Time Series in Linear Programs with Random Right-Hand Sides

Working Paper

TIME S E R E S IN

LDWAR

PROGRAMS WITH RANDOM RIGHT-HAND SIDES

Tomh5 Cipra

May 1986 WP-86-21

International Institute for Applied Systems Analysis

A-2361 Laxenburg, Austria

NOT FOR QUOTATION

WITHOUT

THE

PERMISSION

OF THE AUTHOR

TIME SERIES IN LINEAR P R O G W S WITH RANDOM RIGHT-HAND

SIDES

Tombb Cipra

May 1 9 8 6 WP-86-21

Working Papers are interim r e p o r t s on work of t h e I n t e r n a t i o n a l I n s t i t u t e f o r Applied Systems Analysis a n d h a v e r e c e i v e d only limited review. Views o r opinions e x p r e s s e d h e r e i n d o n o t n e c e s s a r i l y r e p r e s e n t t h o s e of t h e I n s t i t u t e o r of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 L a x e n b u r g , Austria

One of t h e t h e o r e t i c a l p r o b l e m s in s t o c h a s t i c o p t i m i z a t i o n which c a n h a v e im- p o r t a n t c o n s e q u e n c e s f o r p r a c t i c a l implementation c o n s i s t s i n i n v e s t i g a t i n g p r o - g r a m s whose c o e f f i c i e n t s a r e o b s e r v a b l e in time as time series. Some c o n c l u s i o n s d e r i v e d in t h i s p a p e r f o r l i n e a r p r o g r a m s u n d e r r e l a t i v e l y s i m p l e sLaLisLica1 as- s u m p t i o n s o n t h e r a n d o m r i g h t - h a n d s i d e s s t i m u l a l e f u r t h e r r e s e a r c h in t h i s d i r e c - tion.

The work was c a r r i e d o u t within t h e A d a p t a t i o n a n d Optimization P r o j e c t o f t h e SysLem a n d Decision S c i e n c e s P r o g r a m d u r i n g t h e s t a y of t h e a u t h o r as a guesL s c h o l a r a1 IIASA.

A l e x a n d e r B . K u r z h a n s k i C h a i r m a n S y s t e m and Decision S c i e n c e s P r o g r a m

AUTHOR

Assistant P r o f e s s o r Tom63 Cipra from the Faculty of Mathematics and Physics, Charles University, Prague works in the field of stochastic processes, stochastic programming and econometrics. He wrote this p a p e r during his stay a t IIASA in summer 1985.

ABSTRACT

Linear p r o g r a m s s u c h t h a t t h e right-hand s i d e s of t h e i r r e s t r i c t i o n s h a v e t h e form of multivariate time s e r i e s may be useful in p r a c t i c a l applications. Behavior of t h e p r o c e s s e s formed by t h e optimal values of t h e c o r r e s p o n d i n g o b j e c t i v e func- tions i s investigated in t h e following cases: t h e right-hand s i d e p r o c e s s i s ( i ) a normal white noise; (ii) a normal white noise with a l i n e a r t r e n d ; (iii) a normal random walk. Some b a s i c probability c h a r a c t e r i s t i c s of such p r o c e s s e s a r e calcu- l a t e d explicitly.

CONTENTS

1 Introduction

2 Normal White Noise

3 P r o c e s s with L i n e a r T r e n d 4 Random Walk

R e f e r e n c e s

TIME SERIES IN

LINEAR

PROGRAMS WITH WITHOM RIGHT-HAND SIDES Tombs Cipra

1. LNTRODUCTION

Let us c o n s i d e r l i n e a r p r o g r a m s of t h e form

where t h e matrix A ( m , n ) and t h e v e c t o r c ( n , 1 ) a r e deterministic and fb, { i s a m-dimensional p r o c e s s . Such g e n e r a l model may b e a p p l i c a b l e in variops p r a c t i c a l situations. The optimal values q ( b t ) of (1.1) (if t h e y e x i s t ) form obviously a s c a l a r p r o c e s s t h e b e h a v i o r of which we s h a l l investigate.

