the oscillation of stochastic integrals

Z. Walu'scheinlichkeitstheorie verw. Geb. 4, 103--112 (1965)

The Oscillation of Stochastic Integrals By

EUOE~E Wo~G* and MOSHn ZAXAI

1. Introduction

Let Dn = (a -~- t(o n) < t (n) < t~ n ) ' . . < ~k(n)~(n) = b), n : 1, 2 , . . . be a par t i t ion (t(n) . t ~ n ) ) . I t will be assumed throughout of the in terval [a, b], let ~(n) = m a x ~ i+1

O ~ i ~ k ( n ) - - i

this paper t h a t lim ~(n) == 0. Let y(t, co), t ~ O, be the Brownian mot ion process ~t ---> co

(with E {y (t, co)} = 0, E {y2 (t, oJ)} = t, t =~ 0). I t is a well known result of 1 ~ LEVr [7] t h a t

k ( n ) - i

l im /~N" Lv~ru(t(n)i+l, co) - - Y(t! n), co)]2 - - (b - - a) (1) n----~ co i ~ 0

in the mean and if also (from some no on) Dn+l is a ref inement of Dn then (1) holds with probabi l i ty 1.

A similar result, for a class of s t a t ionary Gaussian processes was given by BAXTE~ [1] and later generalized to a class of non-s ta t ionary Gaussian processes by GLADYS~V [4]. An extension to processes with s ta t ionary independent increments was given by Kozi~r [6]. BEI~NASr [2] considered limits of similar forms for t ime-homogeneous diffusion processes.

I n this paper , a result of type (1) will be shown to hold for a general class of processes F( t , co) which can be represented as stochastic integrals ( theorems 1, 1 a) (as in BERMAN'S paper , the l imits are r andom variables). I t then follows tha t the sample functions of such processes are of unbounded var ia t ion (Corollary 1). These results are applied to obta in corresponding results for a general class of diffusion processes (Theorems 5, 5a) and to the mutua l singulari ty of the probabi l i ty measures induced in funct ion space by a pair of diffusion processes.

2. The oscillation of stochastic integrals

Let y(t, co) and Dn be as defined above. Let ~ t be the a-field induced by y (s, co) -- y (a, co), a ~ s ~ t. Le t /(t, co) satisfy:

I1 : / (t, co) is real valued and measurable in co and t (a ~ t ~ b).

12 : for each t in [a, b], ] (t, .) is measurable with respect to !~ t. b

I3: rE{In( t , co)}dt < ~ . ct

Theorem 1. Let / ( t , co) satis/y 11, 12, Ia and 2F(t, co) be defined by the stochastic integral

t

F(t, co)=f/(s, co)dy(s, co); a<- - t<_b . f$

* Work supported in part by the National Science Foundation under Grant GP-2413.

Z. W a h r s c h e i n l i c h k e i t s ~ h e o r i e v e r w . G e b . , B d . 4 8

104 Eu(~E~E WONG and MOSHE ZAXAI:

Then k ( n ) - - i b

lim ~ rr''(n)L k~i~_l, co) -- F(t~n),co)]2= f/~(t, co)dt (2) n - + o o i = 0 a

in the mean. I/, /urthermore, /or some (~ > 0

lira T(n) n s+~ = 0 (3) n---> oo

then (2) holds with probability one. Alternatively, i] I3 and (3) are replaced by I'~ and (3'):

~'~: E {/~(t, co)} __< ~ , a -< t <_ b

lira ~(n)~n 1+~ -= 0 /or some (5 > 0, (3')

then (2) also holds with probability one. The proo] will be based on the following result of ITo [5] : Let ] (t, co) and g (t, co)

b b

satisfy 11, I2 and also .[IS(t, co)dr < r fg2(t, co)dr < r for almost all co. Let a a

t t

G(t, w) = fg(s, w)dy(s, co), F(t, co) -~ ~](s, co)dy(s, co) a @

then, for almost all e) : t t

] g(s, co)dy(s, co) ~/(s, co)dy(s, co) (4) r a

t t t

= fg(s, co)F(s, co)dy(s, co) + ~/(s, 09) G(s, co)dy(s, co) ~- fg(s, co)/(s, o~) ds. a a a

Form now on, when there is no confusion, we will omit the probabil i ty parameter co. Before proving Theorem 1, we state and prove the following lemma.

