A computational study of several problems in stochastic modelling

C

o r r i g e n d a

a n d

A

d d e n d a

A C o m p u t a t i o n a l S t u d y of S e v e r a l P r o b l e m s in S t o c h a s t i c M o d e l l i n g b y D a v i d C.C. B o v e r

page page page page page page page page page page page page page page page page page page page page page page

2, line -2: replace "Mathematical" with "mathematical" 6, line -2: replace "arguements" with "arguments"

8, line -1: replace "3 2x" with " 9 x 2 "

9, lines 9, 11: replace "Plank" with "Planck" 10, lines 16, 18: replace "Plank" with "Planck"

14, line 8: delete "and is the defining p r o p e r t y of a Mark o v p r o c e s s "

15, line 12: add to the end of the sentence "for cases where the c o e f ficients a and b are independent of time"

17, line 12: add to the end of the sentence "corresponding to a M a r k o v process"

17, lines 12 to 16: delete the sentence "This r e sult ... to be applicable."

18, after line 6 insert "In addition, the mu s t be chosen so that w(Ax,t)>0. An example of such a density function is (4.3.1) w h i c h involves the Hermite po l y n o m i a l s . "

19, lines 9, 10: delete "regardless of whether the process po s s e s s e s the M a rkov p r o p e r t y (2.2.1)"

21, delete the last p a r a g r a p h

25, line -1: repl a c e " V (CT , t-v) ” with "V (cr ,t-v)"

U x \. 0 K

25, after line -1 insert "where

V Q (a ,t-v) = / f ( x , v | y , 0 ) V Q (x,t-v)dx / f ( a R ,v|y,0)" a R

26 , lines 16, 25: replace " V 0" with " V 0 27, lines 9, 14: replace " V 0" with " V 0 27, lines 14, 18: repl a c e "Vo" with "Vo 27, line -1: replace "1-Vo (CJ ,

•k R

A )" with 28 , line 3: replace "Vo" with

- v * " V 0" 28 , line 4: replace "Vo" with "Vo" 29, line 4: replace "Vo" with "Vo "

34, replace line -5 with: "For an evo l u t i o n a r y process, stability reqires that the norms of be b o u n d e d i n d ependent of n.

page 45, line -5: replace "such" with "such that" page 47, line 2: replace " (xj-ijij) " with "6 (x x-<f> j) "

page 54, line 6: replace "For stability" with "From the d e f i n i t i o n of stability given in §2.4"

page 54, after equation (2.7.4) insert "for any m a t r i x norm subordinate to the vect o r norm used in §2.4 (i.e. for a matr i x A and vect o r x, the m a t r i x norm is defined as IIAII = IIA x II / llxll, for some v e ctor norm) ."

page •* 1 ^ * L O

line -8 : replace "Wiener" with "white noise" page 69, delete lines 4, 5

page 70, lines - actual

4 to -1: delete the sentences wind and current data."

"But even without page CO line -7 : replace "fron" with "from"

page CD N

)

line -5 : replace "Oceanography" with "oceanography" page 85, line 13 : insert the sentence "This functional form not

only p r o v i d e s a fit to the empirical data w hich is quite a d equate for the p u r p o s e of the study, but also leads to simpli f i c a t i o n s in the analysis in §3.4."

page 89, line -2: replace "analtsis" with "analysis" page 97, add the new pa r a g r a p h to the end of §3.6:

"This example has d e m o n s t r a t e d that a d i f f u s i o n model is a p r a c t i c a l a l t ernative to M o n t e Carlo simulation for p r oblems of this type. The d i f f u s i o n m o d e l a pproach offers c o n s i derable savings in c o m p u t a t i o n a l effort over the Monte C a r l o m odel which, for the P o l y n e s i a n study

in [33], i n volved over 100,000 simulated voyages. However, our computer i m p l e m e n t a t i o n of the d i f f u s i o n model is by no means a g e n e r a l i s e d p r o g r a m and would r e q u i r e some m o d i f i c a t i o n for use in related problems, p a r t i c u l a r l y

if the r e gion of s o lution is changed." page 107, add to the end of the second paragraph:

