NATIONAL BUREAU OF STANDARDS REPORT

1846

SOME PROPERTIES OF STRONGLY CONTINUOUS

STOCHASTIC PROCESSES

by

Eugene Lukacs

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1103-21-5119 August 7, 1952 1846

SOME PROPERTIES OF STRONGLY CONTINUOUS

STOCHASTIC PROCESSES*

by

Eugene Lukacs

*The preparation of this paper was sponsored by the U.S.

Naval Ordnance Test Station, Inyokern

The publication,

unless permission

25. D, C. Such

prepared If that

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

ir In part, Is prohibited

Standards, Washington

I has been specifically

sport for its own use.
,

'
SOME PROPERTIES OF STRONGLY CONTINUOUS

STOCHASTIC PROCESSES

by

Eugene Lukacs

National Bureau of Standards

1 . Introduction

This report is a continuation of an earlier report [l]

dealing with strongly continuous processes . In the following

we use the condition obtained in our previous study and restate therefore this result as theorem 1. We reformulate also

this theorem so as to give it a wording which appeals more to

the intuition. We derive also several new theorems on strongly

continuous stochastic processes . Theorem 2 gives a characterization of the fundamental random process (Wiener process)

while theorem 3 characterizes another stochastic process with

independent increments

2 . Normality of the increments . Theorem 1 : Let y(t) be a stochastic process and assume that

(i) y(t) is a process with independent increments

(ii) y(t) is strongly continuous in the interval

[a,b]

Then y(b) - y(a) is normally distributed.

The concepts ''strong continuity" and "process with independent

increments" are defined in [l] where a complete proof of theorem 1

is given.

H. B, Mann proved recently [2] that a process is strongly

continuous if and only if it can be realized in the space of all

continuous functions [Theorem 3 in reference [2] j. In view of this

fact we obtain the following reformulation of theorem 1.

-2 -

Corollary to theorem 1 . A stochastic process with independent

increments which can be realized in the space of continuous

functions has normally distributed increments.

A formal definition of the realizability of a process in a

function space is given in [2]

,

3Characterization of the fundamental random process . We first establish a property of strongly continuous processes.

Lemma 1If the process y(t) is strongly continuous in the interval [&,b] then it is continuous at every interior point of the

interval, that means pliiji y(t + T ) = y (t ) if a<t<-b.

Proof: We take for the set S the set consisting of the two

points t and t + y , both interior to [a,b]

The event

d* ( cf > £ » s ) is then the event that |y(t+/") - y ( t ) j = £for | r/-£

The strong continuity of the y(t) process implies that for

every £ >0, and E| > 0 there exists a </ = ( 6 , ip such that

P(

j y (t+ r ) - y ( t

)

; | = £ ) ^ | - n for |y|<ff. By the definition

of convergence in probability this means that

plim [y(t+r) - y ( t )

]

= 0, r^o

We consider next a stochastic process y(t) and denote

by f (t , y ) the variance of the increment y (t + y ) - y (t ) . We

prove a lemma concerning the function f(t,y).

Lemma 2 , If the process y(t) satisfies the assumptions of

theorem 1 then the function f(t,7^) is continuous in y . We have clearly

y ( t+ r + A ) - y(t) = [ y (t +r+A) - y (t + ?')] + [y (t + r ) - y(t)].

On account of the assumed independence of the increments on the

right hand side of this equation we have
'
~3 »

(i) f (t,r + A ) - f (t, 7^) = f (t + r, A) . We know from theorem 1 that the increment y(t f/'+A) - y(t +

is normally distributed » Lemma 1 shows that it converges in

probability to zero Its variance f (t + T, A ) must therefore

converge to zero as A goes to zero. We see then from (1) that

lim [f(t,7' + X) - f(t,7*)] - 0 so that lemma 2 is proven.

Definition 1 . A process x(t) is said to be a fundamental random

process (Wiener process) if

(i) it is a process with independent increments and

initial value x(0) - 0,

(ii) the increment x(t + 7") - x(t) is normally

distributed with mean zero and variance c T

.

(where c > 0)

.

We are now ready to characterize the fundamental random

process

.

Theorem 2 . A stochastic process y(t) is a fundamental

random process if and only if the following three conditions

are satisfied

(a) y(t) is a strongly continuous process with independent

increments and initial value y (0) = 0.

(b) The mean value function of the process is identically

zero

(c) The variance of the increments y(t + r ) - y(t) depends

only on the length y of the interval over which the

increment is taken but is independent of the location

of the interval on the time axis.

Proof

The necessity of these conditions follows from well

known properties of the fundamental random process (see [3] )so

that we have only to show that the conditions of theorem 2 are

*4

sufficient. From (a) it is immediately seen that property (i)

of definition 1 is satisfied „ It is seen from theorem 1 that

the increments y(t +T) - y(t) are normally distributed while

(b) shows that the mean of the distribution of the increments

is zero. We have therefore only to show that the variance of

the increment y(t + T') - y(t) is c Y . According to condition

(c) the function , y ) depends only on y so that we may

write f(t,7) - f ( T' ) . From (1) we see that f ( y ) satisfies

the functional equation

(2) f ( r+ h) - f ( r) + f (h).

Lemma 2 states that f ( 7* ) is a continuous function. It is

well known that the only continuous solution of (2) is

f(Y) - c 7. Since f(T) is by definition a variance we

see finally that c ^-0.

