I ntnat. J. Math. & Math. Sci.
No.
(1980) 113-149
113
Vol.
LIMIT THEOREMS FOR SOLUTIONS OF STOCHASTIC
DIFFERENTIAL EQUATION PROBLEMS
J. VOM SCHEIDT and W. PURKERT
Idgenfeurhochschule Zwickau
DDR-95 Zwickau
Germany
(Received September 5, 1978)
ABSTRACT.
In this paper linear differential equations with random
processes as coefficients and as inhomogeneous term are regarded.
imit theorems are proed for the solutions of these equations if
the andom processes are weakly correlated processes.
Limit theorems are proved for the eigenvalues and the eigenfunc-
tons of eigenvalue problems
and for the solutions of boundary value
problems and initial value problems.
KV W0S AP FHASS. Limit Theorem, Stochastic Eigenvalue Problem,
Stochastic Boundary Value Poblem, Differential Equation.
90 K4[HArCS 83Cr CLFCArO COP.
60, 6OH.
114
J.V. SCHEIDT AND W. PURKERT
I. IRTROIEUTION.
At the research of physical and engineering problems it is of great
ortance
the approach to the dierentia1 equations th ochaic
process as eiciemts reectively th ochaic boda or
Int cdiions. ere
a probl.
e
s
a
sees
fir ments of the slution are
fr the fir moments of the
4]). e
oen
calcated
occ
pcess ovi the pbinteest an Implant problem
I. For the applications te is
(see
of papas wch deal th ch
calcation of the die.buttons of the sulutlon pro-
eess fr the dibutions of the involvi pcess is o more
dft. s pblem contains already
diiculties for ve
le
na
pbs (e.g. for the tial value probl of a linear d-
derental equation of he flr order th ochaIc coei-
ents). If one ecializ the ocic pcess olvi that
In a f cas n obtai atents on
he dibion of the luon. A
ele of ch a re,It we
rer t the paper of G.E. beck
L.S. Ornen 8]. ey
probl then one succeeds
d such ochastlc inputs wch do not poss a "dsant eect",
.e. the
ce
.
sss
valuea of the pcess d not
between the obatlon points is
a lation if the
As a result they
c sh tt the solution of a special nitial value problem wth
ch a process without "distant effect" as the gh side is approx-
tvly a -called Oein-lenbeck-pcess,
wch the fir diibutlon ction
s
.e.
a process for
a Gaussian dstrlbuton
ction.
e
pcess without "distt effect" were defined exactly by
the uthors in the paper
6 toh
the pcess class of the "weakly
correlated processes" and they were applied in this paper at the con-
sideraton of ocstic eigenvalue problems d bodary value prob-
115
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
lems. A limit theorem is obtained for the eigenvalues, eigenfunctions
of stochastic eigenvalue problems respectively for the solutions of
stochastic boundary problems, with weakly correlated
coefficients.
Ths
1mlt theorem shos the appro:mate Gaussan dstributon of the first
distribution function of the solutions of eigenvalue problems and
boundary value problems.
In the present paper the conception of the weakly correlated proces
s
defined more gvnerally than in the paper
K6S
(.e. the sta-
tonarty of the process falls out the suppositionS. The correlation
Yenggh
enotes the ,tin,mum distance between bservaton points of a
wly CO:Telate process that the walues of the process do not affect in observation poimts which possess a dstance larger than
la section 2 a few theorems wil be proved about functonals of
weevily correlated
a
rocesse which
re mportant for the apllcaon
eigenalue problems, boundary value poblem$ and nita value
problems in the following sections:
Section
,
deals wit sochasic elgenvalue problems for odinary
dilerentisl equations wth deterministic boundary conditions where
.
the eoefficient of the difTerential opeato are independent, weaEly
eo--elated processes of he eorrelstlon length
egenvalues and the eigenfunctions, as
We prove hat the
550 possess
a Gausslan distri-
bution. For instance the eigenfunctions of the tochasic elgenvalue
problem converge in the distribution as
0 to Gausslan processes.
Methods of the perturbation theory are essentially used. In a general
example it is eferred to a few remarkable appearances.
In section 4 we deal with stochastic boundary problems and we
tan similar results as fo stochastic eigenvalue problems. The case
of a Sturm-Liouville-operator with a stochastic inhomogeneous tem
J.V. -SCHEIDT-a’qI W. PURKERT
i16
(y corelated which was dealt with by .E. Boyce
cluded
n
n []
is
n-
the result of thie section. Soae limltatons relative to the
Iness f the toehastic coefficlents of the operator but not of
the’ nhoageneoue term ae assumed a at the eigenvalue probleae, too.
e ehow a calculation f the correlation fYmctlon by the Ritz-aethod
to eiminate the Green function f the average problea from the correlation fYhnction of the limi pocesa of the sluton of the
bouary value problem.
At last, sectoa 5 deals" with stochastic nitial value problems of
ordinary differential equations. The inhomogeneous temas are weakly
correlated processes. The result
related procesees as
n
this theory of the weakly cor-
It-theory if
0 esemble the results of the
the nhomogeneeus temms are replaced by Gaussan white
m to the I-theor and the formed ItS-equatlon s
noae
accord-
solved according
to the It-theory. The practice by the help of the weakly correlate
procese iffers princpally from the It-theory. One obtains the
lmit theorems by use of the weakly correlated processes f at first
a formula for the a.s. continuous differentiable sample functions of
the solutions is derived (%0) and then we go to the limit (0).
ith it we get an approximation of the sIutfon of the initial value
problem with weakly correlated processes (<<
-theory one goes, to the
limit
,0). In
the
It-
in the differential equation and this
equation is solved by a well worked out mathematical theory. We gt
different results at this different practice in problems of differ-
emtial equations in which the coefficients are weakly correlated pro-
cesees and not the inhomogeneous term. We do not deal with such problems in this: paper.
I% is principally no distinction in the proof of limit theorems
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
117
for the nitial value problems and boundary value respectively eigen-
value problems with weakly correlated processes. A widening of the
l-theory on boundary value respectively eigenvalue problems with
white noises for the coefficients seems to contain a few fundamental
difficulties.
2. WEAELY COREELATED PROCESSES
YNITION I. Let
and 0. A subset
(xl,x2,...,xn)
be a finite set of real numbers
(x1’ x2’’’’’xn)
(xi1’xi2’’’’’x’Ik)
is called -ad-
joining if
xi. j=1,...,k,
is fulfilled for the
which are arranged after the
Xrl,X r2 ,...,Xrk).
quality (these we have termed as
number is always called -adjoining. A subset
(xl ,... ,xn)
(x ,...,x.
is called maximum -adjoining (relative to (x
if it is -adjoining but the subset
joining for
A subset of one
xr (x ,...,xrl\(x
Every finite set
(x ,...,xn)
(xl,...,Xik,Xr)
)=_
,... ,xn)
is not e-ad-
,...,Xik ).