"This e x p a n s i o n was also used by Ledwich in his work on a p p r o x i m a t i n g the p r o b a b i l i t y density of the state v a r i a b l e s by solving equations for the quasi-moments. Our work differs from that of Ledwich in that whereas Ledwich was concerned with a p p r o x i m a t i n g the p r o b a b i l i t y den s i t y function, the p u r p o s e of our study is to follow on from the early wo r k of B e l l m a n et. al. in r eferences [6] and [7] in d evising a meth o d for m o m e n t equation h i e r a r c h y trunca t i o n which is both p r a c t i c a l and t h e o r e t i c a l l y sound."

page 114, line -2: delete "best"

page 125, add the new p a r a g r a p h to the end of §5.1:

"The idea of c o m p u t e r - g e n e r a t e d algebra involving multi-^ d i m e n s i o n a l Hermite p o l y n o m i a l s was also us e d by Ledw i c h in g e n e r a t i n g the o r dinary d i f f e r e n t i a l equations for the quasi-moments. For this purpose, Ledw i c h derived

several results for the polynomials, i n c l uding an equivalent of our Lemma 5.3.3 w hich he used for both the numerical

- 3

-p a g e 126, p a g e 127, p a g e 128, p a g e 131, p a g e 137, p a g e 139,

p a g e 1 4 4 , p a g e 146,

p a g e 149,

l i n e 9: r e p l a c e "a" w i t h " a n " l i n e -5: r e p l a c e "> +< " w i t h ") -< "

l i n e 5: a d d t o t h e e n d o f t h e s e n t e n c e : "as i n T a b l e 5 . 2 . 1 " l i n e -4: r e p l a c e " s o m e " w i t h " a n y "

l i n e 10: r e p l a c e " a n a l a g o u s " w i t h " a n a l o g o u s " r e p l a c e l i n e s 10, 11, 12 w i t h :

" if>(P + l) n

-

I

I

- I

mJ Yi

I

s - 1 n m .3 - 6 .3 /k , n y.

j = 2 •

k = 2 s = 0

n m . -6. , n y . 3 3 'k j - 2 3

l i n e - 1 0 : r e p l a c e " e l e m e m t s " w i t h " e l e m e n t s " a d d t h e n e w p a r a g r a p h to t h e e n d of §5.4:

" T h e m o m e n t e q u a t i o n a p p r o a c h d e s c r i b e d h e r e p r o v i d e s a p r a c t i c a l a l t e r n a t i v e to M o n t e C a r l o s i m u l a t i o n f o r p r o b l e m s o f t h e t y p e (1 . 1 . 2 ) , w i t h i n t h e f o l l o w i n g c o n s t r a i n t s :

(i) t h e c o e f f i c i e n t s a ( x , t ) a n d C ( x , t ) m u s t b e e x p r e s s e d as p o l y n o m i a l s o f t h e s t a t e v a r i a b l e s

(ii) f o r p r a c t i c a l p u r p o s e s in u s i n g a c o n t i n u o u s

s i m u l a t i o n l a n g u a g e , t h e n u m b e r o f m o m e n t e q u a t i o n s g e n e r a t e d s h o u l d b e l e s s t h a n a b o u t 100 .

(iii) t h e q u a s i - m o m e n t s m u s t t e n d to z e r o f a i r l y r a p i d l y w i t h i n c r e a s i n g o r d e r . "

l i n e -1: d e l e t e " i s s u c h t h a t "

p a g e 150, l i n e 5: r e p l a c e

t + A t r| . . d t

13 w i t h (

"t+At n . . dt>

13

p a g e 1 56, a f t e r e q u a t i o n ( A . 14) i n s e r t : " N o t e t h a t c o r r e l a t i o n b e t w e e n t h e i n t e r a c t i o n c o e f f i c i e n t s w i l l h a v e n o d e t r i m e n t a l e f f e c t o n t h e m o d e l a n d w i l l o n l y r e s u l t i n c r o s s - d e r i v a t i v e t e r m s i n t h e d i f f u s i o n e q u a t i o n . " p a g e 1 5 6 , l i n e s -4 , -3: r e p l a c e " 2 . 5 " w i t h " 2 . 6 "

