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UDC 519.217
A. D. Shatashvili
ON THE DENSITIES OF MEASURES CORRESPONDING TO SOLUTIONS OF CERTAIN DIFFERENTIAL EQUATIONS WITH RANDOM FUNCTIONS
(Presented by Academician Yu. V. Linnik on 27 II 1970)
Finding the densities of measures corresponding to random processes is one of the important problems in the theory of random processes (see, on this subject, works \((^{1-10})\)), and the densities themselves are used in the solution of many problems in the theory of random processes: nonlinear extrapolation and filtering, optimal control, problems of mathematical statistics, and computation of distributions of various functionals.
In the present article we consider random processes \(x_1(t)\) and \(x_2(t)\) with values in the \(m\)-dimensional Euclidean space \(E_m\), which are solutions of the differential equations
\[
dx_2(t)/dt + L(t)x_2(t) + f(t, x_2(t)) = \xi(t), \qquad (0 \leq t \leq T),
\]
\[
x_2(0) = \xi(0) = 0;
\tag{1}
\]
\[
dx_1(t)/dt + L(t)x_1(t) = \xi(t) \qquad (0 \leq t \leq T),
\]
\[
x_1(0) = \xi(0) = 0,
\tag{2}
\]
where \(\xi(t)\) is a Gaussian process defined on the interval \([0,T]\) with values in \(E_m\), with zero mathematical expectation, \(M\xi(t)=0\), and continuous correlation matrix \(R(t,s)\) in the domain \([0,T]\times[0,T]\); the function \(f(t,x)\) is defined and continuous jointly in both variables in the domain \([0,T]\times E_m\), takes its values in \(E_m\), and
\[ \sum_{j=1}^{m}\left\|\frac{\partial f(t,x)}{\partial x_j}\right\| < \infty \qquad (t\in[0,T],\, x\in E_m) \tag{3} \]
(here and below the symbol \(\|\cdot\|\) denotes the norm in \(E_m\), and the symbol \((\cdot,\cdot)\) denotes the scalar product in \(E_m\)); and \(L(t)\) is a linear operator continuously depending on \(t\) and acting in \(E_m\).
Let \(\mu_1\) and \(\mu_2\) be the measures generated respectively by the solutions of the differential equations (2) and (1) on the minimal \(\sigma\)-algebra that contains all cylindrical sets of the space of all vector functions defined on the interval \([0,T]\) and taking their values in the Euclidean space \(E_m\).
In the present article conditions are established under which the measure \(\mu_2\) is absolutely continuous with respect to the measure \(\mu_1\), and the corresponding density is written out explicitly. We note that the process \(x_1(t)\) is Gaussian, and its correlation matrix \(B(t,s)\) is simply expressed in terms of the correlation matrix \(R(t,s)\) and the fundamental matrix of solutions \(Y(t)\) of the homogeneous equation
\[ dY(t)/dt - Y(t)L(t) \equiv 0, \qquad Y(0)=I \tag{4} \]
(\(I\) is the identity matrix):
\[ B(t,s)=Y(t)R(t,s)Y_*(s), \tag{5} \]
where \(Y_*(t)\) denotes the matrix transposed to \(Y(s)\).
Denote by \(\mathfrak F_t\) the \(\sigma\)-algebra generated by the quantities \(\xi(s)\) for \(s \leqslant t\), and suppose that the following conditions are satisfied:
1) There exists a matrix function \(Q(t,s)\) such that
\[ B(t,s)=\int_0^T Q(t,u)Q(s,u)\,du . \tag{6} \]
2) There exists an \(\mathfrak F_t\)-measurable Wiener process \(w(t)\) with values in \(E_m\), such that \(w(s)-w(t)\) for \(s>t\) does not depend on the \(\sigma\)-algebra \(\mathfrak F_t\) and
\[ \xi(t)=Y^*(t)\int_0^T Q(t,u)\,dw(u), \tag{7} \]
where \(Y^*(t)\) denotes the matrix inverse to the matrix \(Y(t)\).
3) There exists an \(\mathfrak F_t\)-measurable random function \(g(t)\) with values in \(E_m\), for which, with probability 1,
\[ \int_0^T \|g(t)\|^2\,dt<\infty; \tag{8} \]
\[ f\left(t,Y^*(t)\int_0^t Y(s)\xi(s)\,ds\right) = Y^*(t)\int_0^T Q(t,u)g(u)\,du . \tag{9} \]
Theorem. Let the differential equations (1) and (2) be given in the \(m\)-dimensional Euclidean space \(E_m\), where the Gaussian process \(\xi(t)\), the function \(f(t,x)\), and the linear operator \(L(t)\) satisfy the conditions formulated above, and, in addition, let conditions 1)—3) be fulfilled. Then the measure \(\mu_2\) is absolutely continuous with respect to the measure \(\mu_1\) and
\[ \frac{d\mu_2}{d\mu_1}[x_1] = \exp\left\{ -\int_0^T (g(t),dw(t)) -\frac{1}{2}\int_0^T \|g(t)\|^2\,dt \right\}. \tag{10} \]
The results of this theorem can be applied to solutions of equations of order higher than the first.
