MATHEMATICS

Yu. N. BLAGOVESHCHENSKII and M. I. FREIDLIN

SOME PROPERTIES OF DIFFUSION PROCESSES DEPENDING ON A PARAMETER

(Presented by Academician A. N. Kolmogorov, 21 I 1961)

Consider the stochastic equation

\[ x_t(a,\omega)-x_s(a,\omega)= \]

\[ =\int_s^t \sigma(u,a,x_u(a,\omega))\,d\xi_u(\omega) +\int_s^t m(u,a,x_u(a,\omega))\,du . \tag{1} \]

In equation (1), \(\xi_u(\omega)=(\xi_u^1(\omega),\xi_u^2(\omega),\ldots,\xi_u^n(\omega))\) is an \(n\)-dimensional Wiener process defined on the probability space \((\Omega,\mathfrak M,P)\); \(\sigma(u,a,x)=\{\sigma_j^i(u,a,x)\}_{i,j=1}^n\) is a matrix; \(m(u,a,x)=(m^1(u,a,x),m^2(u,a,x),\ldots,m^n(u,a,x))\) is an \(n\)-dimensional vector. The integrals on the right-hand side of equation (1) are understood as stochastic integrals (see \((^1)\)). Suppose that the elements of the matrix \(\sigma(u,a,x)\) and of the vector \(m(u,a,x)\) uniformly satisfy a Lipschitz condition in \(x\). Then it is proved that there exists a random Markov function \(x_t(a,\omega)\), taking values in the \(n\)-dimensional Euclidean space \(R^n\) and satisfying equality (1) with probability 1. The parameter \(a=(a_1,a_2,\ldots,a_m)\) takes values in some domain \(A\) in \(R^m\).

In the present note we give a number of results concerning the continuity and differentiability of \(x_t(a,\omega)\) with respect to \(a\).

Theorem 1. Suppose that there exists a constant \(C<\infty\) such that for any \(x,y\in R^n\), \(a,\beta\in A\subseteq R^m\), \(u\in[0,T]\), \(T<\infty\),

\[ \sum_{i,j=1}^n \left|\sigma_j^i(u,a,x)-\sigma_j^i(u,\beta,y)\right| +\sum_{i=1}^n \left|m^i(u,a,x)-m^i(u,\beta,y)\right|\leq \]

\[ \leq C\bigl(\|a-\beta\|+\|x-y\|\bigr)^* . \]

Suppose also that \(x_0(a,\omega)\) is continuous in \(a\in A\) for almost all \(\omega\). Then there exists a random function \(x_t(a,\omega)\), satisfying equation (1) and, with probability 1, continuous in \((t,a)\in[0,T]\times A\).

Theorem 2. Suppose that \(\sigma_j^i(u,a,x)\), \(m^i(u,a,x)\) have continuous bounded derivatives with respect to \(a_p\), \(x^r\) \((i,j,r=1,2,\ldots,n;\ p=1,2,\ldots,m)\) up to order \(k+1\) inclusive. Suppose also that \(x_0(a,\omega)\) and \(d^l x_0(a,\omega)/\partial a_1^{l_1}\cdots\partial a_m^{l_m}\), \(l_1+l_2+\cdots+l_m=l\leq k+1\), exist for almost all \(\omega\), are bounded and continuous.

Then, for almost all \(\omega\) and all \(l_1+l_2+\cdots+l_m=l\leq k\), there exist derivatives continuous in \((t,a)\),

\[ \partial^l x_t(a,\omega)/\partial a_1^{l_1}\partial a_2^{l_2}\cdots\partial a_m^{l_m}. \]

If the requi—

\[ {}^* \text{ If } z_i=(z_i^1,z_i^2,\ldots,z_i^k),\ i=1,2,\text{ then } \|z_1-z_2\|=\left(\sum_{j=1}^k |z_1^j-z_2^j|^2\right)^{1/2}. \]

to require the existence of \(d^l x_0(a,\omega)/\partial a_1^{l_1}\partial a_2^{l_2}\cdots \partial a_m^{l_m}\) only in the mean-square sense\(^*\) for all \(l_1+l_2+\cdots+l_m=l\le k+1\), while keeping the same requirements for \(\sigma(u,a,x)\) and \(m(u,a,x)\) as above, then \(\partial^l x_t(a,\omega)/\partial a_1^{l_1}\cdots \partial a_2^{l_2}\partial a_m^{l_m}\) will also exist in the mean-square sense for all \(l_1,l_2,\ldots,l_m\), \(l_1+l_2+\cdots+l_m=l\le k\). The family of random functions \(\partial^l x_t(a,\omega)/\partial a_1^{l_1}\cdots \partial a_m^{l_m}\), \(l_1+l_2+\cdots+l_m=l\le k\), satisfies the following system of stochastic equations:

\[ \frac{\partial^l x_t(a,\omega)} {\partial a_1^{l_1}\partial a_2^{l_2}\cdots \partial a_m^{l_m}} = \frac{\partial^l x_0(a,\omega)} {\partial a_1^{l_1}\partial a_2^{l_2}\cdots \partial a_m^{l_m}} + \int_0^t \frac{\widetilde{\partial^l}\sigma(u,a,x_u(a,\omega))} {\partial a_1^{l_1}\partial a_2^{l_2}\cdots \partial a_m^{l_m}} \,d\xi_u(\omega) + \int_0^t \frac{\widetilde{\partial^l}m(u,a,x_u(a,\omega))} {\partial a_1^{l_1}\partial a_2^{l_2}\cdots \partial a_m^{l_m}} \,du . \tag{2} \]

Here

\[ \frac{\widetilde{\partial f}(a_1,a_2,\ldots,a_m;x^1(a),x^2(a),\ldots,x^n(a))} {\partial a_k} = \frac{\partial f}{\partial a_k} + \sum_{i=1}^n \frac{\partial f}{\partial x^i}\, \frac{\partial x^i}{\partial a_k}; \]

\[ \frac{\widetilde{\partial^2} f}{\partial a_i\partial a_j} = \frac{\widetilde{\partial}}{\partial a_i} \left( \frac{\widetilde{\partial f}}{\partial a_j} \right). \]

The following follows from Theorem 2.

Theorem 3. Let \(x_t^x(\omega)\) satisfy the stochastic equation

\[ x_t^x(\omega) = x + \int_0^t \sigma(u,x_u^x(\omega))\,d\xi_u(\omega) + \int_0^t m(u,x_u^x(\omega))\,du . \tag{3} \]

Then, if \(\sigma(u,x)\) and \(m(u,x)\) have bounded continuous derivatives up to order \(k+1\), inclusive, with respect to \(x^r\), \(r=1,2,\ldots,n\), then for all \(l\le k\), for almost all \(\omega\), the derivatives
\(\partial^l x_t^x(\omega)/\partial (x^1)^{l_1}\cdots \partial (x^n)^{l_n}\), \(l_1+l_2+\cdots+l_n=l\), exist. These derivatives also exist in the mean-square sense.

For the proof of the assertions formulated, we shall need the following generalization of A. N. Kolmogorov’s well-known theorem on the continuity of sample functions of a process.

Theorem. Let \(x_\mu(\omega)\) be a separable random field,\(^ {**}\) defined for \(\mu\in R^m\) and taking values in \(n\)-dimensional Euclidean space \(R^n\). Then, in order that \(x_\mu(\omega)\) be continuous in \(\mu\) with probability 1, it is sufficient that, for some \(\gamma>0\), \(\varepsilon>0\), the inequality

\[ M\|x_\mu(\omega)-x_{\mu'}(\omega)\|^\gamma \le C\|\mu-\mu'\|^{m+\varepsilon} \]

hold.

The proof of this theorem is carried out analogously to the case \(m=1\) (see (1), p. 576).

With the aid of the change-of-variables formula in stochastic integrals (2), when the conditions of Theorem 1 are fulfilled, one can prove the following inequality, valid for positive integers \(n\):

\[ M\|x_t(a,\omega)-x_s(\beta,\omega)\|^{2n} \le C_n\bigl(\|a-\beta\|^{2n}+|t-s|^n\bigr); \tag{4} \]

\[ \alpha,\beta\in A,\quad t,\ s\in[0,T]. \]

From the last inequality, by virtue of the above-formulated generalization of A. N. Kolmogorov’s theorem, Theorem 1 follows. (Independently, Theorem 1 was proved by I. V. Girsanov.)

\[ \underline{\hspace{3cm}} \]

\(^*\) That is, the limit in the definition of the derivative is understood as a limit in the mean-square sense.

\(^ {**}\) Under natural assumptions, for every field \(x_\mu(\omega)\) there exists an equivalent separable field \(\widetilde{x}_\mu(\omega)\).