Lemma 1. 1] ] (t, co) satisfies 11, Is, la then ]or t in [a, b] : t

E {174 (t)} ~ C (t -- a) ~ E {]4 (s)} ds (5) a

where V = (4 + 1/i~1 s. We first prove the lemma under the additional restriction I [ (t) ] ~ M for all t in

[a, b] and almost all co. B y (4):

F4 (t) = (2 j / (s) /7 (s) dy (s) -]- j /2 (s) ds) 2

g 8 ( ] l ( s ) / 7 ( s ) d y C s ) ) 2 ~ - 2 ( j l 2 C s ) d s ) s .

t

Since E{J2(s)/72(s)} ~ MSE{/72(s)} = M s f E{]2(s)}ds < oo, it follows tha t a

t t

E {/74 (t)} < s f E {/~ (s)/72 (s)} as + 2 (t -- a) f E {/4 (s)} ds a a

The Oscillation of Stochastic Integrals 105

therefore E{F4(t)} < oo and

t t

a

Since F(t) is a martingale, F4(t) is a semi-martingale and for a ~ s ~ t ~ b E {F 4 (s)} ~ E {F 4 (t) }, hence

E{Fa(t)} ~ 8 E1/2 {f4(t)} (t -- a) t/2 E{/4(s)}dz -f- 2(t -- a) fE{[a ( s ) }ds .

Let E{Fa(t)} =-x ~, ( t - a ) ~ E { ] a ( ~ ) } d s - #2 then the last inequality becomes t~

x 2 g 8ffx ~- 2ff 2, x > 0, > > 0, hence x g #(4 + ]/18) and (5) has been proved for I/(t)] < M . I f /(t) is not uniformly bounded, let /M(t, CO)-=[(t, CO) for I / (t, co) ] < M, t in [a, b], a n d / ~ (t, co) = M for / (t, 03) > M and/M (t, 03) -= -- M (or t (t, o~) < - M . By (5)

/J } E { y b ( t ) } < c ( t - a ) j ' z { / } ( ~ ) } ~ < C(t--a)E p(~)~8 ;F~( t ) = ~1~(~)dy. ~ (6)

For any fixed t in [a, b], .FM (t) converges in the mean to F (t) as M ~ ~ , hence there is a sequence My such tha~ FM~ (t) --> F (t) almost surely; the result follows from (6) by FA1:olz~s theorem.

We turn now to the proof of the "in the mean'" part of Theorem 1, Let k(n)- - I b

Qn ~--- ~ Lr~/t(n), i+1, CO) ~ F(t~n), CO)J2 - - ~ ]~(8, CO)d8 i = 0 a

- ~=~- ,, 1 (8, ~ ) '/u (s, CO) ~,~~

]~y (4):

Qn = 2 ~ fl(s) l (u)dy(u) dy(s) i = 0 t~'> t~

E{Q~} =- 4 ~ f E 12(s) l (u)dy(u) ds

4 ~ f El~2 {t ~ (~)}" E1/2 l (u) dy (u) d8 i = o t(~ ~) t

By (5):

E{Q~} ~ 41/C ~ [E ue i t ' (s)} . (s --t{")) u~ ([E{/n(u)}du) ds (7)

/ b \ 1 /2 b

\ a / a

Hence E { Qn 2 } converges to zero as n --> c~ and t h e " i n the mean" part of Theorem 1 is proved.

106 EvG]~E Wol~o and Mosn]~ ZAKAI:

B y (7') and the Tchebiehev inequality,

Kl(z(n))ll 2 1 Prob. {[ Qn [ ~ (z (n) n2+~) 1/4} ~ (,(~)n2+O)1/2 - K1 u1+ol2 �9

oo

Since ~ n-(l+x2) < c~, (6 > 0), it follows by the Borel-Cantelli Lemma tha t the ~ 1

set of co for which Qn (co) ~ (z(n) ne+0)l/a for infinitely many n has probabil i ty zero. Therefore, for almost all co, I Qn (co) [ ~ (T (n) n2+O) 1/4 is satisfied only finitely many times. Since we assumed tha t lim T(n) n 2+~ = 0, it follows tha t I Qn I --> 0 almost surely, n-+~

t I f also I a is satisfied then it follows from (7) tha t E{Q~n} g K2~(n). Hence

Kz Prob. {[ Qnl ~ (~(n)nl+O) 1/2} g n l + ~ �9

oo

Since ~ n-(1+~ < r (~ > 0), the almost sure convergence of (2) follows by (3') 1

and the Borel-Cantelli Lemma.