Let \(z_1(t)\) and \(z_2(t)\) be random processes with values in \(E_m\) satisfying respectively the equations
\[ d^n z_2(t)/dt^n+L_1(t)[z_2(t)] +F\bigl(t,z_2(t),z_2'(t),\ldots,z_2^{(n-1)}(t)\bigr)=\eta(t) \tag{11} \]
\[ (0\leqslant t\leqslant T),\qquad z_2(0)=z_2'(0)=\cdots=z_2^{(n-1)}(0)=\eta(0)=0; \]
\[ d^n z_1(t)/dt^n+L_1(t)[z_1(t)]=\eta(t)\qquad (0\leqslant t\leqslant T), \tag{12} \]
\[ z_1(0)=z_1'(0)=\cdots=z_2^{(n-1)}(0)=\eta(0)=0. \]
Here \(L_1(t)\) is a linear differential operator of order \((n-1)\) with coefficients continuously depending on \(t\) and acting in \(E_m\): if \(y(t)\) is an \(m\)-dimensional function with values in \(E_m\), differentiable \(n-1\) times, then
\[ L_1(t)[y(t)]=\sum_{k=0}^{n-1} C_k(t)y^{(k)}(t), \tag{13} \]
where \(C_k(t)\), \(k=0,1,\ldots,n-1\), are matrices of order \(m\). Further, the function \(F(t,z_2(t),z_2'(t),\ldots,z_2^{(n-1)}(t))\) is defined and continuous jointly in all variables in the domain \([0,T]\times E_m\times\cdots\times E_m\), takes its values in \(E_m\), and satisfies the condition
\[ \sum_{j=0}^{n-1}\sum_{i=0}^{m} \left\| \frac{\partial F\bigl(t,z_2^{(0)}(t),z_2^{(1)}(t),\ldots,z_2^{(n-1)}(t)\bigr)} {\partial z_i^{(j)}} \right\|<\infty \tag{14} \]
(\(z_i^{(j)}\) is the \(i\)-th component of the vector \(z^{(j)}\)), and \(\eta(t)\) is the Gaussian process with values in \(E_m\) considered above.
The case under consideration is reduced to the preceding one by introducing the processes \(x_1(t)\) and \(x_2(t)\) in the space \(\mathbf E_{mn}=\mathbf E_m\times\cdots\times\mathbf E_m\), connected with the processes \(z_1(t)\) and \(z_2(t)\) in the following way:
\[ x_i(t)=\bigl[z_i(t),\, dz_i(t)/dt,\ldots,d^{(n-1)}z_i(t)/dt^{n-1}\bigr]. \tag{15} \]
The processes \(x_i(t)\) \((i=1,2)\) already satisfy differential equations of the form (1) and (2) in \(\mathbf E_{mn}\), where the function \(f(t,x_2(t))\) is defined on \([0,T]\times \mathbf E_{mn}\), takes its values in \(\mathbf E_{mn}\), and has the form
\[ f(t,x_2(t))=\bigl[0,\ldots,0,F(t,z_2(t),z_2'(t),\ldots,z_2^{(n-1)}(t))\bigr], \]
the Gaussian process
\[ \xi(t)=[0,\ldots,0,\eta(t)] \]
is defined on \([0,T]\) and takes its values in \(\mathbf E_{mn}\), and the linear operator \(L(t)\) acts in \(\mathbf E_{mn}\) and has the form
\[ L(t)= \begin{pmatrix} (0), & (I), & \ldots, & (0)\\ (0), & (0), & \ldots, & (0)\\ \cdot & \cdot & \cdot & \cdot\\ (0), & (0), & \ldots, & (I)\\ C_0(t), & C_1(t), & \ldots, & C_{n-1}(t) \end{pmatrix}. \]
Here each box of the form \((0)\) is an \(m\)-dimensional matrix with zero entries, and a box of the form \((I)\) is the \(m\)-dimensional identity matrix. If \(\mu_{z_1}\) and \(\mu_{z_2}\) denote the measures generated respectively by the processes \(z_1(t)\) and \(z_2(t)\), then, provided the conditions of the theorem are satisfied for equations (1) and (2) in \(\mathbf E_{mn}\), the measure \(\mu_{z_2}\) will be absolutely continuous with respect to the measure \(\mu_{z_1}\), and
\[ \frac{d\mu_{z_2}}{d\mu_{z_1}}(z_1)=\frac{d\mu_2}{d\mu_1}(x_1), \]
while the density \(\dfrac{d\mu_2}{d\mu_1}(x_1)\) will have the form (10), where the scalar product and the norm must be understood in the space \(\mathbf E_{mn}\), and the random function \(g(t)\) and the Wiener process \(w(t)\) take their values in \(\mathbf E_{mn}\) and are defined, respectively, by formulas (9) and (7).
It should be noted that the results of the present paper remain valid if one assumes that the space \(\mathbf E_m\) is a separable Hilbert space \(\mathbf E\), and equations (1), (2), (11), (12) are differential equations in the Hilbert space \(\mathbf E\).
Donets Computing Center
Academy of Sciences of the Ukrainian SSR
Received
27 II 1970
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