Let us d e n o t e

S

=

Ib ^t R m : q ( b ) i s finite

.

_(1.2)

Then a c c o r d i n g t o [6] or [7] t h e function q ( b ) i s c o n v e x , continuous and piecewise Linear on S . Moreover, S c a n b e decomposed t o a f i n i t e number of convex po- l y h e d r a l c o n e s Sf (i

=

l , . . . , k ) with t h e v e r t i c e s in t h e o r i g i n s u c h t h a t t h e inte- r i o r s of Sf are mutually disjunct a n d q ( b ) i s l i n e a r on e a c h S f . One c a n write

and

where H' are r e g u l a r ( m , m ) m a t r i c e s and g i are ( m . 1 ) v e c t o r s ( t h e v e c t o r s g f need n o t b e mutually d i f f e r e n t ) . One c a n a l s o w r i t e

q ( b )

=

max I g i ' b

\.

b E S

.

i - 1 . . . . , k (1.5)

T'ne explicit f o r m of

&

and g i c a n b e found by means of v a r i o u s algorithmic pro- c e d u r e s ( s e e e . g . [4], [ 5 , p.2761, [8], [91).

EXAMPLE 1 ( s e e [5]). In t h e p r o g r a m

one c a n Lake

The p r o c e s s [ q ( b , ) j o r i g i n a t e s as a piecewise l i n e a r (i.e. n o n l i n e a r in gen- e r a l ) t r a n s f o r m a t i o n of t h e p r o c e s s f b i { . If o n e i n v e s t i g a t e s s t a t i o n a r i t y of

f q ( b , ) j in d e p e n d e n c e o n s t a t i o n a r i t y o f f b t

I

t h e n i t is obvious t h a t frp(bt){ n e e d not b e weakly s t a t i o n a r y when f b l j h a s t h i s p r o p e r t y (i.e. when E b l a n d c o v ( b t , b, -,) d o not d e p e n d on 2).

EXAMPLE 2 Let m = 1, ~ ( b )

=

b' - 2 b - f o r b E

R '

( w h e r e b c

=

maxj0, b

1,

b -

=

min 10, b

I )

a n d bt b e independent random v a r i a b l e s s u c h t h a t

for a r b i t r a r y i n t e g e r r . Then

Eb,

=

0 , v a r bt

=

^1, c o v ( b t , b, -,)

=

0 for s

+

⁰

for a l l t ( i . e . ) b t { i s weakly s t a t i o n a r y ) but

If jbl j is s t r o n g i y s t a t i o n a r y (i.e. t h e joint p r o b a b i l i t y distribution of (btl, . . .

.

bti) i s e q u a l to t h a t of ( b t l + , ,

. . .

, bti +,) f o r a l l i ,

t i , . . .

, t i , s) t h e n [rp(bl)j shouid n a v e t h e same p r o p e r t y but one must b e a r in mind t h a t rp(bl) i s n o t finite f o r bt

<

^S. Moreover, t h e e x p l i c i t calculation of b a s i c probability c h a r a c - t e r i s t i c s of rp(bl ) { (e.g. t h e mean value and a u t o c o v a r i a n c e s ) may b e v e r y difficult even in simpie s i t u a t i o n s . In o r d e r t o d e m o n s t r a t e i t t h e c a s e with a two- dimensional normal white noise [ b t { i s studied in s e c t i o n 2. The d e r i v e d formulas f o r E rp(bl) a n d v a r rp(bt) are s o complicated t h a t i t t u r n s up r e a s o n a b l e t o recom- mend t h e simulation a p p r o a c h of Dedk 131 f o r a more g e n e r a l case. The case of a s t a t i o n a r y p r o c e s s lbt { with a c o n s t a n t mean value seems t o b e n o t v e r y useful in p r a c t i c a l situations. T h e r e f o r e a m-dimensional normal p r o c e s s [b, with a I i n e a r t r e n d i s c o n s i d e r e d in s e c t i o n 3. Finally, in o r d e r t o p r o v i d e potential generaliza- tion t o t h e nonstationary i n t e g r a t e d p r o c e s s e s of Box a n d Jenkins which are c a p a - ble to model t r e n d s in a s t o c h a s t i c way ( s e e [I]) w e d e a l with a one-dimensional normal random walk [bl { in s e c t i o n 4.