Corollary 1. Under the conditions o] theorem 1, let F (t) be the continuous version t

o/~/(s) dy (s). Then with probability 1, either F (., co) =- 0 in [a, b] or F (., co) is o/

unbounded variation in [a, b]. Proo/:

k ( n ) - I k ( n ) - I

~[F(t~)--F(t~))]2< max IF(t~)--F(t!~))l. ~ ]F~t(~)~ F(t~))] = ~ i + l J - - �9

= 0 0 _<~ _-< k ( ~ ) - 1 i = 0 ( 8 )

Consider only the continuous sample functions F (., co). Because of the continuity of F(-, co); max [ F ( t ~ l ) - - F(t~))[ converges to zero as z(n) ->0 . Le t nn be

0 ~ i _ - - < k ( n ) - - I

sequence of parti t ions such tha t the "a lmost sure" par t of Theorem 1 holds. I f for a given col(t, co) -~ 0 for almos$ all t ill In, b] then F( . , co) ~ 0, ff not then the right hand side of (2), hence the left hand side of (8) are strictly positive. Hence it

k ( n ) - - I

follows from (8) t ha t ~ I/7(t~1) - - F(t~n))l must diverge to oo as n ~ c~ and F( . ) i = 0

is of unbounded variation. We consider now the following generalization of Theorem 1. Le t Y(t, o9)

= (yl (t, w) . . . . . yq (t, w)), t ~ 0, be a q-dimensional Brownian motion (that is, the components yP (t, co), 1 g p ~ q, of Y (t, w) are one dimensional Brownian motions and yP(t, co), 1 ~ p ~ q, are independent processes). Le t ~q be the a-field induced by Y(s, co) -- Y(a, co), a ~ s g t.

Theorem l a . Let each ]p(t, 09) satis]y I1, I2 (with respect to ~q), I~ and let F (t, w) be defined as

q t

F(t, co) = ~ ~]~(s, co)dy~(s, co);a----- t ~ b. p = l a

Then k ( n ) - I q b

lira co) - F(t co)] = ft (t, co)dt (2a) n--+oo i ~ 0 p = l a


in the mean. I/ , also,/or some (~ ~ 0

lim ~(n) n2+~ ~ 0 (3a) n---> oo

then (2 a) holds with probability one. Furthermore, i/ /or some finite M

I'~': E(/~(t)} g M , a ~ t _ < _ b , ] ~ = p g q

and i / /or some (~ ~ 0 lira ~(n) n1+5 _~ 0 (3'a)

n ---> oo

then (2a) holds with probability one. Proo/:

k( 1 t~+l 2 __

.[ 1~ (t) dy~ (t) i /~ (t) dt i ~ o p 1 t~ n) p ~ ] t~ n)

k ( n ) - - I ( n ) \ 2 t~nl )

i=0 [p=lL\tI ) / t, ) (n) (n)

k ( n ) - - I q q t~+1 t(+l

+ ~ ~ y. fl,(t)dyr(t), fl~(t)dy~(t) = Q~)+ R~). i=o r*p tp) t~ ~)

From equation (7') it follows tha t E{(Q~)) 2} g Ka(v(n)) I/~ (where Ka is independent of n). F rom (7) it follows tha t under I~', E{(Q~)) 2} g T(n) �9 K~.

In order to evaluate E {(R(nq)) 2} we use the following result of IT5 [5]: for r ~: s b b

.[t,.(t). dyr (t). ~l~(O) dy'(O) a a

- - f / r ( t ) ( j / s ( u ) d y s ( u ) ) d y r ( t ) ~ - S / s ( O ) ( j / r ( ~ t ) d y r ( n ) ) d y s ( O ) "

Following the steps tha t lead to equations (7) and (7') we obtain: E{(R~)) 2} H

<--__ (7;(n))l/2Kc and under I3 , E{(R(nq)) 2} _~< "c(n)Kd;K~ independent of n. The rest of the proof is the same ~s the proof of Theorem 1.

3. Some related results

Theorem 2. Let / ( t , to) and F(t, to) be as in Theorem 1. Let Dn be a sequence o/ partitions such that: (a) /tom some no on Dn+l is a refinement o /Dn , (b) (2) holds with probability one. Let g (t, to) satis/y 11, 12 and 13 and let the sample/unctions g (., to) be continuous in [a, b] with probability one, then

k(n) 1 b lim ~. g (t! n)) [ F ( t~ 1) -- F (t! n)) ]2 = f g (t)/~' (t) dt (9)

n --> oo i ~ 0 et

with probability one. BElCMA~ [2] considered the special case/ ( t ) = 1 and g(t) ~ ~f(y(t)) where ~p(.)