-adjoining subset s.
DEFINITION 2. A stochastic process f(x,) with
Ef{x,) =0
(f(x,)>
is called weakly correlated of the correlatfon length
when the relation
..
<zx )...f(Xn)>=<z(x ...(xp ) (f(x1 }...f(xep2)>
<fx
s
tisfied for the nth moments (for al n ] if the set
x
,...,xn) lits n
If the pces
leth
,
-
split unique in disjoint maximum
th
f(x,)
mam
is
k
-adjoini subsets
wey
correlated with the
then we get for its corlaton ction
(pi=n).=
corlation
J.V. SCHEIDT AND W. PURKERT
118
f(x}ry)p
(x,y)
=
0
where
K Cx)- yg R
for
y Kt(x)
for
y K(x)
|x-y|a.
The existence of especially
has been proved in the paper
aatonary weakly correlated processes
6.
In this paper it is also proved
that weakly correlated processes with smooth sample functions exist.
In the following a few theorems will be proved about weaEly
cot-
related processes. These theorems will be used essentially in the
applications at equations of the mathematical physics.
THEOREM I. Let
fE (x,)
be a sequence of weakly correlated pro-
cesses as 0 with the correlation functions
<fCx)f(y) =
[
(x’y) for y K(x)
for
0
y K (x)
where is fulfiled
R (x,x+y)dy _i
lm
o
a(x)
foy in x. her on let
and a(x)
gi(x’Y)’ i=1,2,
ferentable ctions relative to x
0
be in
( 0b i > a)
[a-,bi+
dif-
and
i,x,
en
G2ffi[(xl,x2):
Ix 1-x21e, aaxi
i (x,) =
lira
$o
il
then we get for
i f (y,)g(y,x)dy
a
the relation
<rl )r2 (x)}
(x
b
=
n(
R00F. We v set b 1 b and
2
II
11<rI (x 1) r2 (x2)>= g
a
,b 2)
a(Y)gl (Y’Xl)g2 (Y’x2)dY"
b 1. We substitute n
(Yl)f (Y2)>gl (Yl ,Xl )g2 (Y2,x2)dY12
119
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
wih h(z
,z2):$ (z ’=1 +z2)g! (=1 ’Zl )2 (z1+=2’z2)
{==,
lhCz
,z2) A Cz
lm
"
F.
/V
)$(z1+=2,z1+))1/2C2
,z
12 =lml
gl
(=1
)
it follows
Ng(=I l+2}2(1+=2x2)d=1
At laa e obtain
=
o I12
]
=
gl
lira
f the
a
RS(z1 ,1+=2)g2(z1 ,x2}+O(z2)] 8z2dzl
(=1 ’Xl
gl
(=1 ’Xl )g2(Zl ’z2) lira
ction g2(1+z2,z2)
developed
relative
o =2 a
z2=O. e
theorem is proved.
Before we denote the more important theorem 3, we prove a simple
theorem.
[(z1,...,zn}EWn:(x1,...,xn) E-adjoinin. Let (x1,...,zn) be a
cton which is in [(x1,...,xn) :a- x bi+,i=1 ,...,n limited:
(x 1,...,xn)
C. en the ntegral
g(x1,...,xn)dx 1...d is
G
at lea of the order
PROOF. By
I
for
n(bl,b2,...,bn)
g(1 ’" ,Xn)1
where the
a
d
Cnl
x+ 2"’"
x
n
EO.
it is
I
a
Xn_1+
S
Xn_
x
dx2"’"
-I
n=Cnl
denotes the vole of
(-a)
J.V. SCHEIDT AND W. PURKERT
120
and we get
H1...n
1...n
G
points in Gn with x i
results from
On
-
G 1"’’in marks the set of the &-adjoining
xi
x.
z2
vW(1 ,...,n
Then the above given inequality
n
)I17;(il....noomn
i1"’’in
)
and wth it the tatement of the theorem2
f (x,)
THEOEEM 3. Le%
(:esees
and
as
e O.
cj.(x) Cj.
be a sequence of weakly correlated pro-
The absolute moments
Let
gi(x,y),
<If(x,)l J>=cj(x)
are to exist
[a-,bi+ ]
i=1,2,...,n, be in
differenti-
able functions relativ to x (q > O,b i> a) and
i,x,y
Then we have for
ri (x,)=
if (Y’)gi (y,x)
dy
the relation
l.m<rlt (xI )r2 (x2)-.-rn(Xn)
’
(i I,i 2),...,(in_
=
=
Iim<r.Xlg ri..>lim<r
,o
5r
1,in)
..
0
eends for all splittis of
The
th
ir 4 1
equal wch differ
toh
"’’,(n-l,in
e
"n-1 t
for n even
for n odd.
(1,...,n)
in pairs
par (ir,il).
rn
(I’i2)’
Splttis re
the sequence of the pairs.
pof of this theorem subts silar to the proof of theorem
6.S
7 in
fer every
lim<r
go
by the help of the above given theorem
the proof of the
and it also bmi%s
followi theorem 4 from
the proof of the theorem
flz (x,),"’,fn(X,9)
be sequences of independent
s n [].
THEOREM 4. Let
weakly correlated processes as aO. Let
be in
[a-,bi+],i=1 ,...,k,
gij(x,y),
=1,...,k,j=1,...,n,
differentiable fUnctions relatv to x and
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
c-n
I...
2
p
e
in the te for Ap
CI,...,i,P )
r; p)
f
r( )
p
ip
p
J
121
) rCP
lira<tOP
i
o
\
"
eends for all splitters of
in pairs like in teorem
As an application of the theorems
to 4 we will prove the
theorem 5.
THEORE 5. Let
fi (x,w),...,fn(X,W)
be sequences of independent,
weakly correlated processes with eontlnuous eample functions as the
correlation length 0. Let
[a-,b+]x[a,b
gj(x,y)
be differeniable functions in
relative to x with the condition8 of he limited
like in theorem 4. Then the stochastic vector process
r(x,) = (ri (x,))T
with
r(x,)
=
j=1
converges in the distribution as
)
a
gij(y’x)fj ,(y )dy
0 to a Gausslan vector process
I) Matrices are denoted by underlining.