Let us explain the proof of Theorem 2 in the case \(n=m=l=1\). The random functions \(x_t(\beta,\omega)\) and \(x_t(\beta',\omega)\) are solutions of equation (1) for \(\alpha=\beta\) and \(\alpha=\beta'\), respectively. The function
\[ y_t^{\beta\beta'}(\omega)=\frac{x_t(\beta,\omega)-x_t(\beta',\omega)}{\beta-\beta'} \]
satisfies the equation
\[ \begin{aligned} y_t^{\beta\beta'}(\omega)=y_0^{\beta\beta'}(\omega) &+\int_0^t \biggl[ \frac{\sigma(u,\beta,x_u(\beta,\omega))-\sigma(u,\beta',x_u(\beta,\omega))}{\beta\beta'}+{}\\ &\qquad\qquad +\frac{\sigma(u,\beta',x_u(\beta,\omega))-\sigma(u,\beta',x_u(\beta',\omega))} {x_u(\beta,\omega)-x_u(\beta',\omega)} \,y_u^{\beta\beta'}(\omega) \biggr]\,d\xi_u(\omega)+{}\\ &+\int_0^t \biggl[ \frac{m(u,\beta,x_u(\beta,\omega))-m(u,\beta',x_u(\beta,\omega))}{\beta-\beta'}+{}\\ &\qquad\qquad +\frac{m(u,\beta',x_u(\beta,\omega))-m(u,\beta',x_u(\beta',\omega))} {x_u(\beta,\omega)-x_u(\beta',\omega)} \,y_u^{\beta\beta'}(\omega) \biggr]\,du . \end{aligned} \tag{5} \]

Equation (5), together with equation (1) taken for \(\alpha=\beta\) and \(\alpha=\beta'\), forms a system of stochastic equations for the random function
\[ z_t(\beta,\beta',\omega)=\bigl(x_t(\beta,\omega),\,x_t(\beta',\omega),\,y_t^{\beta\beta'}(\omega)\bigr). \]
Using the fact that \(\sigma(u,\alpha,x)\) and \(m(u,\alpha,x)\) are differentiable, it is not difficult to prove that the coefficients of equation (5) satisfy conditions ensuring the existence of a solution of the system of stochastic equations for the function \(z_t(\beta,\beta',\omega)\). To prove the existence of the derivative
\[ dx_t(\beta,\omega)/d\beta=\lim_{\beta'\to\beta} y_t^{\beta\beta'}(\omega), \]
it remains to verify that \(z_t(\beta,\beta',\omega)\) is, with probability 1, continuous in \((t,\beta,\beta')\) for \(t\in[0,T]\), \(\beta,\beta'\in A\). The latter assertion follows from the generalized theorem of A. N. Kolmogorov by means of inequalities analogous to (4). From simple estimates for stochastic equations it follows that \(dx_t(\beta,\omega)/d\beta\) satisfies the equation
\[ \begin{aligned} \frac{dx_t(\beta,\omega)}{d\beta} &=\frac{dx_0(\beta,\omega)}{d\beta} +\int_0^t \left[ \frac{\partial\sigma(u,\beta,x_u(\beta,\omega))}{\partial\beta} +\frac{\partial\sigma(u,\beta,x_u(\beta,\omega))}{\partial x} \frac{dx_u(\beta,\omega)}{d\beta} \right]\,d\xi_u(\omega)+{}\\ &\quad+\int_0^t \left[ \frac{\partial m(u,\beta,x_u(\beta,\omega))}{\partial\beta} +\frac{\partial m(u,\beta,x_u(\beta,\omega))}{\partial x} \frac{dx_u(\beta,\omega)}{d\beta} \right]\,du, \end{aligned} \tag{6} \]
which is the limit of (5) as \(\beta'\to\beta\). From inequalities similar to (4), it is not difficult to derive that \(y_t^{\beta\beta'}(\omega)\) converges to \(dx_t(\beta,\omega)/d\beta\) also in mean square.

Remark 1. The coefficients of the equations for higher derivatives, generally speaking, grow faster than \(\|x\|\), and therefore the existence of a solution of system (2) has to be proved separately.