The following denotation will b e used in t h e p a p e r : a ' a n d A ' f o r t h e t r a n s p o s e of a v e c t o r a and a matrix A ; Ila(l

=

f o r a ^E R m ; d e t A f o r t h e d e t e r m i n a n t of a s q u a r e matrix A ; s g n ( z )

=

1 f o r z

>

0 ,

=

0 f o r z

=

0 and =-1 f o r z

<

0 : z t

=

max 10, z { , z -

=

min [O, z{.

2. N O M A L WHITE NOISE

Let [bt j b e a two-dimensional normal white noise, i.e

where C i s a positive d e f i n i t e v a r i a n c e matrix. Let T b e a lower t r i a n g u l a r m a t r i x with positive elements on t h e main diagonal s u c h t h a t

(Cholesky decomposition) a n d l e t us d e n o t e

= H < T ,

where t h e matrix

Q '

h a s t h e elements denoted as q:, and t h e row v e c t o r s of t h e t y p e ( 2 , l ) denoted as q: (u , v

=

1.2).

LEMMA 1 It h o l d s

PROOF If using t h e method of substitution we h a v e

k

= C JJ

( 2 n ) - l r e x p ( - r 2 / 2 ) d r dl9

i = I IT, a : r r 0, ~ ' ( r cos 6, r sin 6)' r 01

=

( 2 n )

C

k

J

^dl9

i = l ~ ~ : Q ' ( c o s 19, sin d), > 01

The l a s t i n t e g r a l s a r e e q u a l t o t h e values of t h e convex a n g l e s between a n d

-

s o t h a t (2.4) i s obvious now.

W e c a n p r o c e e d t o t h e caiculation of Erp(bt) a n d v a r rp(bi). S i n c e t h e p r o b a - bility (2.4) c a n b e l e s s t h a n o n e in g e n e r a l t h e conditional values E(rp(bl)(bi E S ) a n d v a r (rp(bi)Jbt E S ) h a v e s e n s e only.

THEOREM 1 U n d e r t h e p r e v i o u s a s s u m p t i o n s i t h o l d s

w h e r e P ( b i E S ) is g i v e n i n (2.4) PROOF W e c a n w r i t e

k

p ( b i ~ S ) E ( r p ( b l ) l b l t S ) = C

r

g " b ( ~ n ) - l ( d e t C ) - ~ ' ~

i = l l b : $ b > O {

e x p ( - b'C-lb / 2)db

k

= C

g i ' T

J

(COS 19, sinl9)'(2r) - l r 2

i = l Ir, d : r s 0, ~ ' ( r cos d, rsln d), r o j

e x p ( - r 2 / 2 ) d r d 19 k

=

( 2 6 ) - '

C

^{g i ' T}

J

(cosl9, s i n 19)'d 19

i = l 16: Q'(cos 6, sln d)' s O!

The v a r i a b l e 6 i s bounded by t h e a n e i e s c o r r e s p o n d i n g t o t h e couples of v e c t o r s ( Q : ~ , - q i l ) ' a n d ( - q t 2 . q : ~ ) ' (if d e t

Q( =

q t 1 q i 2

-

Q : ~ Q ~ ~ > O ) o r ( Q : ~ . - Q : ~ ) ' and

-

^{q q (if d e t gi}

<

0). Since J c o s 6 d 6

=

s i n 6 and

J

sin 6 d 6

=-

c o s 6 w e have

which i s equivalent t o (2.5).

REMARK 1 The formulas (2.4) and (2.5) c a n b e r e w r i t t e n t o t h e form

where h:(u

=

^1. 2 ) a r e t h e row v e c t o r s of t h e t y p e (2.1) of

H'.

I t i s obvious t h a t random v a r i a b l e s rp(bt) a r e mutually independent; t h e follow- ing theorem e v a l u a t e s t h e i r (conditional) v a r i a n c e .