p a g e 1 6 2 , l i n e 3: r e p l a c e " a r g u e m e n t " w i t h " a r g u m e n t " p a g e 164, l i n e 6: r e p l a c e " s u q a r e " w i t h " s q u a r e "

p a g e 1 6 4 , l i n e -3 : r e p l a c e " s i m i l a r i t y " w i t h " s i m i l a r i t y p a g e 185 , l i n e s 2, -4: r e p l a c e " P l a n k " w i t h " P l a n c k " p a g e 1 8 9 , l i n e s 7, -2: r e p l a c e " P l a n k " w i t h " P l a n c k " p a g e 190 , l i n e 11: r e p l a c e " P l a n k " w i t h " P l a n c k "

S

P

S

M

by

D a v i d C. C. B o v e r

A t h e s i s s u b m i t t e d to the A u s t r a l i a n N a t i o n a l U n i v e r s i t y for the d e g r e e of D o c t o r of P h i l o s o p h y

(i)

P

r e f a c e

Some of the w o r k in this thesis was c a r r i e d out in c o l l a b o r a t i o n w i t h Dr. M R Osbor n e . In p a r t i c u l a r , C h a p t e r s 2 and 3 c o n t a i n r e s u l t s w h i c h were e s t a b l i s h e d jointly.

Some of the m a t e r i a l f r o m C h a p t e r s 3 and 4 has b e e n p u b l i s h e d as B o v e r [9] and B o v e r [8], r e s p e c t i v e l y . W h e n p r e s e n t i n g that m a t e r i a l here the text of these p a p e r s has b e e n f o l l o w e d closely.

A

c k n o w l e d g e m e n t s

The w o r k for this thesis was u n d e r t a k e n at the C o m p u t e r Centre, A u s t r a l i a n N a t i o n a l U n i v e r s i t y . I g r a t e f u l l y

a c k n o w l e d g e the f i n a n c i a l a s s i s t a n c e of a C o m m o n w e a l t h P o s t g r a d u a t e R e s e a r c h A w a r d d u r i n g this period.

I am i n d e b t e d to M i k e O s b o r n e for his help and s u p e r ­ v i s i o n of my w o r k and for his c o n s t r u c t i v e c o m m e n t s on

the d r a f t of this thesis. All those who w e r e m e m b e r s of the C o m p u t e r C e n t r e d u r i n g my time t h ere have b e e n of help to me in some way and p r o v i d e d a m o s t c o n g e n i a l and i n s p i r i n g w o r k e n v i r o n m e n t . In p a r t i c u l a r I am v e r y g r a t e f u l to

Bob W a tts and F r a n k de H o o g for t h e i r i n t e r e s t in my wo r k and their h e l p f u l c o m m e n t s and s u g g e s t i o n s r e l a t e d to it. I am also m o s t g r a t e f u l to P r o f e s s o r R G W a r d of the Dept. H u m a n G e o g r a p h y , A u s t r a l i a n N a t i o n a l U n i v e r s i t y , for

p r o v i d i n g the w i n d and c u r r e n t data u s e d in the study of the s e t t l e m e n t of P o l y n e s i a , d e s c r i b e d in this thesis.

(iii)

A

This thesis i n v e s t i g a t e s the c o m p u t a t i o n a l a s pects of two d i f f e r e n t a p p r o a c h e s to the study of m a t h e m a t i c a l m o d e l s c o n s i s t i n g of s y stems of s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s . T h e s e two a p p r o a c h e s , p r e s e n t e d as a l t e r n a t i v e s to the simp l e bu t u s u a l l y e x p e n s i v e m e t h o d of M o n t e C a rlo s i m u l a t i o n , are:

(i) c a l c u l a t i o n of the a p p r o p r i a t e p r o b a b i l i t y

d i s t r i b u t i o n of the state v a r i a b l e s by s o l u t i o n of the p a r t i a l d i f f e r e n t i a l e q u a t i o n of a r e l a t e d d i f f u s i o n model, and

(ii) s o l u t i o n of systems of d e t e r m i n i s t i c o r d i n a r y

d i f f e r e n t i a l e q u a t i o n s for the m o m e n t s of the state v a r i a b l e s .