Remark 2. With the help of certain additional constructions, in Theorem 2 one can dispense with the requirement that the derivatives be bounded: it is sufficient that, for some \(N<\infty\), they grow no faster than \(\|x\|^N\). If one is not concerned with convergence in mean square, then it is sufficient to require of \(x_0(\alpha,\omega)\) the existence and continuity, for almost all \(\omega\), of all partial derivatives with respect to \(\alpha_p,\ p=1,2,\ldots,m\), up to order \(k+1\) inclusive.

Theorem 3 is strengthened analogously.

Remark 3. If \(x_t^\alpha(a,\omega)\) is a one-dimensional random Markov function satisfying equation (3), then the equation for \(dx_t^\alpha(\omega)/d\alpha\) has the form
\[ \frac{dx_t^\alpha(\omega)}{d\alpha} = 1+\int_0^t \frac{\partial\sigma(u,x_u^\alpha(\omega))}{\partial x} \frac{dx_u^\alpha(\omega)}{d\alpha}\,d\xi_u(\omega) + \int_0^t \frac{\partial m(u,x_u^\alpha(\omega))}{\partial x} \frac{dx_u^\alpha(\omega)}{d\alpha}\,du . \]

This equation can be solved in explicit form:

\[ \frac{d x_t^a(\omega)}{da} = \exp\left\{ \int_0^t \frac{\partial \sigma(u,x_u^a(\omega))}{\partial x}\,d\xi_u(\omega) + \int_0^t \frac{\partial m(u,x_u^a(\omega))}{\partial x}\,du - \frac{1}{2}\int_0^t \left[ \frac{\partial \sigma(u,x_u^a(\omega))}{\partial x} \right]^2 du \right\}. \]

Let us indicate some applications of the results obtained. Let \(x_t^x(\omega)\) be the solution of the stochastic equation (3); \(x_0^x(\omega)\equiv x\); and let \(f(x)\) be a \(k\)-times continuously differentiable function. Then, if the conditions of Theorem 3 are satisfied, the function \(u(x,t)=M f(x_t^x(\omega))\) has continuous partial derivatives with respect to \(x^r\), \(r=1,2,\ldots,n\), up to and including order \(k\). On the other hand, it is known that if the function \(u(x,t)=M f(x_t^x(\omega))\) possesses continuous partial derivatives up to and including second order with respect to \(x^r\), then

\[ \frac{\partial u}{\partial t} = \sum_{i,j=1}^{n} a_{ij}(t,x)\frac{\partial^2 u}{\partial x^i \partial x^j} + \sum_{i=1}^{n} b_j(t,x)\frac{\partial u}{\partial x^i}, \tag{7} \]

where

\[ a_{ij}(t,x)=\sum_{k=1}^{n}\sigma_k^i(t,x)\sigma_k^j(t,x), \qquad b_i(t,x)=m^i(t,x), \qquad i,j=1,2,\ldots,n. \]

Thus, if \(\sigma_j^i(t,x)\), \(m^i(t,x)\) have continuous bounded partial derivatives with respect to \(x^r\), \(r=1,2,\ldots,n\), up to and including order \(k+1\), and \(f(x)\) has continuous partial derivatives up to and including order \(k\), then the solution of the Cauchy problem for equation (7) also has continuous partial derivatives with respect to \(x^r\) up to and including order \(k\) (the matrix \(a_{ij}(t,x)\) may also be degenerate).

In conclusion we formulate the following theorem, whose proof is based on Theorem 2.

Theorem 4. Let \(x_t(a,\omega)\) be a random function satisfying equation (1), the coefficients of which possess continuous bounded partial derivatives up to and including second order. Denote by

\[ \tau_D^a(\omega)=\inf\{t:x_t(a,\omega)\notin D\}, \]

where \(D\) is a domain in \(R^n\) whose boundary \(\Gamma\) has a continuously rotating normal. Then, if

\[ \det\left|\{\sigma_j^i(t,x)\}_{1}^{n}\right|\ne 0 \]

for \(x\in \Gamma\) and \(t\ge 0\), then with probability \(1\)

\[ \frac{\partial \tau_D^a(\omega)}{\partial a_i}=0, \qquad i=1,2,\ldots,m, \]

for all \(a\), except for some set \(\Lambda(\omega)\in R^m\) of Lebesgue measure zero.

The authors express their gratitude to E. B. Dynkin for posing the problem and for a number of substantial comments.

Moscow State University
named after M. V. Lomonosov

Received
14 I 1961

REFERENCES

1. J. L. Doob, Stochastic Processes, IL, 1956.
2. K. Itô, Nagoya Math. J., 3, 55 (1951).