THEOREM 2 U n d e r t h e p r e v i o u s a s s u m p t i o n s i t h o l d s

w h e r e

E(rp(b,)lbt E S ) is g i v e n i n (2,5), I is t h e (2.2) u n i t m a t r i z and

1 i i

P:

=

( - q 1 Z s qil)'- P:

=

(422s - ~ 2 1 ) ' .

PROOF One can w r i t e analogously as in t h e p r o o f o f Theorem 1

=

(T;)-'

2

k ^{g t ' ~}

J

( c o s 19, s i n 19)'(cos 19, s i n 19)d I9 T~~~

.

t = I [ ~ : P ' ( c o s 3, sin u ) , > 01

S i n c e

J s i n I 9 c o s r ~ d r ~ = 1 s i n 2 I 9

= L

- L c o s 2 $

2 2 2

a n d s g n (det Q' )

=

s g n ( d e t Hi d e t T)

=

s g n (detHi ) we s h a l l g e t (2.9) similarly as in t h e proof of Theorem 1.

3. PROCESS

WITH

LINEAR TREND

Let Ibt

I

b e a m -dimensional p r o c e s s of t h e form

where a a n d b a r e (m

.

1 ) fixed v e c t o r s (a

+

0 ) and Ict

I

i s a m -dimensional normal white noise, i.e.

-

^{i i d N,,, (0, C), C}

^{z o}

^{. (3.2)}

The l i n e a r model (3.1) i s t h e usual model of multivariate time s e r i e s used f r e q u e n t - ly in p r a c t i c e .

I t i s obvious t h a t in t h i s situation t h e b e h a v i o r of t h e p r o c e s s Iq(b,)l depends substantially on t h e position of t h e v e c t o r a with r e s p e c t t o t h e sets Sf. If i t i s a f S t h e n obviously a f t e r c e r t a i n time t h e p r o c e s s ) q ( b t ) j will n o t be finite with a i a r g e p r o b a b i l i t y . W e s h a l l e x c l u d e t h i s c a s e from f u r t h e r considerations.

Now Let u s i n v e s t i g a t e t h e situation when a is a n i n t e r i o r point of a s e t Sf.

Then due t o t h e p r o p e r t i e s of t h e convex polyhedral cone St when time t p r o c e e d s t h e p r o c e s s ]lp(bt)

1

will h a v e t h e f o r m [ g i ' b t

1

with a probability which grows in

time a n d i t e n a b i e s to draw some conclusions on t h e b e h a v i o r of t h i s p r o c e s s . The following t h e o r e m e v a l u a t e s t h e time p e r i o d a f t e r which i t i s g u a r a n t e e d with a given probability t h a t lrp(bt)l l i e s in St. Let t h e denotation (2.2) a n d (2.3) b e p r e s e r v e d .

THEOREM 3 Let 0

<

a

<

1 be a g i v e n n u m b e r , l e t a be an i n t e r i o r p o i n t of St a n d l e t & ( a ) be t h e c r i t i c a l v a l u e of t h e c h i - s q u a r e d d i s t r i b u t i o n w i t h m

d e g r e e s offreedom o n t h e level a (i.e. P ( ~ : r x:(a))

=

a ) . Then f o r t f u l f i l l i n g

t h e v a l u e s bt l i e i n St (i.e. v ( b t )

=

g i bt) w i t h t h e p r o b a b i l i t y a t l e a s t 1

-

a

17

(one c a n a l s o u s e l/q:(l

=

d ( h , Zh,)).

PROOF I t holds f o r a l l t

According t o [ 2 , Theorem 11 t h e ( 1

-

a ) 100 p e r c e n t confidence r e g i o n

lies in St if a n d only if

S i n c e a i s t h e i n t e r i o r point of St i t i s hza

>

0 f o r u

=

1,

. . .