(I)
J.V. SCHEIDT AND W. PURKERT
122
n
proceseee
Cx,)= (Cx,))T &
=I ,... ,n,
A m’
are ndependen d
aj(Y)gpj (Y, Xl )gqj(y,x2)dY)
.e.
all the dibution ctions of
the adequate distribution
(x,). I
0,
E(x,)
p,q m’
0 to
converge a
ctons of the Gausan vector process
is for the weakly corlated pcesses
fig (x,e),
Zo
A f we calculate he l of he k-h
the theorem
a
i=I ,...,n,
momen
and 4
<rat& (Xl 1a2& C2 )" "rak&()>
d we obtain by
with
ap
r.(x,)= r: :(x,),
X
=’
then
n
ak
#o
v_=o
"-nk
II splittings
of (1,...,k) in
(t ,...,i
:
(
,...,
n
I
o
A
VlO
xp a,b
[I ,2,...,rot,
n
j=Ira
An
t
in
2v
(2)
n
2vi
It is
p
$(x:[p)ra.p
+/-1
:12
,r
and from theorem
aiP
P
(x.
",wp
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
lim<ra .(xi) raj p (xj)> =
123
mn(x i x.)
Sap Cy) gaiP Cy x)g Cy, xj)dy.
ajp
#o
i
a
We introduce the Gaus prcess
P(x’@)ffi(ip(X’))T_1iam with the
ments
<P(*I )P(x2)T>
min(x
a
ffi
and obtain
ip
i
p
i
x
2ap (y)gip(y’x)gjp(y,x 2)dy)
P
i
2v
are pd ndepende fm (2) wth (x,)1 (,,W),...,n(x,)
T
( (x,))the
IAm
fola
nm <
en om
#x
.
<%,
k
(5) we obtain
x ...x s (tl ’’"’)=P(r(xt) I ’’’’’(xs)
(r,...,)
%or
)
d
dmtes the distbu%on fetion of the Gaussian
(x ),... ,(:s) ). e
heor 5 is
OC EIAL 0
.I. We ard
L()U &
the ochaic
(-I)m[fm(x)u(’)
u (k) (0)
- u (k) (I)
] (.) +
egvaue
pbl
(-I )r[f(x,w)u()] (r) . u
No
(4)
O, 0,I ,... ,I.
r(X’W)r(x’)-<fr(x)>
for 0, r4 m-1 be ndependen,
weany
correlated protests h the correlati leith E d a.s. all the
ajetoee of
fm(x)
r(x,)
be continuous.
e
detenistic ction
ha continuous m-th rder devatives. Fther
S)(x,)q
is asked
o
be with a I
0,1,...,r
0,I,...,m-I,
for
a 0,I,...,I. e nctions Cr(X)<fr(X)>,
ssess such ppeles that the avered problem
%o (4) (see
[4])
124
J.V. SCHEIDT AND W. PURKERT
m-1
wCI)CO)
.
= w Ck) CI) = O, kffiO1,...,m-1
is positive definite. Then L() is also a.s. positive
definite for
small
Let
()
be the eigenvalues of Eq. C4). Further on let
the
egenvalues and wl(x) the eigenfUnctlons of the
averaged problem.
1
We assume that the eigenvalue
de.elopment of
()
() = 3 + |I ()
with
11()--bll()
/
are simple. Then we have the
21 () +
(convergence a.s.)
and of the eigenfUnctons of Eq.(4)
ulCx,) = wI(x) + UllCX,) + u21(x,) +
(convergence in L2(O , 1) a.s.) with
_I b
u11(x’’) =
(see .4S). It is
piji-j and bjC):CL ()wi,wj) e
L ()u
e
L()u-{Lu =
(-I)r[rCX,)uCr)] (r).
o
followi imposer theorem 5 is proved in the paper [5].
6. We get for the tes l(x’) d
of the
2()
develoents
of
()
I ("’ =r ,.., rk:o
( ) where
!"’, (Y’)"’’rk (Yk)r ..r
(Y,, "’", Yk) dY,
’(P)
(xy, .-.,yklC C([0,II I)
1...r(X;Yl
,’’’,Yk;-%1...r
k
for Opm-1
and
...rCY,.-.,Yk
Paica
we obtain
C(O,]).
dYk
f
G(x,y} s
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
125
the generalized Green function to the operator
L-
conditlons u (k) (0=u 0t (1 =0 for =0, |,... ,m-!.
For the following we define the notatfon
and J:he
boundar
IC)
C) A
rot
= o
for i= I, 2,... are given numbers. Then we obtain, as
giTen above,
i) =
wen
we
each
o ki
()
(convergence a.s. )
additonall3 ase the conveence
[0,I. Hence we ve
of
(,)
a.s. for
(lr
w
these
neons we
l"*’r(X[;Yl’’’"Y)
for
pve the follo heorem 7.
quences of ndependent, wesley correlated pPocesses with the
correlation lemgth $$ 0 where
0
and lm
o
(x,x+y}dy = ar(X]
for
y l[tCx )
foy In x.
en
the nd
ct
coteries
e
n
te dstbuion to a
rdom vector
au
rdom vector
It(Zlil ,...,tlsis} T, t=0,1 ,...,I, are
ndependent d we
ve
Iq
(y) dy)lp,qs"
J.V. SCHEIDT AND W. PURKERT
126
PROOF.
e calculate
/C^ Cnl.
the k-th moment
(A Cn
a
wth
n
laa laka
aq [,...,s_,1 ) ()I_
()
li
d_
ts
show that
men
converts to the adequate k-th moment of
AI fir we calculate the order of a te of the fo
as
0 if
...r
tiies the condition
..rp
Because of the independence of the pcess
i(x’)’
we
mu
deal
wth tes of the fo
(5)
.
when w wt to calculate the order of P
o )
L...
since
0
0
o
d
It follows
.d
is bonded. By the use of
l(eqa(,) fq( ,pq )>I d, dPq =
(e the proof of the adequate theorem
ts paper) we obtn
(
)/2
al
/UAI
Q&
for p odd.
ak
consideri
a
theorem 6
k k
n
pq
to the theorem
for p even
With ts result we can now show
n
n K6
fo
Pp
[;(p/2)
t(
al al
0(
estimation of the order of
we obtain the estimate of
I1
O(
odd
o
and hence
127
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
AI
i
al
0()
o($) for k even
for k odd
d hence
lira
.((n)]a_Aola,.
(x(n).
).
)>
-A__
[ <AalAa2 ...ak
l’m
Zo
=
0
q
fo k even
o
o
% (Y)t , i () d and
Inq]aq
=onsidmrstons as in the pof of theorem
we obtain the theorem
We will show in the followi considtions that the theorem 7
nce
.
(n)
-Ai)
imE<(Ai
n@
ifoy in
bounda
)T.
s-
’IiI
)T
s-
m
inead of
For this we denote the conve
= o
We will deal with the idea of ts proof at the
value problems because we use for the proof dental
estimations from the perturbation
Then from ts convergence in
theol.
