Mathematics

A. D. Venttsel

On Continuous Additive Functionals of a Multidimensional Wiener Process

(Presented by Academician A. N. Kolmogorov, 23 X 1961)

Theorem. Let \(X=(x_t,\infty,\mathcal M_t,P_x)\) be a multidimensional Wiener process; \(\varphi_t^s,\ 0\leq s\leq t<\infty\), a continuous almost homogeneous additive functional of \(X\) (see the definition in \((^1)\)). Then there exist: a) an increasing sequence of closed sets \(K_n\) such that \(x_t\) leaves, in finite time, only a finite number of these sets (the process is bounded relative to the system of sets \(\{K_n\}\)); b) a sequence of numerical functions \(F_n(x)\) and c) a sequence of vector functions \(G_n(x)\) such that, with probability 1, up to the moment of exit from the set \(K_n\),

\[ \varphi_t^s=F_n(x_t)-F_n(x_s)+\int_s^t G_n(x_u)\,dx_u . \]

The theorem generalizes to a broad class of diffusion processes and to functionals that are continuous from the right and have a left limit at every point (in the case of a Wiener process all such functionals are continuous). In the case of a one-dimensional Wiener process the results take an especially simple form. Namely, every continuous almost homogeneous additive functional can be written in the form

\[ \varphi_t^s=F(x_t)-F(x_s)+\int_s^t G(x_u)\,dx_u, \]

where \(F(x)\) is a continuous function; \(G(x)\) is a function square-integrable on every finite interval. Conversely, to every pair of such functions there corresponds a functional of the described form.

For simplicity we shall carry out the proof for a three-dimensional Wiener process and for functionals that remain constant after exit from a certain ball (the proof in the general case can be reduced to the consideration of such functionals). Since a three-dimensional Wiener process sooner or later leaves any ball, the functional \(\varphi_t^s\) can naturally be extended to \(t=\infty\).

For the proof it is enough to construct a sequence of functionals \(\varphi_t^s(n)\), having bounded expectation and variance and such that if, for all \(u\leq t\), \(x_u\in K_n\), then \(\varphi_t^s(n)=\varphi_t^s(n+1)=\cdots=\varphi_t^s\). For the functional \(\varphi_t^s(n)\) put \(F_n(x)=-M_x\varphi_\infty^0\); then \(\varphi_t^s(n)-F_n(x_t)+F_n(x_s)\) is a functional having bounded variance and expectation equal to zero; hence

\[ \varphi_t^s(n)-F_n(x_t)+F_n(x_s)=\int_s^t G_n(x_u)\,dx_u \]

(see \((^2)\)).

The main difficulty consists in constructing the functionals \(\varphi_t^s(n)\). It is overcome with the aid of an idea due to A. V. Skorokhod \((^3)\). Put

\[ \Phi_\lambda(x)=M_x\exp\left\{-\lambda\sup_{0\leq s\leq t<\infty}|\varphi_t^s|\right\}. \]

We shall state a lemma belonging to Skorokhod.

Lemma. If everywhere \(\Phi_\lambda(x)\geqslant \varepsilon>0\), then the functional \(\varphi_t^s\) has bounded expectation and variance.

Let us outline the proof. Choose \(N\) such that \(e^{-\lambda N}<\varepsilon\). Denote by \(\tau_1\) the first moment when \(|\varphi_t^0|=N\) (\(\tau_1=\infty\) if for all \(t\), \(|\varphi_t^0|<N\)); by \(\tau_2\), the first moment after \(\tau_1\) when \(|\varphi_t^{\tau_1}|=N\), or \(\infty\) if \(|\varphi_t^{\tau_1}|<N\), and so on. It is proved that

\[ P_x\{\tau_n<\infty\}\leqslant \left(\frac{1-\varepsilon}{1-e^{-\lambda N}}\right)^n . \tag{1} \]

(We note for later that for \(n=1\) the formula is true if \(\Phi_\lambda(x)\geqslant\varepsilon\) not for all \(x\), but only for the \(x\) appearing in the formula. This is a simple consequence of Chebyshev’s inequality.) Formula (1) proves that the distribution of

\[ \sup_{0\leqslant s\leqslant t<\infty}|\varphi_t^s| \]

is majorized by an exponential distribution; therefore this random variable, and still more \(\varphi_t^s\), has all moments, moreover bounded with respect to \(x,s,t\).

