]_1\) in formula (2) has the same meaning as the scalar product appearing in the formulas for
\(d\mathscr P_2/d\mathscr P_1[\xi(t)]\) in \((^8,^5)\). In the case when, under the measure \(\mathscr P_1\), the process is a Wiener process, the definition of \([A(s,t),\zeta(s,t)]_1\) is equivalent to the definition of the multiple stochastic integral of Itô \(((^7),\ \text{cf. }(^ {11}))\). In the particular case when \(R_2(s,t)-R_1(s,t)\) is a positive definite kernel, condition 2) (together with another, less general formula for \(d\mathscr P_2/d\mathscr P_1[\xi(t)]\)) was indicated by Parzen in \((^9)\).

1. Let \(\xi(t)\), \(t\in[a,b]\), with respect to the measure \(\mathscr P_1\) be a g.m. process, \(M_1\xi(t)=0\). The set of points of nondegeneracy of the process \(\{t:D_1\xi(t)\ne0\}\) is the union of a countable number of intervals, on each of which \(M_1\xi(t)\xi(s)=R_1(s,t)\) is represented in the form \(R_1(s,t)=\)

\[ = \varphi(s)\varphi(t)\min[\psi(s)/\varphi(s),\psi(t)/\varphi(t)], \]
where \(\psi(t)\) and \(\varphi(t)\) are continuous functions such that, within the interval under consideration, \(\varphi(t)>0\), \(\psi(t)>0\), and \(\psi(t)/\varphi(t)\) is nondecreasing. Since the values of the g.m. process \(\xi(t)\) on different intervals of nondegeneracy are mutually independent, we may restrict ourselves to considering one such interval \((a,b)\).

Let, with respect to the measure \(\mathcal P_2\), the process \(\xi(t)\), \(t\in[a,b]\), be a g.m. process and
\[ M_2\xi(t)=0,\qquad M_2\xi(t)\xi(s)=R_2(s,t)=\theta(s)\theta(t)\min[\rho(s)/\theta(s),\rho(t)/\theta(t)], \]
where the functions \(\rho(t)\), \(\theta(t)\) have the same properties as \(\psi(t)\), \(\varphi(t)\). Let \(g(u)\), \(a_1=\psi(a)/\varphi(a)\leq u\leq \psi(b)/\varphi(b)=b_1\), be the function inverse to \(\psi(t)/\varphi(t)\), i.e. \(g[\psi(t)/\varphi(t)]=t\). Introduce the notation:
\[ \xi[g(u)]/\varphi[g(u)]=\zeta(u),\qquad \rho[g(u)]/\varphi[g(u)]=\mu(u),\qquad \theta[g(u)]/\varphi[g(u)]=\nu(u), \]
\[ A(u,v)=\int_{a_1}^{b_1}\frac{\varphi'(u)}{\varphi(u)}\,e(s,u)\min(t,u)\,du +\int_{a_1}^{b_1}\frac{\varphi'(u)}{\varphi(u)}\,e(t,u)\min(s,u)\,du -\int_{a_1}^{b_1}\left[\frac{\varphi'(u)}{\varphi(u)}\right]^2\min(s,u)\min(t,u)\,du, \qquad e(s,t)=\frac{\partial}{\partial t}\min(s,t). \]

Theorem 2. \(\mathcal P_1\sim\mathcal P_2\) if and only if:

1) \(\mu(a_1)=0\), if \(a_1=0\), and \(\mu(a_1)\ne0\), if \(a_1\ne0\); \(\mu(u)>0\) for \(u>a_1\), \(\nu(u)>0\) for \(u\geq a_1\);

2) The function \(\mu(u)\) is absolutely continuous in \(u\), the function \(\nu(u)\) is absolutely continuous in \(u\), and for \(u>a_1\),
\[ \nu^2(u)\int_{a_1}^{u}\mu'^2(u)\,du+\mu^2(u)\int_{u}^{b_1}\nu'^2(u)\,du<\infty \]
for every fixed \(u\in[a_1,b_1]\);

3)
\[ \mu'(u)\nu(u)-\mu(u)\nu'(u)\equiv1; \]

4)
\[ \int_{a_1}^{b_1}\nu'^2(u)\left(\int_{a_1}^{u}\mu'^2(v)\,dv\right)du<\infty. \]

In this case
\[ \frac{d\mathcal P_2}{d\mathcal P_1}[\xi(t)] = C\exp\frac12\left[ \int_{a_1}^{b_1}\int_{a_1}^{b_1} \frac{\partial^2}{\partial u\,\partial v}A(u,v)\,d\xi(u)\,d\xi(u) + 2\frac{\xi(a_1)}{a_1}\int_{a_1}^{b_1}\frac{\partial}{\partial u}A(a_1,u)\,d\xi(u) + \frac{[\xi^2(a_2)-a_1]}{a_1^2}A(a_1,a_1) \right], \tag{3} \]
where
\[ C=\exp\left\{-\frac12\int_{a_1}^{b_1}u\left[\frac{\nu'(u)}{\nu(u)}\right]^2\,du\right\} \]
and the multiple stochastic integral is defined in the sense of Itô. If, in addition, the condition is satisfied:

5) \(\displaystyle \lim_{u\to b_1}\nu(u)\) exists and is not equal to \(0\),

then
\[ \frac{d\mathcal P_2}{d\mathcal P_1}[\xi(t)] = \left[\frac{\nu(b_1)}{\nu(a_1)}\right]^{1/2} \exp\left\{\frac12\left[ \xi^2(a_1)\left(\frac1{a_1}-\frac1{\nu(a_1)\mu(a_1)}\right) +\int_{a_1}^{b_1}\nu'(u)\,d\frac{\xi^2(u)}{\nu(u)} \right]\right\}; \tag{4} \]
if \(a_1=0\), then in (3) and (4) we put
\[ \frac{\xi(a_1)}{a_1}\int_{a_1}^{b_1}\frac{\partial}{\partial u}A(a_1,u)\,d\xi(u) = \frac{\xi^2(a_1)-a_1}{a_1^2}A(a_1,a_1) = \xi^2(a_1)\left(\frac1{a_1}-\frac1{\nu(a_1)\mu(a_1)}\right) =0. \]

This theorem contains, as a special case, the main result of the paper \((^{12})\).