L2-me
it follows the
conveence
in probability ifoy in
(n)
P-lira
n.=--
li
-Aoli
=(Ali -Aoli
and then
ifoy in
.
We have sho in a first part of this proof
(n)(t I’’’’’ t s ) =$ s (t I’
lira "s
$o
s)
for
a
na
if we set
F)(t1,.. .,is)
pC
#s(t,,...,ts) = P(?
1(A(n)111_AOll I t1’’’’’Alsis(n)
)&
i
< t,,...,{
Isi s
t s ).
(9)
J.V. SCHEIDT AND W. PD-RKERT
128
--Fes(t1’’’’’t s)
Let
_
be the distribution function of
)T
"’’’sis-lS.FEs()(tl,...,t s) -Fs(tl,...,ts
Hence we obtain from relations as in Eq.(8)
n.no() uniformly in
s(tl ,...,ts)- __._I F[s(t
every 0. Then it follows
o s I’’’’’ s
for all
for
)
and with Eq.(9)
,...,t s)
Ss(tl ,...,ts)+
’’’’’ts)
s
(10)
and by it the theorem 7.
Now we consider a few important special cases of theorem 7:
()
(()-t%
n
,o
the distribution_
o,
random variable with the parameters
m-1
=
= O,
at(y)(
’12o>
<Io
(b)
(ul(x,)-wl(x))
wl(t
in the,
where
Io
is a Gaussian
(y))4dy.
distributin&o
l(X),
where
l(X)
a Gaussian process with the parameters
l(x)> O,
<l(X)l (y)>
=
at (z)- Ozt
t=o
(Wl((z))
t"
dz.
By theorem 7 the distribution function of the random vector
(Ul
(x
)_(Xl ),... ’Ul (Xk)_WI (xk))T
dsributon function
converges
0 to the
random vector
k) of
Xl...Xk(tl,...,t
vector wth
Gaussian
the
(11’’’’’Ik)Tm-1which is a
random
m-1
q
P
t=o
t (Xq (wl(t) 2dy.
m-1
=o oat G (Xp’y)---ly
11-11,’’’,Ir-I ’r ulr+1(x)-wlr+1(x)’’’’’ulk(x)-wlk(x))T
=
(y)
y)
I
(y))
Hence the case (b) is proved.
(c)
in the distribution_
o
where
(
(x)=(11o,.. .,IrO,Ir+ (x), I.
’IrO’ll(’ Ik (x))T
,Ik(X ))T is a Gaussian
o’
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
process with
<_(x) =
= ICwl(t)(z)w
m-1
h
s
o
P
at(z)
0 and
(t)(z)12 )1p’qr(V tqP (y
zll
lq
(Wpq (x,y,z))
I pk
1qar
n
)
r+lqk1pr
dz
t
t
(Vpq (x,z))
t
129
(x,z)
(w(t) ())
wt
I p, qk
()
ease (c) also follows from theorem 7.
3.2. The results of this section can also be used of eigenvalue
problems of the form
Lu
Ui[ul
%h(x,)u,
(11)
O, i=1,2,...,2m, Ox&1.
The operator L is a deterministic differential operator of the order
m
(r) (r)
2m
(12).
]
Lu
(-1)r[fr(X)U
where the coefficient
fr(X)
is continuous differentiable of r-th
order. It is for the stochastic process
ho(x)+g(x,). Let
ho(X]
h(x,)
be positive d let the
weakly correlated process of the correlation
g(x,)
conditions
(
sufficient small).
Ui[u]=O,i=1,2,...,2m,
e
h(x,)=
process g(x,) be a
the equation
eth
with
deterministic bounda
are constituted in that maer that
the problem (11) is sefadjoint and positive definite.
By the made suppositions the averaged problem to (11)
Ui[w]=O,i=1,2,...,2m,
0
I 2
wl(x)
Let al eigenvalues
be simple. The eigennctions
are asked to be ohonosl in
L2(O, 1)ho,
i.e. we have for
k,l=1,2,...
(Wk’Wl )ho
We define as
n
LhoW
possesses enterable many positive eigenvalues
k(x)wl(x)hO(x)dx
w
o
section 3.1.
1"
J.V. SCHEIDT AND
130
u
0,.
with x i
li()
lxi,)
.
PURKERT
for
Thn a dvlopmnt xists
llil)
for which we asse the ,onvergence of
Ali=
I
for i:0
where
Gl(X,y)
denotes the generalized Green nction to
wl(x i) for i#O
and
the boundary conditions
b (w)
IIi- o
li=
Gl(x,y)g(y,W)wl(y)dy
Ui, i=1,2,...,2m,
(w ())2g(y,) dy.
g (Y’)i (Y) Y’
It is
for i$0
ho
d
d
By
I
G. (y)=
-I (Wl (Y)) 2
for i=O
G(xi,y)wl(y)
for i@O
we can foulate a to theorem 7 adequate theorem.
TOS 8. Let x 1,x 2,... be from 0,I and let g(,) be a
quence of weakly correlted process with the correlation leith
and
for
(a(x)O)
limo
y K(x)’
a ,x+y)dy=a(x
uniformly in x. Then the random vector
s-oI sis
)T
i
i ,’"
11
11
formed from the stochastic eigenvalue problem (I
’41
with
g(x,)
converges in the distribution to a Gaussian random vector
11ii"’’’ Isis
q pi
p
)T
with
0,
pi plqi q
=
o
a(y)IGp (y)lqGiq (y)dy.
RE,AEK. We can also prove such a theorem for the more general
stochastic eigenvalue problem
Lu + L ()u
A(hoU
+ M
(W)u),
Ui[u I
= O, i=I ,2,...,2m, O gx
Here the operator L is a deterministic differential operator like
in Eq. (12) and
m-1
L ()u
_-6aok(X,)u (k)
m-1
M ()u
=bk(X,) (k)
I.
131
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
with
L ()B = (M ()
=0 where the coefficients
ak(x,) bk(X,)
are independent, weakly correlated processes. Perturbation serieses
relativ to L
in
(), M () form
the basic. These serieses were deduced
5S.
3.3. We regard the stochastic eigenvalue problem
-u" + a(x,)u = (I + b(x,))u, u(O) u(1) = 0
where
a(x,), b(x,@)
denote independent, weakly correlated processes
of the correlation length
-w"
=w,
.
The averaged problem to (13) is
w(O) = w(1) = O.