Suppose that \(\Phi_\lambda(x)\geqslant\varepsilon\) in some domain \(A\). In this case A. V. Skorokhod [3] showed that one can construct a functional \(\psi_t^s\), equal to \(\varphi_t^s\) up to the exit from some smaller domain and having all moments. The functional \(\psi_t^s\) is constructed as follows. Let \(f(x)\) be a twice continuously differentiable function, equal to zero outside \(A\) and to one in a somewhat smaller domain. According to the change-of-variables formula in the stochastic integral [4],

\[ f(x_t)-f(x_s)=\int_s^t g(x_u)\,dx_u+\int_s^t h(x_u)\,du . \tag{2} \]

(Here \(g(x)=\operatorname{grad} f(x)\), \(h(x)=\tfrac12 \Delta f(x)\).) The functional \(\psi_t\) is constructed by the formula

\[ \psi_t^s=f(x_t)\varphi_t^s-\int_s^t \varphi_u^s g(x_u)\,dx_u-\int_s^t \varphi_u^s h(x_u)\,du . \tag{3} \]

This gives a functional that remains constant while \(x_t\) is outside \(A\).

However, it is not always the case that \(\Phi_\lambda(x)\geqslant\varepsilon\) in some domain; the set where this inequality holds may be nowhere dense, and then a smooth function \(f\), equal to zero outside this set, is identically zero. It turns out that here one can use nonsmooth functions for which a formula analogous to (2) holds.

Put \(f_i(x)=\min(3-3\Phi_\lambda(x),i)\) (\(i=1,2\)). It is easy to show that \(f_i(x)\) is a nonnegative superharmonic function tending to zero at infinity. Since the function \(f_i\) is bounded, it is possible (see [5]) to construct a nonnegative functional \({}_i\chi_t^s\) by the formula

\[ {}_i\chi_t^s=f_i(x_s)-f_i(x_t)-\int_s^t g_i(x_u)\,dx_u, \]

where \(g_i(x)\) is the gradient of \(f_i(x)\). In addition, \(M_x\int_0^\infty g_i^2(x_u)\,du\) and \(M_x\,{}_i\chi_\infty^0\) turn out, for all \(x\), to be less than 20. Now put \(f(x)=1+f_1(x)-f_2(x)\), \(g(x)=g_1(x)-g_2(x)\), \(\chi_t^s={}_2\chi_t^s-{}_1\chi_t^s\). There is a formula analogous to (2):

\[ f(x_t)-f(x_s)=\int_s^t g(x_u)\,dx_u+\chi_t^s . \tag{4} \]

Now we construct the functional \(\psi_t^s\) by the formula

\[ \psi_t^s=\varphi_t^s f(x_t)-\int_s^t \varphi_u^s g(x_u)\,dx_u-\int_s^t \varphi_u^s\chi(du). \tag{5} \]

(\(\chi_t^s\) is the increment of some function of bounded variation; integration with respect to \(\chi(du)\) means integration with respect to this function of bounded variation.) Using formula (4) and the properties of the stochastic integral (see (†)), it is easy to prove that \(\psi_t^s\) is again a continuous almost homogeneous additive functional. Moreover, up to the exit from the set \(\{x:\Phi_\lambda(x)\geqslant 2/3\}\), where \(f(x)=1\) and \(g(x)=0\), \(\psi_t^s=\varphi_t^s\), while as long as \(x_t\) is in the set \(\{x:\Phi_\lambda(x)<1/3\}\), the functional \(\psi_t^s\) remains constant.

Let us estimate
\[ \Psi_1(x)=M_x\exp\{-\sup_{0\leqslant s\leqslant t<\infty}|\psi_t^s|\} \]
from below. Suppose first that \(\Phi_\lambda(x)\geqslant 1/3\). Put \(\overset{0}{\varphi}_s=\varphi_s^0\) if \(\sup_{0\leqslant u\leqslant s}|\varphi_u^0|<N\), and \(\overset{0}{\varphi}_s=0\) otherwise, and consider the expression

\[ \overset{0}{\psi}_t=\overset{0}{\varphi}_t f(x_t)-\int_0^t \overset{0}{\varphi}_s g(x_s)\,dx_s-\int_0^t \overset{0}{\varphi}_s\chi(ds). \tag{6} \]

The first term of this expression does not exceed \(N\). The second term is a martingale, and
\[ M_x\left(\int_0^\infty \overset{0}{\varphi}_s g(x_s)\,dx_s\right)^2\leqslant 80N^2; \]
hence, by Kolmogorov’s inequality,

\[ P_x\left\{\sup_{0\leqslant t\leqslant\infty}\left|\int_0^t \overset{0}{\varphi}_s g(x_s)\,dx_s\right|\geqslant \frac{9N}{\delta}\right\}\leqslant \delta^2. \]