, m a n d (3.6) i s equivalent t o

s o t h a t t h e t h e o r e m i s p r o v e d

REMARK 2 Theorem 3 c a n b e formulated f o r more g e n e r a l t y p e s of p r o c e s s e s f o r which o n e i s c a p a b l e t o c a l c u l a t e t h e confidence r e g i o n in t h e form of a n elip- soid as in (3.5) anci t h e t r e n d of which s t a y s in a convex cone with t h e v e r t e x in t h e origin contained (with e x c e p t of t h e v e r t e x ) in t h e i n t e r i o r of St. Specially s u c h n a t u r a l generalization may b e d e r i v e d f o r t h e p r o c e s s e s t h e t r e n d of which h a s b e e n estimated by means of t h e r e g r e s s i o n technique ( s e e 123).

If a i s v e r y small t h e n f o r t fulfilling (3.3) o n e c a n a p p r o x i m a t e t h e probabili- t y c h a r a c t e r i s t i c s of t h e p r o c e s s I v ( b , ) j by t h e o n e s of t h e p r o c e s s f g i ' b t j , e.g.

E(rp(bt)(bt E S )

-

~ g l ' b *

=

g t ' ( b

+

a t ) , (3.7)

v a r ( p ( b , ) l b , E S )

-

v a r g i ' b ,

=

gi' g i ( 3 . 8 ) In some situations o n e c a n a l s o d e s i r e t h e evaluation of t h e a c c u r a c y of such ap- proximations. In t h e following t h e o r e m such evaluation i s d e r i v e d f o r t h e a p p r o x i - mation ( 3 . 7 ) of t h e mean value.

THEOREM 4 Let u n d e r t h e a s s u m p t i o n s of T h e o r e m 3 t f u l f i l l (3.3). L e t u s d e n o t e c

=

& ( a ) , Q t h e d i s t r i b u t i o n f i n c t i o n o f t h e s t a n d a r d n o r m a l d i s t r i b u - t i o n N ( 0 , 1 ) a n d

. -

v = max d g j ' Z g j j = 1 , . . . , k

T h e n i t h o l d s

( 1

-

a ) g i ' ( b

+

a t ) 5 E ( p ( b t ) J b t E S ) S g i ' ( b + a t ) ,

+

^{L [ a max} f g j ' ( b + a t ) ! + v V m ( c )

1 - a j = ~ ,

. . . ,

^k

I

w h e r e

v z ( c ) = [ I - 2 1 4 ( 6 ) - 1 / 2 j ]

+ 6

e x p ( - c / 2 ) ,

. . .

+ m - l ) ( m - 3 ) . . . 2 ( e x p ( - c / 2 ) f o r o d d m ~ 3

PROOF Let us d e n o t e f t t h e probability density of t h e d i s t r i b u t i o n Nm(b

+

a t , Z ) . A s t h e lower bound in ( 3 . 1 0 ) i s c o n c e r n e d it i s obviously

The upper bound in (3.10) can be derived in t h e following way:

E ( 9 ( b r )lbt

6s)

⁵

-

_{l - a -} 1

r

q ( b t ) ~ t (bl )dbl R"

-

¹

ⁱ

--. J

g f ' b t f l ( b l ) d b t +

f

^{max ~ ~l f t j}

( e l

^{)dbt ~}

j

^{b ~}

[ ~ ( a ) R m ; P ( a ) j

.

.

.

^{I k}

=

g f ' ( b + a t )

+ -

^{[ a max} l g j r ( b

+ a t ) !

1 - a j = 1 ,

. . . ,

k

5 g f ' ( b

+

a t )

+ -

^{[ a max} i g j ' ( b

+

a t ) { 1 - a j = 1 . . . . , k

exp ( - r Z / 2 ) r m -'cos 1 9 ~ cosZ 1 9 ~

. . .

cosm

-'

^19, - l d r d 1 9 ~ . . . dI9,

I

, where t h e last inequality holds due t o t h e fact that outside the elipsoid

i y E Rm : y 'y 5 c

I

t h e graph o f the function max [ g j ' T y

I

can be dominated by t h e surface o f the cone C in R m with t h e vertex in the origin o f the form

where max [llT'gjlIl

=

max J g j ' ~ g j

=

v ( t h e description o f the mentioned surface

in the polar coordinates is used with the Jacobian r m

cos 1 9 ~

. . .