This averaged problem possesses the simple eigenvalues
and the eigenfunctions
Wl(X=sin(lrx).
a(x,) and b(x,)
aCx)a(y)
Re(x,y), <bCx)b(y)>
(Ra(X,y)=Rb(x,y)=O for y K a(x)) and
lira
I. Ra
(I)
(x ,x+y)dy = (x)
l=(llr)
2
It is for the processes
=
RbCX,y)
.Rb(X,x+y)dy = (x).
lim
o
By the supposition that a(x,) and b(x,)| are smal we obtain
the development (with the notation as above given)
Ali()
li=
i
the formula
=
()
li
k=o
where
for i=O
is and with
(x i) for
0
i
(y)
_li= o--(a(Y)-b(Y))(y)dY s
1
l(wl(y))2
_G
for i=O
l(x i,y)wl(y)
for i$0
right. It is
sn(Ivx)-xcos(11x) sin (11y)+ (l-y) cos(11y) sin(lx
Gl(x,y):
for OAxy1
1
s n(ll")/(-x)es(l’’x) in(l’y)-es(l’y)
.iiiGaussian
converges in the distribution to
By theorem 7 the random vector
the
(I 1’’’" ’lss )T wth
s-
random vector
)T
J.V. SCHEIDT AND W. PURKERT
132
q
o
q
Particulary we obtain:
llpip
P(Io t’ko ’’ s)
(Gaussian distribution) we have
Ik(t,s)
_)wwdy) p,q
(<PO)P, q I, k=( (+
lira
with
=0,
->
<.pu
io
Particulary for
eigenvalues
f
bO
and
q
p
Flk(t,s
!
1,
wide-sense stationary, weakly correlated processes
__
aCx,), b(x,) are
wide-sense stationary, weakly cor-
relsted processes. This is reflected in a melt of the eigenvalues
with
inereasi nber (see
--,--
Fig. 2).
-#
#
/
Fig. 2
i
#
By b-zO the variation of the limit distribution does not dependlon
the number of the egenvalues. In this case the corre]ation
loko=
for
k#l
is independent of k and I. This effect is expli-
cable from the fact that the operator L(@) determines the eigenvalues respectively the eigenvalues
()
for a
the random operator L(@) the other eigenvalues
o determines about
()
for this
o"
STOCHASTI DIKENIAL EUATION PROBLEIS
(b) The random vector
dstributioh as
wh
<,[p(x)>
for p, q
:
Ik,.
(ul(x)-wl(x),uk(x)_wk(x )T
133
converges in the
I0, to the Gaussian vector process (I(x)’k(x))T
O,
P
o
P
We can also calculate the correlation values of ths Gaussian
vector process from
<p(X)(x)>
Ulp(X,) denotes the
Wp(X) relativ to a(x,)
x
if
= lm
Zo
<Ulp(X)U q (y)>
first term in the development of u (x W)
p
and
b(x,w). One obtains
o
PP o(1-t)WW’p..pdt
after a few c a Iculations with
(x,)=a(x,w)-pb(x,)
P
-w (x)
(14) the correlation function of the limit process of
(14)
and then from
],(Ul-WI)
x) )w. (x)w
o PURKERT
J.Vo SCREIDT AND
The figure 3 shows this variation for the parameters =I,2,3,4.
e
ea ske very good statements about the behaviour of the limit
pro(C)esm
of the egsnfunctions because of the limit process
l(x)
is
a auealan process.
4. ;. We, cnsider the stochastic boundary value problem in this
aection
u[u]--o, --,,...,m,
The boundary conditions are constituted so, that
m
m
t
(r)
r=o o
r=o o
f right for all permissible functions u,v (i.e. functions, which
pseeas 2m continuous derivatives end fulfil the boundary condi-
tions), i.e. the boundary terms of
-
the integration by parts
mue
be zero. Then the stochastic operator L() is symmetric reltiv to
aI1 permssdble functions.
r(z,) fr(z,) -fr(Z,)),
r m-;
(0
(,)
(,)
are assumed to be stochastic independent end weakly
correlated with the correletion length
and
fm(x)$O
must be e
eermintstic continuous function. Further on (s)
aasumsd to be ith a small
,
s=O,,...,m-;,
and the processes
-
appearing must be almost everywhere differentieble. The boundary
vaIue problem () cen be written
n
()u=(x,)/(x,),
n()u=(Du/.
mm-;
e assume that
0.
{L()}w=O, Ui[w=O
,, ,
the following form:
u u=O,
possess only the trivial solution
135
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
We make for the solution u(x,) of (I 5) the statement
uCx,
when
=
.
k=o
denotes the homogeneous part of k-th order of
uk(x,)
in the (C)oefficents
r
and
Substituting this statement in (15)
eads to the boundary value problem for
(Lu
u(x,)
Uk(X,)
Kg>;
o
L>u
<L>uk
-L
U[Uol 0
L 1()uo; Ui[Ul] = 0
(W)Uk_ Ulink] = 0
Let G(x,y) be the Green
(i=1,2,...,2m)
for k=2,3,...
function corresponding to
<L>
(17)
and to the
boundary conditions U.[.=O then we obtain
(x,y){g(y,)dy,
u o () =
ioeo
o
is a deterministic funetiono ih (16) fr
Uo()
u()
(18)
m-1
o
G(x,y)(y,)dy-
rCY,)u
(r)
(y) rG,(x’ y) dy
yr
r=o o
obtained.
We get the following theorem that is right analog to the theorem
6 for eigenvalue problems:
THEOREM 8. It holds
u
and for
=k
(x
o(x)
=
O
k=-1,2,...
, =r1, m-J =o !’"lrl (Yl) "r(Yk rl
rk
)
+
"
r 1,-.rk_ I=o
Uk(X,)
for the tes
’"
o
rk (x;Yl
"’’ Yk)dY dYk
r (Yl)’"?rk_ Yk-1 )g(Yk
Hr ..rk(X;Y, ,k) dyl -..dY
2
k
of the development of the solution
u(x,)
the boundary value problem (I 5) with
r(X;y)
=
r
uo(r o.r
2
Hr(X;y) =
G(x,y)
and
of
-
J.V. SCHEIDT AND W. PURKERT
136
..
l"r
2
(x;yl’" ’yk)
:
(z;y ,... ,y)
ffi
(xmYk))’Hrl
(x,yk_
rk-1
))
C;...) C1)C0,1).
Cx;...) and
Cx;...)
00F. he atement followm for 1 vm C18). Ater e hate
mbaihed the fIa for k-1 then t followv for
O
frCy,)
:
r
II
I
I
Cy)-
,..,>-_ s-- rl ()"" r-2 (y_}r(y)(y_)
:
_2:o o
rO(’Y)(r)
(Y;Yl "" ’Yk-1 )dYl"" Yk-1 dy.
The theom 8 is ped.
Now we deal th questions of the convergence of he delopment.