of s e v e r a l i t e r a t i v e m e t h o d s of s o l u t i o n of the fin i t e d i f f e r e n c e e q u a t i o n s . In a d d i t i o n we d i s c u s s s t a b i l i t y of the f i n i t e d i f f e r e n c e scheme. We d e m o n s t r a t e the use of a d i f f u s i o n m o d e l in a s t udy of the q u e s t i o n of the . s e t t l e m e n t of P o l y n e s i a . The a im of the s t u d y is to c a l ­ c u l a t e p r o b a b i l i t i e s of o c e a n - d r i f t v o y a g e s to and b e t w e e n v a r i o u s i s l a n d gro u p s in P o l y n e s i a , in o r d e r to test

H e y e r d a h l ’s h y p o t h e s i s of the s e t t l e m e n t of P o l y n e s i a from

the A m e r i c a s .

In so m e cases of c o n t i n u o u s s t o c h a s t i c s y s t e m s it

m i g h t be s u f f i c i e n t or more a p p r o p r i a t e to s o l v e for m o m e n t s of the s t a t e v a r i a b l e s . H o w e v e r , e x c e p t for m o d e l s c o n ­ s i s t i n g of c o m p l e t e l y linear s y s t e m s of s t o c h a s t i c d i f f e r ­ e n t i a l e q u a t i o n s , the m o m e n t e q u a t i o n s u s u a l l y f o r m an i n f i n i t e c o u p l e d s y s t e m in w h i c h e q u a t i o n s for m o m e n t s of any g i v e n o r d e r inv o l v e m o m e n t s of h i g h e r o r d e r s . Then to f a c i l i t a t e s o l u t i o n it is n e c e s s a r y to a p p r o x i m a t e the i n f i n i t e s y s t e m w i t h a finite c l o s e d s y s t e m of e q u a t i o n s . We i n v e s t i g a t e one m e t h o d of a c h i e v i n g this w h i c h in v o l v e s

the use of q u a s i - m o m e n t s , w h i c h are the e x p e c t a t i o n s of

mu 11 i - d i m e n s i o n a 1 H e r m i t e p o l y n o m i a l s in the s t a t e v a r i a b l e s . This m e t h o d is s h o w n to be s a t i s f a c t o r y in t h e o r y b u t

(V)

T

a b l e

of

C

o n t e n t s

P R E F A C E (i)

A C K N O W L E D G E M E N T S ( Ü )

A B S T R A C T (iii)

C H A P T E R 1: I N T R O D U C T I O N

1.1 A C l a s s of M a t h e m a t i c a l M o d e l s 1 1.2 The C h o i c e b e t w e e n I to and S t r a t o n o v i c h

C a l c u l u s 6

C H A P T E R 2: D I F F U S I O N M O D E L S OF C O N T I N U O U S S T O C H A S T I C S Y S T E M S

2.1 I n t r o d u c t i o n 7

2.2 A s s u m p t i o n s and I m p l i c a t i o n s in the Use

of D i f f u s i o n M o d e l s 14

2.3 The I n t e r i o r F i r s t P a s s a g e P r o b l e m 22 2.4 C o m p a r i s o n of a D i s c r e t e R a n d o m W a l k

w i t h the F i n i t e D i f f e r e n c e S o l u t i o n

of a D i f f u s i o n E q u a t i o n 30

2.5 B o u n d a r y C o n d i t i o n s 36

2.6 A F i n i t e D i f f e r e n c e Sch e m e for the N u m e r i c a l S o l u t i o n of the K o l m o g o r o v

E q u a t i o n 41

(vii)

C H A P T E R 3: A D I F F U S I O N M O D E L OF O C E A N - D R I F T M I G R A T I O N

3.1 The P o l y n e s i a n P r o b l e m 64

3.2 The D i f f u s i o n M o d e l 68

3.3 S u r v i v a l C h a n c e s 73

3.4 The M u l t i - I s l a n d P r o b l e m 75

3.5 Imp lernentat ion of a D i f f u s i o n M o d e l for

the " P o l y n e s i a n P r o b l e m " 79 3.6 R e s u l t s f r o m the D i f f u s i o n M o d e l 92