cosm

, , ,

⁾ The final form o f the upper bound can be derived using the formulas

J

cos l J z cos2 193

[ o r d l <2n, - n / 2 r d z r n / 2 , . . . , - n / z S d m - l S n / 2 j

. . .

cosm -2 19

. .

dlJm - 1

- -

^-1 T ; a l a z ' ' ' a,,, -2 ,

n/ 2

at

= J

cost z d z ,

0

i.e. a l

=

1, a 2

=

X / 4 a n d

a, = ( i - 1 - 3 ) 2 - 2 ) . . . l j f o r o d d i 2 3 ,

=

( 2 - 1

-

^{3 ) . . . 1/ l i ( i}

-

^{2 ) . .}

.

21 f o r even i r 4 ;

- j

r m e x p ( - r 2 / 2 ) d r

7;

REMARK 3 F o r m

=

1 o n e c a n calculate E(rp(bt)lbt E S ) exactly. If e.g.

S

=

R1 (i.e. S1

=

( - -, 01 and S2

=

10, -)), rp(b)

=

g l b -

+

g 2 b t f o r b E R1 (where g l , g z ER1), b1 - N ( & c?) (where w = b + a t ) a n d P ( a )

=

[cl, c2] (where

- -

^{< c l}

^<

^{c Z}

^<

- ) t h e n

E(rp(bl)lbt E S )

=

Erp(bt)

= B I [ W I @ ( C ~ )

-

@(C1)j + u ( ~ T T ) - ~ / ~ ~ B x P ( - ~ : / 2 ) - e x p ( - ~ $ / 2 ) j ] f o r c 2 5 0 ,

=

g l i w l l / 2

-

a ( c l ) i + 0 ( 2 x ) - ~ / ~ l e x ~ ( - c:/z) - e x p ( - (w/ 0 ) ~ / 2 1 ] + g Z I ~ l Q ( C 2 ) - 1 / 2 l +

c r ( ~ r ) - l / ~ [ e x p ( - (w/ u12/ 2

-

^{e x p ( - C: /2)1] for c l}

^<

⁰

^<

^{c 2 ,}

=

g2[wf@(C2)

-

@(C1)j

+

c r ( ~ r ) - ~ / ~ ~ e x p (- c f / 2 ) - e x p ( - ~ $ / 2 ) ] 1 f o r c l 2 0 , where Ct

=

(c,

-

w)/ a, i

=

1 , 2.

In Table 1 t h e r e a r e given Vm ( c ) for some values m if a

=

0.05 and a

=

0.01 (C

=

,ym(a)). For l a r g e r even 2 m t h e f i r s t term in t h e corresponding formula f o r t h e calculation of Vm(c) c a n b e omitted since then @(&)

-

1 (e.g. f o r m 2 4 if a

=

0.05).

Table 1 Values Vm ( c ) if a ) a

=

0.05 and b ) a

=

0.01.

Now let us consider the case when a is a relative interior point o f a ( m

-

1)- dimensional face in which two cones St

=

lz E

R m

: H i z 5 O { and S j

=

fz c

R m

: H j z 5 01 adjoin. One can assume (renumbering t h e rows o f Hi and H j I f it is necessary) that t h i s f a c e has t h e form

where h:

=

A h { f o r some negative scalar X

E X A M P L E 3 In the situation described in Example 1 e.g. the vector a

=

( 0 , 2.

5)' is t h e relative interior point o f t h e two-dimensional face fz ^E : z l

=

0 , z 2 5 3 , z 3 5 O { in which t h e cones S 1 a n d S 5 adjoin. In t h i s case it is h:

=-

h:

=

( 1 , 0,O)' so that it is not necessary t o renumber t h e rows o f t h e matrices H 1 and H 5 . The following theorem can be proved quite analogously as Theorem 3.

THEOREM 5 Let 0

<

a

<

1 be a given number a n d let a be a relative i n t e r i o r point of the ( m

-

1)-dimensional face (3.13) i n w h i c h two cones St a n d S j ad- join. Then for t f u l f i l l i n g

t 5 max 2

v = i , j u = l , .