is continuous. The proof of Theorem 2 is precisely the same as the proof given by

108 E v ~ W o ~ o and M o s ~ ZaKgz:

BnRMAN and i~, therefore, omit ted, I t follows by the same a rguments t ha t if r h~s a e o r ~ i n ~ s der iva t ive th~. , a. ~ ,

k(n) - t b lira ~ (O(F(t~n+)L)) -- r = ] (qS' (F(t l))2t2(t)dt .

n--> oo i ~ q a

Theorem 3. Let g (t, co) satis/y I1, 12 and let E {g2 (t, co)} be continuous in [a, b]. Then

/ c (n) - - I b ] ~ ~. g(t! n)) ( y ( t !~ ) -- y(t~n))) 2 = fg ( i )d t (10)

in the mea~, k (n) - 1

Proo/: ~ " q(~h itch) t! n)) converge~ in the mean to j" g (t) (it, .f_~Yx i I \ ~ i + 1 - i = 0 a

L e t T!n) § __ t!n) v~n) __ . , i t ( n ) , y (t~n)) ~ i + l ' - - Y t i + l ! - - �9 ( / k ( n ) - i }

f k ( n ) - - i } Consider a typical t e rm in the double ~um, assume ] > i. Then ~7~ n) is independent of all the other r andom variables in this t e rm and E {(V~n/) z - - z~)} = 0. There- fore the e~:p~ctatmn of the double ~ur~ is ~vro, Since v]! ~ i~ independent of g (t(~)) we have

E ~_:(t~ n)) [(~!~)) ~ - - ~]~ = ~E{g2( t~n))}E ( ( ~ ) ) ~ - - "t'!n)) 2}

i = i = 0

k (n) -- 1 /c (n) -- 1 2 ~, E {g2 (t~'))} (~))~ ~ 2 ~(n) ~ E (g2 (t! n)) z~n)}

i = 0 i = 0

which converges to ~ero. b

Th(~orem 4, Let y (t, ~ ) eali~/y I~, I~ ([or almq,~t aZ t in [a,5] and f E (g~ (~)} 8l < oo~ a

Let gn (t, co) be a ~equence o/ approximations to g (t, co) with the [ollowirgl properties ~(n) = b [re/erence 3, p. 439] : (1) For each n exists a partition a = t (n) < t (n) < ... < ~(~o

(independent o/ co) such that gn(t, o ) = gn(t (n), co), t~ n) ~ t < t~n)+~ and ~(n) __> 0 as n -> co. (2) gn (t, co) satisfies I~ and I~, (3) :

b lim ] E {(g (t) -- g~r (t)) 2} dt = O.

Then

and

/r b l . i .m ~gn( t ! ~q) ~i+l"~nL -- t! n)) = .fg(t) dt n--+ c~ i = 1 a

k ( n ) - - I b

] . i . m ~ ~ n (t~ n)) ( Y (t!~ 1) - - Y (t~n))) 2 = f g ( t ) ~/t:

n --~ c~ i = 0 a


Proo/: ~ / k ( n ) - - l t ( ~ ) 2 1 k(n)-- 1 t~(~

S t ( ~ f(g(t)--gn(t))dt _<_(b--a)~ jE{(g(t)--gn(t))2}dt. ( \ i = o t~ ) i = o t~)

This proves the first assertion. F rom the proof of Theorem 3 it follows tha~

r/k(n) -1 )2} El i i~= go n(tl)[(y(t!n+)l) -- Y(t!n)))2- (t~vl - - t!n))] ~

k(n)- - I < 2 v(~) ~ E ~,,~ ~t (~)~ ~t (~) t~ n)) == ~ [ Y n ~ i / J ~ i + l - -

i = O

which �9 -~ ~(n) 0 and converges to zero smc~ ~ -~ k(n)- - I b b

E {g~ (t~ n)) ,/t(~)i+l - - t~n)) } ~- .[ E (g~ (t)} dt =< 2 .f E (g2 (t)} dt ~- i ~ 0 a a

b ~- 2 f E {(gn(t) -- g(t))2}dt

a

t~MA~K : The results of this section can be wri t ten in the symbolic form: b b

Sg(t) (dy(t))2 = Sg( t )d t g ~

b b

ol g (t) (dF (t) )2 = f g (t) I~ (t) dr. a a

4. Application to the sample functions of diffusion processes

Lemma 2. Let /p(t, co), 1 ~ p g q, satis/y the conditions o/ Theorem la. Let g (t, ~,J) satis/y 11 and 13. Let

t

~(t) - . ~g(~)~s ct

and

then H(t) -~ G(t) + F(t)

k ( n ) - I q b lim ~ [ H ( t ! ~ l ) - - H(t!~))] 2 = ~ ]I~(t)dt (11)

n - . + c ~ i = O p = l a

in the mean. Furthermore, i] /or a given sequence o] partitions Dn, (2) conve~y~ with probability I then (11) also converges with probability 1.