C H A P T E R 4: S O L U T I O N OF M O M E N T E Q U A T I O N S

4.1 I n t r o d u c t i o n 99

4.2 The M o m e n t E q u a t i o n s 101

4.3 The Q u a s i - M o m e n t s 107

4.4 M o m e n t E q u a t i o n s for the S t o c h a s t i c

C u b i c O s c i l l a t o r 115

C H A P T E R 5 C O M P U T E R G E N E R A T I O N OF M O M E N T E Q U A T I O N

A P P R O X I M A T I O N S F O R ITO D I F F E R E N T I A L S Y S T E M S

5.1 I n t r o d u c t i o n 124

5.2 G e n e r a t i o n of M o m e n t E q u a t i o n s 126 5.3 G e n e r a t i o n of Q u a s i - M o m e n t H i e r a r c h y

T r u n c a t i o n A p p r o x i m a t i o n s 131

5.4 A P r e p r o c e s s o r to a C o n t i n u o u s Sys t e m s S i m u l a t i o n L a n g u a g e for S o l u t i o n of

A P P E N D I X A:

A P P E N D I X B

A P P E N D I X C

A P P E N D I X D

A P P E N D I X E

A S t o c h a s t i c M o d e l of I n t e r a c t i n g S p e c i e s

S t a b i l i t y A n a l y s i s of the O n e - D i m e n s i o n a l D i f f u s i o n E q u a t i o n

D i s t r i b u t i o n s of W i n d and C u r r e n t D a t a for the P a c i f i c O c e a n

P r o g r a m for the S o l u t i o n of the D i f f u s i o n E q u a t i o n for the S i n g l e - I s l a n d P r o b l e m

An I n d e x i n g S y s t e m for M u l t i - D i m e n s i o n a l M o m e n t s

148

158

163

174

180

1

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h a p t e r

1

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n t r o d u c t i o n

1,1 A C

l a s s

of

M

a t h e m a t i c a l

M

o d e l s

This thesis is concerned with a numerical investigation of the class of mathematical models which can be specified in terms of systems of coupled stochastic ordinary differ -ential equations

dx

— T = a (x ,t ) + C ( x , t ) Z ( t ) . (1.1.1)

dt ~ ~ ~

Here x(t) is an n-dimensional vector of state variables, a(x,t) is an n-dimensional vector representing the d et e r ­ ministic influences in the model and Z(t) is an m - d i m e n s i o n a 1 vector of random processes which influence the model through

the nxm matrix C ( x , t ) . Since the elements of C(x,t) can be functions of the state variables as well as of time

this class of models includes equations with random c oe f f ­ icients as well as those with random inhomogeneous terms. The effects of the mean parts of the random processes can be included in a ( x , t ) , thus ensuring that the elements of Z(t) all have zero mean.

Classification of mathematical models can be made in various ways. For example we can classify models according to whether they are completely deterministic or contain

some random element, whether they are continuous or discrete with respect to time and whether the state variables may

c o n t i n u o u s wi t h r e s p e c t to time and c o n t i n u o u s w i t h r e s p e c t to state v a r i a b l e ranges. The c o n t i n u i t y of state v a r i a b l e ranges m i g h t w e l l be a m a t t e r of s c a l i n g and r e q u i r e d

a c c u r a c y . For e x a m p l e it w o u l d be a p p r o p r i a t e to f o r m u l a t e a c o n t i n u o u s m o d e l of w o r l d p o p u l a t i o n , d e s p i t e the fact that the real s y s t e m is d i s c r e t e b o t h w i t h r e s p e c t to time

(births and deaths o c cur at i r r e g u l a r i n s t a n t s of time) and w i t h r e s p e c t to the range of the s t a t e v a r i a b l e

" p o p u l a t i o n " (the p o s i t i v e i n t e g e r s ) . A c o n t i n u o u s m o d e l w o u l d be a p p r o p r i a t e b e c a u s e i n d i v i d u a l b i r t h s and deaths do not have a s i g n i f i c a n t e f f e c t on the p o p u l a t i o n level, w h e r e a s the a g g r e g a t e d p o p u l a t i o n c h a n g e over some time

i n t e r v a l (for e x a m p l e a day) w o u l d be s i g n i f i c a n t and t h e r e ­ fore w o u l d be of i n t e r e s t in f o r m u l a t i o n of the model.