4 „ Processes with factorable variance function .

In this section we give an example which shows that in

a strongly continuous process with independent increments

the variance function f (t

,

7* ) need not be independent of t.

Theorem 3 „ Let y(t) be a stochastic process, defined

for t = 0 and denote by f (t, ^

)

the variance of the increment

[y(t +T) - y(t)] 0 Assume that

(i) y(t) is a process with independent increments and

is strongly continuous in every finite interval

[a,b] where 0 = a<-b,

(ii) the mean value function of y(t) is identically zero,

i ,e o Ey (t) s 0,

-5

(iii) the initial value is zero, y(Q) = 0,

(iv) f(t,r) > 0 for t = 0 and r ^ 0

(v) f (t,r

)

= h(t> g( r)

Then y(t) is a Gaussian process with covariance function

(3) <Ty (t x )y (t 2

) y (t

)

c e

K -1

where t ~ rain (t-^,t 2

).

By means of theorem 1 we conclude from assumptions (i) and

(iii) that the process is Gaussian, assumption (ii) determines

its mean value function. The purpose of assumption (iv) is to

exclude some "improper” processes. To see this we assume that

there exist values t Q and YQ ^ 0 such that f(t Q , 7"0

) = 0. From

the independence of the increments and the additivity of the

variance it follows then that f(t0 ,7"x) =“ 0 if T = Y and

also f(t

Q + 7^ » T'g ) "fe/t) if 7^ + X, = j The increments y (t 0+?^+ ?^>

)

y(t + ) have therefore a degenerate distribution if

+ 71 = Y * We conclude therefore from (ii) that y(t) =

J- 2, o

y(t_) if t. ^ t = t + T » Assumption (iv) excludes o o o o

therefore processes which do not change over a fixed interval.

We proceed now to derive formula (3), From assumption (v) and

from (1) we see that

(4) h (t ) [g( /+ \) - g(T')] = h (t + X) g(A)

Further we conclude from (iv) that h(t) ^

0 for all t = 0

while g( A ) ^

0 for allA>0. We may therefore rewrite (4) as

(5) g( r + A) - g( T) h(t + r)

g( A ) h (t

)

The left hand side of this equation is independent of t but

depends on >* and A , which the right hand side of (5) is a function of A and t but is independent of A. This is only

6

possible if the quotient on either side is a function only

of T'. We denote this function by f (?) and obtain

(6) h (t +r) = f ( r) h (t >

If we set t = 0 we have h ( 7") - h (0) j°(7") hence

h(t+y) - h(0) J

3 ( t +Y). Substituting these expressions

into (6) we obtain

h(t +r) = h(o> f (t+ r) - f(r)h(t) = />(r) h(o> f(t) or

(?) f(t +r) - f(t) f <r> . Since g(T') a continuous function of T the same is true

for f('Y) so that the only solution of (7) is

(8) f(t) - e Kt

and therefore

(9) h(t) = h(0) eKt

the

From (5) and (8) w@ obtain/functional equation for g( Y)

(10) g(T+A) = g(r) + e Kt

g( X)

In case K= 0, this reduces to

(10a) g(Y+X) « g('T) + g(A)

and the process y(t) is then a fundamental random process

We may therefore assume in the following that K 0. We

obtain easily from (10)

g(ti+t2 ... +t

n

) = g(t

1

) + eKt l g(t

2

) + e E ^ t l +t 2^g(t3

) + .

+ e

If in particular t

1 = t = .

n-1 2

Kst

(11) g(nt) = g(t) V* e

1' 2

K(tx+t2 4- ...+ t Qa5 i) g(t

n

) r

s=0

g(t)

t we have

e

Knt -1

„Kt _i

~7~

Setting here t = — we obtain for integer n

n

l x , eK/n -1

g( n

) - g(l

)

for integers ra and n

( 12 ) g( £ ) = gd) n

. In the same manner we obtain

mK/n _

•K

-i

From the continuity of the function g( P\) we conclude

finally that for any real

AK

(13) g( X) = g(l)

eK -1

(if K £

0)

The variance f (t , T ) of the increment y(t+T) - y(t) is

now immediately obtained from assumption (v) and from (9)

and (13) and is

(14) f(t, r) = c e Kt

e

K -1

where we wrote for brevity c = h(Q) g(l) = f(0,l).

By assumption (iii) y(t) = y(t) y(0) so that the variance

of y (t) is obtained by substituting t = 0 and T = t into (14)

_ _Kt 1 hence <5~ = c v . The formula (3) for the covariance

y (t

)

e

K

„i

function follows then easily from the independence of the

increments

The Gaussian process defined by theorem 3 could offer

an alternative to the fundamental random process, which is

different from the Ornstein - Uhlenbeck process. Its main

significance lies however in the fact that it shows together

with theorem 2 that there exist strongly continuous processes

with independent increments which- are different from the

fundamental random process
. . - v \ • - .
-8 TM

[1] Eugene Lukacs , A property of strongly continuous

processes. NBS report 1665, May 1952

[2] H. B. Mann, On the realization of stochastic processes

by probability distributions in function space.

Sankhya 11, p. 3-8,(1951)

[3] H, B. Mann, Introduction to the theory of stochastic

processes depending on a continuous parameter.

NBS report 1293, to be published as AMS 24.
. : : , S. „

’
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