OSG.(x’)
r,0,,...,2m with 0,s&2m s continuous
n O,xy respectively in Oyx, it follows for
(x,y) O,x[0, and the stated r,s rS
C, ther on
Sie
we have
for
(r)(x)l
C0
Uo
(x)
Irp(r)(x W)
C,C 0
a
for O&r&2m and
for x
E [0,11
x [0,I
(19)
and
almost rely for
Op&m-1,0ra,x[0,1].
and a are constants. By properties of the Green"
ction
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
u(2m)(x,&; = o -,.ox2m
u(P) (x,)
=
E(x’Y)
=
o
and therefore
m-1
o
for aost alI U then
r
=
lu ) =,=
d
Hence
0&x
I
o
when
condition. t
"",
C o o (v
v=o
o
=(c*)c o
almost all
=E[0,1].
(x,u)I converges
tt ((C+a))-1 for
(P)(x,@)
L %-, (z)
p ,.--, 2m-I
%-I (y)dy
r -(v) (2r-v)
(v)fr
Uo
throh induction for
for Op2m and
f=z
xafo,] t%-I
Hence we obtain th
IL luo(x)I
LI_
137
almost surely and ifoNy in
almost all
also converges unifoy in
Uk(X,)
0x1 th this
be calcuted by (17) then
luton
(x,@)u
(5). Ro-e folae
is the
of the =tochastic
u(x,)=
boda
81ue pblem
he heoem 9.
TO 9. If
e1
f(x), (x)
8e
sequences of ndepenen,
coelated pocess h he correlation
(x,y)
0
for
for
y$ (z),
Iim
ao
(u(x
,UI_uo(
ag
t+o
then the dom vector
,’’"
and
Zor
0
((x)() =
leh 0
,U(Xs,)_u
(Xs))T
which has been constituted by the solution
u(xu)
(x E [0,11 ),
of (15) and the
J.V. SCHEIDT AND W. PURKERT
138
solution of the averaged problem to (15), converges in the dsribution to a Gaussian random vector
-
(x,w})=(x,)+(x,).
Io
with
0,
,... ,m-,
Further on the random vectors
(r(
are independent Gaus rdom vectors with
]. This theorem 9 proves the
u(x,)
problem to (]5) u
o(x)
convergence
n
the diri-
(u(,)-uo(x)) whCeh has been constituted
bution of the processes
by the solution
(z))2d)i,j s.
zr
r
< (x, )> =o
,---,
of (]5) and the solntion of the
(x,)
to the Gaussian process
avered
th
and
o
m-1
rG(x,z) rG(y,z )(
+ Sar (z)r
r
u (r) ()) dz.
O O
n
(x,)KUk(X,9).
o
PROOF. We put
we see with similar
eons
derations as in the pof of theorem 7 d with theorem
)-% Xa )... ((xa )-uo Xap ) )
P
for p even
I
m$o <( (Xa
=
whe
ith
{ lim<u
@o
aq E {1,...,s.
m(x,) =
br(x, )
(Xal) u1(Xap)
0
If we set u
,
-I
for p od
1(x,)=e(x,)+br(x,)
No
G(x,y)(y,)dy
=_
r(y,)uo(r) ()
d
it follows as in the proof of the theorem 7
lm
+o
P
u
(Xa )...u
(Xap)B. = [
o
for p even
for p odd
(see (18)
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
,
where
lira
$o
A
lg
z
I splitt-ngfi
(1,...,n) in
of
v
v_+
A
o
139
Am-
.:1[vg i
i 2v
i?!
i m-1
eee
i=0
-vj=
(i
,
, go
!" im-1
Vm_
and
ig
ilg
2Vg
(i
ig’) ,.., ig
J2
’X2Vg
(Xaig.
a2Vg
2Vg-1
#o
and appropriate for Ar
"’’I
en
with b r instead of c.
the folas
lira
o
<br(X)br(Y) =
to g
lira
o
arZ)
G(x"Z)0z G(y.Z)r (ur)
r
ily the statement of the theorem for
n
r
U(X )-u (x s)
((x )-uo(x
The complete pof of ts theorem follows as
(z)) 2dz
)
In theorem 7 from
the
ifo convergence relativ to
lira
We consider
._(Up(X)Uq(X) with p,q kN
Then we obtain with theorem 8 d the
I
(X)>l’ ’r,
2q=’O
to pve foula (20).
follo
0
estitions
dy
0
(21)
Let h(p,o be a wetly corlated process t the corIatt
Ieh
,
h(y,)
ao ehe
d
I 2.
we estimate
J.V. SCHEIDT AND W. PURKERT
140
an integral of the fom
O
O
For
I2 1/(2)
an
for
I
the
2
we
ve
/ 2 g)
deMgne the vle of the points
-Im
adjoini spliti is
lYl Yl
lYl -Y21
and we .btain for all 1
0
o.
der he ondto
because the process
for which
Since
.
=I
2
...l<h(y 1)...h(yI)dy 1...dyI
0
(1...,yl) of ,I] I
(I )’ (Y2)’’’’, (Yl)"
22
(22)
fcor of te fo a in (22) h
r(X,),
(x,)
are independent. Hence it
foll fo (2)
<=p(X)Uq(X)1A 2(mo)q-2o2(q)2[m2C2+
ses
of the
p,q=l
(q)2(mc)q for
o0
l(p1 )(2)> dP1dE2
ImCi Z I.
The theorem 9 is
preyed.
2. Let
miic operator,
fr(X’)O
en
for 0, I,...,m-I,
.e.
let L a deter-
the statement of theorem 9 is right without
seone ction of the weakly correlated pcess (x,) lat-
e to
the aless.
oh
s
folows from the proof of theorem 9 or
a direct use of the theorems
and
on the solution of
this boundary value problem
u(x,) =
uo(x)
+
t (x,:y)(:,)e
0
with uo(x) =
IG(x,y}<g(y)>dy.
0
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
(u(x, )-uo(x))
Hence we obtain in this case that
distribution to a Gaussian process
(x)
converges in the
wh
I ag(z)G(,z)G(y,z)dz.
(x)(y)
= O,
(x,)
141
0
.E. Boyce deals ih this case of a sochasic bounda value
problem n [] and he shoed hat (u(x,)-Uo(X)) is in the limit
a Gauss random vaable
3. Let
[0,I.
for any x
be the eigenfctions of the
avered
i(x)wj(x)dij
w
d
the eigenvalues of
o
we can calculate the correlation ction of the limi process
operator
en
KL>
wi(x)
(x)
of
th
(u(x’)-u(x)) toh
w
m-1
(X)Wq(
with
(Z)Wp(Z)Wq(Z)dz
bpq
crpq A at(z) (uo(r) (z))
d
Wp (Z)Wq
Specially we have for a wie-sense stationary, weakly correlated
process
-r
<(x)(y)
th
ese
-r
ag
pW (X)Wp(y)+ nard---- pq
P
o
p,q=;
(r)(z))
(z)dz.
p (Z#Wq
statements can be proved easly wth
Cpq
(see also
(u
Cpqw p (X)Wq(y)
o
G(x,y)=
[7]).