Proo/: k(n)- - I k(n) - - I ]~(n)-- 1

(H(t!n+)l) -- H(t!n)))2 -- ~ (G(t!~l) -- G(t~n)))2 + ~ (F(t!n+)l) -- F(t~n)))2 + i = 0 i ~ 0 i ~ 0

k (n) -- 1 -~- 2 ~ (G(t~x) - - G(t!n)))(F(t!~O1) - - E(t!n))).

i ~ 0

The expectat ion of the square of the last sum is dominated by

4E1/21C(,I~--o(1F(t~n+)I) -- F(t!n)))2)21" E1/21(k(:~o 1 (t:!ig(t) dt y)2 } �9

110 EUGENE WoNa and MOSH~ ZAKX~:

Therefore, by Theo2-em 1 a., i t is ~ufficient to show tha t

(flat (12)

converges to zero in the mean and with probabi l i ty 1. Since

[ ~=~ [/:~(S) t = < [ '= n~- ' ~ i 4 -1 - - t!n))f~

=< ~ '~i+ll'(n' __t!n))2. ~ ( ~ 2 ( 3 ) d8) i = o i = 0 \tj~ ]

b

@

If. follows tha t (12) converges in th~ mean and with probabil i ty one, which comple- tes t~e proof ~f the ]emma.

Theorem 5, Let x (t, 03) be the solution to the stochastic di~erential equation t t

~(t) = x(a) + ]~(x(s), s)ds + ] ~ (x (s), s) d~ (s) 03) a @

where

lm(x,t) ~ m(~,t)l <~ k Ix- - ~l ;lm(x,t) l ~ k (1 + ~z)l/a

and let {~(a)} < oo.

Then

04)

k(n)--I b 1ira ~ (x(t~n+)l) - - x(t~))) 2 = ~ a~(x(t), t)dt (15)

n - - + o o i = 0 a

in the mean~ and ~] lira T(n)n 1+~ ~ 0 ]or some s > 0 then 05) holds with probability

Proo/: I f E(z~(t)} is bounded m In, b] then the resul~ [oliows b y apply /ng L e m m a 2 to the right hand side of equat ion 13. I t remains aniy to show tha t under the assumpt ions of the theorem, E{x~(t)} is bounded in [a~ b]. The proof of this will follow closely the proof of existence and uniqueness of the solution to (13) by successive approximat ions (reference 3 p. 281).

Le t x0 (t) be any process satisfying I2, I2 and

Consider the sequence of successive zpproximat ions t t

Xn. l (t) : x (a) ~- f m (xn (~), s) dis A- f a (x . (s), s) dy (s) . ct @

* After this paper was submitted for publication we learned that a similar theorem h ~ been proved by S. l~m~i~: Sign-invariant random variables and ~tov~hastic processes with sign invariant increments, Theorem 4A, To be published in Trans, Amer, math. Soe.


Then xn (t) converges with probabil i ty one to x(t) which is the solution to (13). I t .follows from (14) and Lemma 1 tha t ifE{x4n(t)} is bounded in [a, hi, so are

E{(m(xn( t ) , t ) )4} , E{((~(xn(t),t)) 4} and E{x~n+l(t)}. Let

A ~x (t) = x~ (t) - x~_~ (t)

z] nm (t) = m (xn (t), t) -- m (xn-1 (t), t) ; A n(~ (t) = (~ (xn (t), t) -- a (xn-1 (t), t).

by (14): [l]nm(t)] =< K[zJnx( t )[; [Z]n(~(t)] =< K ]Anx(t)].

{(! /(! E {(Anx(t)) 4} <= 16E An- lm(S)ds -t- 16E An- l~(s )dy (s )

t t

< 16 (t - aVK4 J 'E { (~_ ~x(s))4 ds} + 16 (t - - a) r I E ( ( ~ ~x(s))4ds g g

t

< K~ j" E {(~ n- ix (s))4} ds . a

By (16)

Now

K~ E { ( J n x ( t ) ) 4}=< K2Kr(tnl-a)" =<K2 n!~ ," a=<t_<_b.