As for the last term, we have
\[ M_x\int_0^\infty |\overset{0}{\varphi}_s|\,|\chi(ds)|\leqslant 40N, \]
and

\[ P_x\left\{\sup_{0\leqslant t\leqslant\infty}\left|\int_0^t \overset{0}{\varphi}_s\chi(ds)\right|\geqslant \frac{40N}{\delta}\right\}\leqslant \delta. \]

Thus, for any \(\delta>0\) and \(N>0\) there exists a constant
\[ A=N+\frac{49N}{\delta} \]
such that
\[ P_x\left\{\sup_{0\leqslant t<\infty}|\overset{0}{\psi}_t|\geqslant A\right\}\leqslant \delta+\delta^2. \]
Now choose \(N\) so that \(e^{-\lambda N}<1/3\), and \(\delta\) so that
\[ 1-\frac{1/3}{1-e^{-\lambda N}}-\delta-\delta^2>0. \]
With probability not less than
\[ 1-\frac{1/3}{1-e^{-\lambda N}} \]
(see formula (1)),
\[ \sup_{0\leqslant t\leqslant\infty}|\varphi_t^0|<N; \]
in this case \(\psi_t^s=\overset{0}{\psi}_t-\overset{0}{\psi}_s\). Hence we obtain

\[ P_x\left\{\sup_{0\leqslant s\leqslant t\leqslant\infty}|\psi_t^s|\geqslant 2A\right\}\leqslant \frac{1/3}{1-e^{-\lambda N}}+\delta+\delta^2, \]

\[ \Psi_1(x)=M_x\exp\{-\sup_{0\leqslant s\leqslant t<\infty}|\psi_t^s|\} \geqslant e^{-2A}\left[1-\frac{1/3}{1-e^{-\lambda N}}-\delta-\delta^2\right] =\varkappa>0. \]

We now show that also for \(\Phi_\lambda(x)<1/3\), \(\Psi_1(x)\geqslant\varkappa\). Denote by \(\tau\) the first exit time from the set \(\{x:\Phi_\lambda(x)<1/3\}\). It is easy to prove that with probability \(1\), \(\tau<\infty\). Clearly, for \(s,t<\tau\), \(\psi_t^s=0\). Therefore
\[ \sup_{0\leqslant s\leqslant t<\infty}|\psi_t^s| = \sup_{\tau\leqslant s\leqslant t<\infty}|\psi_t^s|, \]
whence, using the almost homogeneity of the function—

of the functional \(\psi_t^s\), one can infer that \(\Psi_1(x)=M_x\Psi_1(x_\tau)\). Since \(\{x:\Phi_\lambda(x)<1/3\}\) is an open set, \(x_\tau\) does not belong to it; thus, \(\Phi_\lambda(x_\tau)\ge 1/3\) and \(\Psi_1(x_\tau)\ge \varkappa\). Therefore \(\Psi_1(x)\) is everywhere \(\ge \varkappa\).

Now choose a sequence \(\lambda_n\to 0\) and take as the set \(K_n\) the set \(\{x:\Phi_{\lambda_n}(x)\ge 2/3\}\); the functional \(\varphi_t^s(n)=\psi_t^s\) coincides with \(\varphi_t^s\) up to the time of exit from \(K_n\) and has bounded mathematical expectation and variance. It remains to prove that the process \(X\) is bounded with respect to the system of sets \(\{K_n\}\). It is easy to prove that the probability that \(x_t\) ever leaves \(K_n\) does not exceed \(3(1-\Phi_{\lambda_n}(x))\). As \(n\to\infty\), the probability of this event tends to zero, since \(\Phi_\lambda(x)\to 1\) \((\lambda\to 0)\); therefore the probability of the process exiting all \(K_n\) is \(0\).

Moscow State University
named after M. V. Lomonosov

Received
18 X 1961

REFERENCES

\(^{1}\) E. B. Dynkin, Theory of Probability and Its Applications, 5, no. 4, 441 (1960).
\(^{2}\) A. D. Wentzell, DAN, 139, no. 1 (1961).
\(^{3}\) A. V. Skorokhod, Theory of Probability and Its Applications, 6, no. 4 (1961).
\(^{4}\) K. Itô, Nagoya Math. J., 3, 55 (1951); Sbornik: Mathematics, 3, 5, 131 (1959).
\(^{5}\) A. D. Wentzell, DAN, 137, no. 1, 17 (1961).