Now we regard the bodary value problem (;5) with the conditions
given above. It denotes
energetic space
u(x,) of
H<L >.
i
a system of fctions of the
is system be complete in
HL >.
The solution
the equation L()u=g() is the minimum of the energetic
ctional (Lu,u)-2(g,u)
n
the energetic space. The Ritz-method
with the co-ordinate fctions
i
conducts to the
foowi stem
J .V. SCHEIDT AND W. PURKERT
142
n
(L=),)z
n) =
C,),
(2D)
=1,2,...,.
n
It is u
n(x,)=_1_ x(n) ()Tk(X)
solution
u(x,u)
the n-th Ritz-approximation of the
of L()u;g(). We can write the frula (23) in the
following form:
(n)
with
A_o =
_B()
( + BC))_z
((Lk,j)11k
-
_bo + _c()
b-o
jan
()k, j) )1k, j.n’
(L
A_o is
--o;(+/-j) li ,.n
=
-()
((g’j))Tjn’
= ((’j))Titian.
regulr through the conditions for
The matrix
n);b_;’tx(n)
Let
’on(n) ) be then e btin
fr the n-th 8it-pprxitin Un(X)
n (n
n
(x) +
+) k
Un(X,) = k=l ok
+ tes of higher order th first in
Kx
respectively with
)k
K(I(-n)
-
Xok k(X)
Uon(X)
k
n
+ tes of higher order thn fSrst Sn
h
((h
(z,x),h2(z,x))z-
that
((x,)-Uon (x ))
the
Gausan process
n
<n(Z)n(y)
(z
x)h2(z
Paicularly we
converges in the distribution as $ 0 to
n(X,)
th
n(X)_
=
0 d
%jpqXol Xoi Tk(X)p
obtain for i(x)i(x) (see
a
(r)
=
ag k(X)wk(Y) *
1
n
() 2_ (r)
+Karr=ok,((un) ’wk
"I
(24)
)
remark 3) and wide-sense
ationa, weakly correlated procesaes
n
<n(X)n(y))
b,
x)dz). From theorem 5 follows
,j,p,qkjpqk(X)p(Y)(agj,q
+>
Nok, j, p, q=
Z
I=I
)Wk(1)Wl (y)
143
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
an approximation of the correlation function how it has been represented in the remark 3. The formula is used by an approximation
of the correlation function
explicit calculation of
<q(x)(y)>
((x)(y)
and (24) is suitable for a
if the _Green function is
ealculable difficult.
.2. YAFLE. Let g(x,) be a wide-sense stationary, weakly
correlated process then the simple boundary value problem
-u" + bu = g(x,),
u(O) = u(1) = 0
with b=const, possesses the averaged problem
-w" + bw = O,
w(O) = w(1) = O.
__
This problem has only the trivial solution when b$-(kl) 2 is and the
Green fUnction
Gb Cx,y)
Gb(X,y)
=
is
sh(,x.).sh({(1.-y)
for 0
sh(y) ah((1-x)
for
H(’)
o
x<y
y x
Hence we obtain the correlation function for the limit process
b(X,)
from theorem 9
o
d the variance
2(x,z)d z
with
f(x)
= sh
=
a
2((I-x)) (sh(2x)
(f(x)
x). a is
helots to the wide-sense stationa, wetly
g(x,). I iafor bO and
th
f(x):-sin((-x))
bo
in(2x)-x).
b-(
+
f(1-x))
tie
pareter wNch
correlated process
Eecialy we have
b
In the following Fig.4 the variance of the limit process
b(X,9)
is
J.V. SCHEIDT AND W. PURKERT
144
potted for ,ome values of the parameter b. Since
we have plotted ths function for O
values submi for
x1/2 only.
<b2(x) =<b2(1-x)>,
The following
b= by contrast to it:
04
05
5,2382 .7,2570
0004
5. STOCHASTIC INITIAL VALUE PROBLEMS
We consider a system of ordinary differential equations of the
first order
dx
-
A_C t )x_
+
_C ,)
with the initial condition
matrix,
(26)
_X(to)o.
A(t)=(aij(t))lli,jan,
x
s
A(t) is a nxn deterministic
the vector x(t)=(x
o(i)’iAn^ and f(t,)=(fi(t,))TI, in is a
cess. Let
fi(t,)
continuous and
(.e.
stochastic vector pro-
be processes of which a.s. the trajectories are
Q(t,t o) be the principal matrix associated with A_(t)
_Q(to,to)=_l
A(t)Q_(,t o )),
i(t))T olin’
(I is the dentity matrix) and
(t,o)=
then the unique solution a.s. of the nitial value
problem (26) may be written in the form
STOCPASTIC DIFFERENTIAL EQUATION PROBLEMS
t
x(t,) =
Q(t,to)(o
_Q-1(s,t o)f(s,)ds)
+
to
the
The ntegral is defined by
145
integral of sample functions. In
generally we cannot calculate the distribution of the solutfon
x_(t,) f we
know the distribution of
Gausslan vector process, then is
vector process and in the same
system of linear
differential
Q_-I
to
way
f(t,). Let _f(t,;) be a
also a Gaussian
(s,t)_f(s,)ds
o
the solution
x(t,) of
this
equations (see /2/). The first
moments of the solution are
x(t)
Q(t,to)(o
R(t ,t 2)
to
-l<Ctl )-_(t
t
_
= _Q(t
with
K(t
I Q-1(S,to)_f(s)>ds)
+
,t 2)
provided that
)] [_x(tp)-(x(t2) T
,to) Q-I (s ,to)_K(s
to
-" f(t
s2)_Q-T(s2,to)dSl as2QT(t2,to)
)-_f(t ) [_f(t2)-f(t2)T,
<ll_f(s)llds
to
and
_f(s )fT(s2)l>dSldS2a_
to
o
t o (1.| denotes the Elidian no of Rn).
:for all t 1,t 2
In the folowing we consider the solution (26) if f(t,)=
E(t,)
denotes a vector process with independent, weakly correl-
ated processes
gi(t’)
as components. This leads to the following
theorem.