.

. , m

d

x m ( a ) llquYII

-

h , ~ ' b ] / h c ' a

the v a l u e s v ( b t ) li e i n St or S j w i t h t h e probability a t least 1

-

a

This theorem enables again t o approximate t h e probability characteristics o f t h e process [ v ( b t ) { f o r t fulfilling (3.14) i f a is small. E.g. we can write f o r t h e mean value

where f t ( b t ) i s t h e density function of Nm (fi, C) with fi

=

b

+

at. Let R be a ( m , m ) matrix s u c h t h a t C

=

RR' a n d the f i r s t row of R - l h a s t h e same d i r e c t i o n as t h e v e c t o r h i (such matrix R c a n b e always c o n s t r u c t e d ) . Then h : ' ~ h a s t h e same d i r e c t i o n as t h e v e c t o r ( 1 , 0, .

.

^. , 0). Let u s d e n o t e

Then i t holds e . g .

=

( ~ r r ) - ~ / ~ d i e x p ( - v f / 2 )

+

g i ' u ) l

-

@(- vl)l Altogether we s h a l l o b t a i n

a n d similarly

var(rp(bt)lbt ^E S )

-

( 2 / rr)1/2(d:gi

-

d l g j ) ' f i e x ~ (-v:/2) (3.17)

4 . RANDOM WALK

Random walk i s t h e simplest c a s e of t h e i n t e g r a t e d p r o c e s s e s ARIMA of Box a n d Jenkins. These p r o c e s s e s a r e nonstationary b u t t h i s n o n s t a t i o n a r i t y c a n b e r e - moved easily b y a i f f e r e n c i n g t h e originai p r o c e s s . S i n c e t h e s e p r o c e s s e s a r e v e r y useful f o r p r a c t i c a l p u r p o s e s i t is important t o i n v e s t i g a t e w h e t h e r t h i s t y p e of nonstationarity i s p r e s e r v e d a l s o f o r t h e p r o c e s s e s [ q ( b t ) l . W e s h a l l confine o u r - s e l v e s t o t h e one-dimensional case with a normal random walk f b t

I

of t h e form

where i c t

1

i s a normal white noise, i.e

Let t h e function q ( b ) b e f i n i t e f o r a l l b ^E

R 1

s o t h a t i t h a s t h e form

where g l and g Z are given r e a l numbers. W e s h a l l i n v e s t i g a t e t h e b e h a v i o r of t h e p r o c e s s [ q ( b t +1)

-

v ( b t ) l ( t h e p r o c e s s ibt + l

-

bl j

=

i c t

+11

i s s t a l i o n a r y ) .

THEOREM 6 U n d e r t h e p r e v i o u s a s s u m p t i o n s i t h o l d s

w h e n t

-

^m.

PROOF I t i s

s o t h a t

Hence t h e a s s e r t i o n of t h e theorem follows.

THEOREM 7 U n d e r t h e p r e v i o u s a s s u m p t i o n s i t h o l d s f o r a r b i t r a r y k 5 0 cov Iq(bl +t + l )

-

^{q(bL + t ) . ~ ( b t +I)}

^-

q ( b t ) l (4.6)

w h e r e

and C ( t . t

+

k )

=

C ( t

+

k , 1 ) . D ( t , t

+

k ) = D ( t

+

k , t ) . M o r e o v e r , it is w h e n

t

--

COV ~ ~ ( b ~ + , + 1 )

-

rp(bt

+ , I ,

rp(bt + 1 )

-

rp(bt)l

-

( a 2 / 2 ) ( g : + g $ ) f o r k

= o

^, ( 4 . 9 )

-

^{0 f o r k > O .}

REMARK 4 Specially it holds var Irp(bt + I )

-

rp(bt

P R O O F Let us denote C ( t

+

k , t ) = E Y + Z + , where

The joint density f (y , z ) of Y and Z h a s the form

w h e r e

~ ( ~ 1 2 ) = ( 2 n k ~ ~ ) - ~ / ~ e x p 1 - ( Y - z ) Z / ( 2 k u 2 ) I is t h e c o n d i t i o n a l d e n s i t y of Y f o r f i x e d Z a n d

h ( ~ )