1. Let the random function \(\xi(s,t)=\xi(P)\), \(P=(s,t)\in[0,1]\times[0,1]\), be, with respect to the Gaussian measure \(\mathcal P_1\), a Wiener field in the sense–

(13), i.e. \(M_1\xi(s,t)=0,\ M_1\xi(s,t)\xi(u,v)=\min(s,u)\min(t,v)\), while \(\mathcal P_2\) is a Gaussian measure such that
\(M_2\xi(s,t)=m(s,t),\ M_2[\xi(s,t)-m(s,t)][\xi(u,v)-m(u,v)]=\min(s,u)\min(t,v)\) in Theorem 3, or
\(M_2\xi(s,t)=0,\ M_2\xi(s,t)\xi(u,v)=R(s,t;u,v)\) in Theorem 4. Denote

\[ \int_0^s\int_0^t b(s',t')\,ds'\,dt'=\int_0^P b(P')\,dP',\quad \min(s,u)\min(t,v)=\min(P,Q), \]

and so on.

Theorem 3. \(\mathcal P_1\sim \mathcal P_2\) if and only if
\[ m(P)=\int_0^P b(P')\,dP', \]
where
\[ \int_0^1 [b(P)]^2\,dP<\infty. \]
Moreover,

\[ \frac{d\mathcal P_2}{d\mathcal P_1}[\xi(P)] = \exp\left\{ -\frac12\int_0^1 [b(P)]^2\,dP + \int_0^1 b(P)\,d\xi(P) \right\}. \]

Theorem 4. \(\mathcal P_1\sim \mathcal P_2\) if and only if:

1) the function \(R(P;Q)\) can be represented in the form

\[ R(P;Q)=\min(P;Q)+\int_0^P\int_0^Q b(P';Q')\,dP'\,dQ', \]

where

\[ \int_0^1\int_0^1 [b(P;Q)]^2\,dP\,dQ<\infty; \]

2) the equation

\[ \int_0^1 b(P;Q)f(Q)\,dQ=\lambda f(P) \]

has no eigenvalue \(-1\) in \(L^2\).

In this case

\[ \frac{d\mathcal P_2}{d\mathcal P_1}[\xi(P)] = C\exp\left\{ \frac12\int_0^1\int_0^1 A(P;Q)\,d\xi(P)\,d\xi(Q) \right\}, \]

where the stochastic integral is understood in Itô’s sense, and the function \(A(P;Q)\) is determined from the condition that

\[ f(P)=g(P)-\int_0^1 A(P;Q)g(Q)\,dQ \]

is a solution in \(L^2\) of the equation

\[ f(P)+\int_0^1 b(P;Q)f(Q)\,dQ=g(P). \]

The constant \(C\) can be determined through the eigenvalues \(\lambda_k,\ k=1,2,\ldots,\) of the equation

\[ \int_0^1 b(P;Q)f(Q)\,dQ=\lambda f(P) \]

by the formula given in item 1, putting \(\mu_k=\lambda_k+1\).

The proof of these theorems is based on the result of Parzen–Hájek \((^5,^8)\), which was discussed in the remark after Theorem 1, and on Theorem 1, whose reformulation for the case of fields \(\xi(s,t)\) is obvious.

Theorem 3 generalizes Theorem 3 of paper \((^{13})\), and Theorem 4 is similar to results of Shepp \((^{11})\) and of the author \((^1)\), pertaining to the one-dimensional case.

I take this opportunity to express my deep gratitude to A. M. Yaglom, who supervised the execution of this work, and to L. A. Shepp, who kindly sent me his interesting work \((^{11})\).

Moscow State University
named after M. V. Lomonosov

Received
8 XII 1965

REFERENCES

1. Yu. Golosov, DAN, 166, 7 (1966).
2. Yu. Rozanov, Probability Theory and Its Applications, 7, 84 (1962).
3. Yu. Rozanov, DAN, 165, 1000 (1965).
4. N. Aronszajn, Trans. Am. Math. Soc., 68, 337 (1950). Transl. collection Mathematics, 7, 2, 67 (1963).
5. I. Hajek, Czechoslov. Math. J., 12, 404 (1962); Transl. collection Mathematics, 7, 3, 97 (1963).
6. I. Feldman, Pacific J. Math., 8, 699 (1958).
7. K. Itô, Japan. J. Math., 22, 63 (1952).
8. E. Parzen, Proc. 4-th Berkeley Symp. Math., Stat. and Probab., 1, Los Angeles, 1961, p. 469.
9. E. Parzen, Time Series Analysis, ch. II, N. Y., 1963.
10. I. Segal, Trans. Am. Math. Soc., 88, 12 (1958).
11. L. Shepp, Radon—Nicodim Derivatives of Gaussian Measures, preprint, 1965.
12. D. Varberg, Pacific J. Math., 11, 751 (1961).
13. J. Yeh, Trans. Am. Math. Soc., 107, 409 (1963).