THEOREM I0. Let
processes for
gi(t)gi(s)}
$ 0
:
(t,)
a sequence of weakly correlated vector
with independent components
Rit(t’s)
0
for
s K(t)
for
sK(t)’
then the solution
t
with
gi(t’)
and
lmo E. Ri (t t+s)dsai(t)
146
J.V. SCHEIDT AND W. PURKERT
y(t)
_
_Q(t,t o) (o +
:
t
-Q-I (s’to)h(s)ds)
t
of the initial value problem
= _ACt)_x + _hCt) +
x
= -o
X(to,)
C,),
(27)
converges in the distribution to a Gaussian vector process
4(,) =
(t) =
wth
(t,)
y(t) +
0 and
min(l ,t 2)(ak(s)qik(tl,t ;s) qjk
(tl)(t2 )T =
o
to
(t
2’ t o ;s) I_ i ,jn de"
(qlj(t,to;s))|.i,n; -(t’to )Q--(a’to)
It is
PROOF. The proof is following from theorem.5 then we obtain
with the dfinition of the qi
n t
x(t,) = [(t) +
(qik(t,to;S)gk$(S,))iAindS
o
with
and from this for the limit process
n
the notations of theorem 5
t
(t,)=(i(t,))1i
n
i (tl)j(t2)
=
<ik(tl )jk(t2 )
k__n mnlt1’t2)ak(s)qik(tl,to;S)qjk(t2,to;s)ds.
In generally
(t,)
process while
on
t2-t I.
o
does not denote a wide-sense stationary vector
(tl)T(t2)
is not a function which only depends
We assume that the processes
giz(t,)
are wide-sense sta-
tionary and weakly correlated and A(t)=A is a constant matrix, the
of which have negative real parts. Then we obtain
egenvalues
T
for t
way the
formula Q(t,t)o=e(-tO)o->
t
eA (t_s)
)s
solutions
the
(h_(s)
eA(-)((s)+(.))s of
(s)
0 and in the same
and
the system of differential equations
differ by a solution of the homogeneous sy’tem of (2"/) which is near
by zero for t Z Tt o. Hence the solution of (27) is described by
_Z(t,w)
=
h(s)ds +
(s,w)ds.
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
147
eA(t-s)(s,)ds is wide-sense stationary as we
Then the term
obtain from
4 t-c,s>
:
o
o o
theorem 11 can be proved like the theorem 10.
e
THEOREM 11. Let
gZ(t,w)
be a sequence of wide-sense stationary,
weakIy correlated vector processes for
0 with independent
component s and
<g (t)gi (s)> = t
(ai#O),
(t-s) for
It-s.
for
It-s
0
then the solution
lim
(s)ds = a.ee
:
of the system of differential equations
d=
(t)
+
+
(t,)
converges in the distribution to a Gaussian vector process
t
& (t-s) h(s)s + (t,).
(t,) =
(t,)
with
is a wide-sense stationa, weakly correlated vector process
(t)> =
0 and
<<t1Zt> =
Indeed
(t,w)
we obtain for
rain t I, t 2
is a wide-sense
tl e t2
<t1-
min(I_. t2(t l-S))ik((t2-s))jk
<t-s
ds)
1i,j
en.
stationa vector process because
s <t-t+
=
" the followi we describe the coecton with the It
In
O
differential equations. The It8 differential equation
148
J.V. SCHEIDT AND W. PURKERT
(]’_it = (A_.()Xt + h(t))dt +
_Xt ()=_.xo
o
(27) as 0 where
_G(t)dWt,
corresponds to the initial value problem
G(%)=(i a-)1,i,, n
and
(28)
Et=(Wt)I,, n s %he ene process
independent components. This statement
s
followng from
correspondence of he integral equation
x(t)
=o t
(A(s)x(s) /h(s)ds
/
/
with (2V and of the It integral equation
t
_() = x + (A(s)X () + h(s))ds +
"
t
o
(s,ds
a(s)__s
with (28) if we take into consideration the convergence
(29)
n
the
distribution from theorem 5
Z,o
We note the
to
relation
,1
2G( st)d-ws)T> =
<toG--(s)dW-s
(%o-
for the It8 integral
the
It8 differential
tl ’t2)
mino
_G(s)_G(s) T ds
G(s)dWs. It is following from the theory of
to
equations that the solution of (28) (the solu-
tion of the integral equation (28)) is a Gaussian vector process
t
with
= Q(t,to)(o +
<Xt>
and
<(_Xt1-_Xtl > (Xt2_Xt
_Q-I (S,to)h(s)ds)
t)T>
in(t-?,2)Q-’(s t
)G(s)G(s) TQ-T (s t )dsQT(t t
oo
2’ o
to
A comparison of the limit solution (t,) from theorem 10 with this
solution from (28) shows the correspondence of the solution pro-
Q(tl,t o)
cesses:
_X(@)=(t,O).
Hence the same solutions of the initial value
problem (27) is obtained if one takes up the limit value
equation and the
It8 differential equation solves which one obtains
and if one solves (27)
limit value
&0
0 in the
n
the sample functions and
in the solution.
akes up
the
STOCHASTIC DIFFERENTIAL EQUATION PROBLEMS
149
We can make many applications (e.g. the number of threshold crossings
and other) because the limit process
_(t,) or [(t,), respectively,
is a Gaussian process. Hence we obtain approximately a good general
view of the behaviour of the solution
_xt(t,)
or
z_(t,w),
respective-
ly, if weakly correlated processes come in the differential equation.
REFERENCES
I. Boyce, W. E. Stochastic nonhomogeneous Sturm-Liouville problem,
J.Franklin Inst. 282 (1966) 206-215.
.
2. Bunke, H. GewOhnliche Differentialleichunen mit zufIRigen
Parametern, Akademie-Verlag, Berlin, 1972.
4.
urkert, W. and vom Schedt, J. Zum Mittelungsproblem bei stochastischen Eigenwertaufgaben, Beitre zur Analysis 10
(1977) 47-61.
W. and vom Scheidt,
Purkert!
Mittel.un_sproblems fGr
J. Zur approximativen LSsung des
die
Eienwerte
stochastischer Dif-
ferentlaloperatoren, ZMM 57 (1977) 515-526.
5. Purkert, W. and vom Scheidt, J. Stochastic eigenvalue problems
for differential equations, Reports on mathem, physics,
Torun (in print).
6. Purkert, W. and vom Scheidt, J. Ein Grenzverteilungssatz f3r
stochastische Eigenwertprobleme, ZAMM (in print).
7. Purkert, W. and vom Scheidt, J. Randwertprobleme mit schwach
korrelierten Prozessen als Koeffizienten, Transactions
of the 8th Prague Conference on Information Theory,
Statistical Dcision Functions, Random’ Processes, Prag
( 9’78) o’7-is.
8. Uhlenbeck, G. E. and 0rnstein, L. S. On the theory of
motion, .PhYsical revie.w 36 (1930) 823-841.
Brownian