=

( 2 ~ t u ~ ) - ~ / ~ e x p

I -

z 2 / ( 2 t c 2 ) 1 is t h e m a r g i n a l d e n s i t y of Z. H e n c e it i s

e x p

1-

( y

-

z ) ' / ( Z k d ? )

-

z 2 / ( 2 t u 2 ) j d y d z

w h e r e

,-I

=

( k t ,z)-~1[-

: ;:I

We c a n w r i t e

w h e r e

If using t h e method of s u b s t i t u t i o n a n d t h e f o r m u l a s (2.10) a n d (3.12) we c a n f u r t h - e r w r i t e

t

I

/ + ( d k i /

( t +

k ) ) u e x p

/ -

( u 2 + u 2 ) / ~ i d u d v

+

u Z m / (7.7;)

JJ

u v e x p

1-

( u 2

+

v 2 ) / 2 i d u d v lu, v : u 2 0, v 2 - ( d l / t ) ~ 1

-

^{n/ 2}

=

u2t / ( 2 n ) J r 3 e x p ( - r 2 / 2 ) d r

J

c o s 2 + d +

0

-

-arctg dt / k

- -

+

& m / ( ~ n ) J u e x p ( - u 2 / 2 ) [

- J

^{v e x p ( -} v 2 / 2)dv i d u

o

-(dl / k )u

which coincides with (4.7). I t can b e shown similarly t h a t D ( t

+

k , t ) defined as

coincides with (4.8). If w e notice t h a t

then a f t e r some a l g e b r a i c manipulation t h e formula (4.6) follows. Finally, i t i s pos- sible t o show (e.g. by means of 1'Hospital r u l e ) t h a t

lim [ C ( t + k , t ) - ~ % / 2 \

=

lim D ( t + k , t ) = 0

t

--

^t

--

s o t h a t (4.9) follows.

One c a n summarize t h a t t h e p r o c e s s frp(bt

-

rp(bt)l w h e r e [bt i s t h e r a n - dom walk (4.1) i s not (weakly) s t a t i o n a r y b u t approximately f o r l a r g e t one c a n t a k e i t as t h e ( s t a t i o n a r y ) white noise with t h e v a r i a n c e ( d ? / 2 ) (9:

+

9:).

REFERENCES

1 Box. G.E.P., Jenkins, G.M. Time S e r i e s Analysis, Forecasting a n d Control. Holden Day, San F r a n c i s c o 1970.

2 C i p r a , T. Confidence r e g i o n s f o r l i n e a r p r o g r a m s with random coefficients. IIASA Working P a p e r WP-86-0, Laxenburg, Austria 1986.

3 Dedk, I. T h r e e digit a c c u r a t e multiple normal probabilities. Num. Math. 35, 1 9 8 0 , 369-380.

4 Gal. T.. Nedoma, J . Multiparametric l i n e a r programming. Management S c i e n c e 18.

1972, 406-422.

5 Nogizka. F.. Guddat. J . , Hollatz. H., Bank. B. T h e o r i e d e r l i n e a r e n p a r a m e t r i s c h e n Optimierung. Akademie

-

Verlag. Berlin 1974.

6 Walkup.

D.,

Wets. R . Lifting p r o j e c t i o n s of convex polyedra. P a c i f i c J . Mathem.

28, 1969, 465-475.

7 Wets. R . Programming u n d e r uncertainty: t h e equivalent convex p r o g r a m . J . SIAM Appl. Math. 1 4 . 1966, 89-105.

8 Wets, R . S t o c h a s t i c programming: solution techniques and approximation schemes. In: Mathematical Programming: The S t a t e of t h e A r t (A. Bachem, M.

G r o t s c h e d . B. K o r t e eds.). S p r i n g e r . Berlin 1983. 566-603.

9 Wets, R. L a r g e s c a l e l i n e a r programming techniques in s t o c h a s t i c programming.

IlASA Working F a p e r WP-84-90. Laxenburg. Austria 1984.