A3x(t) < J (A3.x) \ ] = 1

/ m \8 m =</22-, / y22,( ,x)4

\ j = l / i=1 H e n c e

E{(xm(t ) -- xo(t)) 4} =< K2 ~. (22Ka)n < K2e 41r~ n! - =

1

Hence, for some K4 < oo E {x4~ (t)} < K4 for all t in [a, b] and all m. Since Xm ( t)-+x (t) with probabil i ty one, it follows from FATOU'S theorem tha t E{x4(t)} is bounded in [a, b].

Theorem 5a. Let Y(t , o~) = (yl(t), . . . , yq(t)), t >= O, be the q-dimensional Brownian motion. Let X( t , ~o)= (xl(t) . . . . . xq(t)) be the solution to the vector stochastic di~erential equation:

t q t

x~ (t) = x~ (a) + f m~ (X(s), slds + 2 S ~ r (X(s), s) dyr(s) p = 1, 2 . . . . . q; a r = l a

where q

I1 x (t)tl ~ = y (x~ (t))~ p = l

[m~ (x~ , t ) - m~ (X2, t) I =< K 11 X1 -- X2 II; 1 __< p =< q

I (~pr(Xl, t) - - o'P r ( / 2 , t ) I =< K I] X t - - X2]I; 1 ~ / ) , r =< q and let

Then E {ll x (~)ii~} < 0o.

k (n ) -- 1 q b

lim ~(xpt§ ',~i+1, - - xP(t~n))) 2 = 2 f ( apr(x( t ) ' t l )2d t (17) n - + z o i = 1 r = l a

112 E u o ~ , W o ~ and Mos~. ZAKAI: The Oscillation of Stochastic Integrals

in the mean, and i/ lim T(n) n 1+~ -~ 0/or some a ~ 0 then (17) holds with probability one. The proo/ is the same as the proo/ o/ Theorem 5 and is there/ore omitted.

BAXTE~'S resul t [1] w~s ~ppl ied b y S ~ e I A ~ [8] to the p rob lem of de tec t ion of noise- type signals. A similar app l i ca t ion of Theorem 5 is the following. Consider the processes x~ (t, ~o) and xz (t, co) defined b y

t t

x~(t, co )=x~(a)+ ]m~(x~(s),s)ds-~ ](r~(x~(s),s)dy(s), a ~ t g b a t~

t t

x~it, ~) = x~(a) + [mAx~(s),s)ds + ~f(~(x~(s),s)dy(s), ~ <_ t <-- b , a a

where x~(a), x~(a), ma, m~, a~, a~ sat isfy the condit ions of Theorem 5. Le t a~, a~ t0 to

be such t h a t there exists a f ixed to in [a, b] for which ~ a~ (x (s), s ) d s . ~ a~ (x (s), s) ds a a

for al l cont inuous x (s). A sample z0 (t), a ~ t ~ b, is ava i lab le to an observer and i t is known to h im t h a t x0 (t) is e i ther a sample f rom the sys tem wi th subscr ip t ~ or with subscr ipt /~. The observer is to decide whether x0 (t) came from ~ or f rom/~. Let p~ (p#) be the p robab i l i t y t h a t the observer decides t h a t the sample came from

(/~) when indeed i t came from/~ (~). I f the resul t of Theorem 5 is used as a t es t (in the in te rva l [a, to]), then p~ ~ pa = 0; namely , the p robab i l i t y measures induced b y x~ (t) and x~ (t) on the space of funct ions ~re m u t u a l l y singular.

A~knowledgement. The ~uthors would like to thank Prof. K. ITO and Prof. ~r LOEVE for instructive and illuminating conversations.

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522--525 (1956). [2] B~MA~, S. M. : Oscillation of sample functions in diffusion processes. Z. Wahrscheinlieh-

keitstheorie verw. Gebiete 1, 247--250 (1963). See also correction in review of this paper: Math. Reviews 28, p. 132 (1964).

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Amer. math. See. 8, 960--963 (1957). [7] Lv, vv, P. : Processus Stochastiques et Mouvement Brownien. Paris: Gauthier-Villars 1948. [8] SLEPIAN, ]).: Some comments on the detection of Gaussian signals in Gaussian noise.

IRE Trans. Inform. Theory IT-4, 65, 1958.

Department of Electrical Engineering University of California, Berkeley

and Applied Research Laboratory Sylvania Electronic Systems

Waltham, Mass.

(Received June 6, 1964)

the oscillation of stochastic integrals

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