P h e n o m e n a that have be e n m o d e l l e d in terms of e q u a t i o n s of the type (1.1.1) range from the r e l a t i v e l y s i m p l e stu d i e s of B r o w n i a n m o t i o n and the s t o c h a s t i c l i n e a r o s c i l l a t o r

by C h a n d r a s e k h a r [15] and M o y a l [4l] and the w o r k on s t o c h ­ astic n o n l i n e a r o s c i l l a t o r s by C a u g h e y [14], S t r a t o n o v i c h [ 55] , Liu[ 35] , B e l l m a n and W i l c o x [ 7] , S a n c h o [ 52] and M o r t o n and C o r r s i n [ 40] to the m o r e c o m p l i c a t e d p r o b l e m s of a s t o c h a s t i c m o d e l of i n t e r a c t i n g speci e s , c o n s i d e r e d by Goel, M a i t r a and M o n t r o l l [ 20] and the s e t t l e m e n t of P o l y n e s i a by o c e a n voyag e s , m o d e l l e d by L e v ison, W a r d and We b b [33].

3

d x ( t ) = a ( x , t ) d t + C (x , t ) d W X t ) , (1.1.2)

w h e r e W ( t ) is an m d i m e n s i o n a l v e c t o r of i n d e p e n d e n t G a u s s i a n -d i s t r i b u t e -d W i e n e r p r o c e s s e s ( -d e f i n e -d in § 2 . 1 ) , e a c h w i t h

t h e p r o p e r t i e s

< W. (t)> = < dW. (t)> = 0 ( 1 . 1 . 3 )

l l

a n d < ( d W i ( t ) ) 2> = Cf2 d t, ( 1 . 1 . 4 )

f o r s o m e r e a l c o n s t a n t s a . , a 0 , . . . , a . H e r e a n d t h r o u g h o u t

i z m

t h i s t h e s i s t h e a n g l e b r a c k e t s o p e r a t i o n (<f)(x,t)) o n a f u n c t i o n (p (x , t ) is t h e e x p e c t a t i o n , d e f i n e d b y

( cj> (x , t ) >

/CO / 00

' -co ■'— oo

[00

<J)( X , t ) w ( x , t I x , 1 0 ) dx J d x 2 . . . dx , (1.1.5)

,— oo ~ ~ ~ n

w h e r e w ( x , t | x 0 , t 0 ) is t h e t r a n s i t i o n d e n s i t y of t h e s t a t e v a r i a b l e s . T h i s is d e f i n e d s o t h a t

w (x ,t I x 0 , t 0 ) d x jd x 2 . . . d x n

is t h e p r o b a b i l i t y t h a t a t t i m e t t h e s t a t e v a r i a b l e s t a k e v a l u e s in t h e r a n g e C x , x + d x ] , g i v e n t h a t x = x Q at t i m e t = t Q .

S t a t i s t i c a l p r o p e r t i e s of the state v a r i a b l e s . P s e u d o - r a n d o m n u m b e r g e n e r a t o r s are u s e d to p r o v i d e the s t o c h a s t i c e l e m e n t s and the statistical p r o p e r t i e s are c a l c u l a t e d from a large n u m b e r of trials. An e x ample is the P o l y n e s i a n s t u d y of L e v i s o n et. al. [33]. An a l t e r n a t i v e a p p r o a c h u s i n g a w h i t e n o ise g e n e r a t o r on an a n a l o g u e c o m p u t e r has b e e n g i v e n in the s t udy of a n o n l i n e a r o s c i l l a t o r by M o r t o n and C o r r s i n [ 40] .

The t r o u b l e w i t h M o nte C a r l o s i m u l a t i o n is that it t y p i c a l l y r e q u i r e s a large a m o u n t of c o m p u t e r r e s o u r c e s

to p e r f o r m s u f f i c i e n t trials to o b t a i n res u l t s of s t a t i s t i c a l s i g n i f i c a n c e . In a d d i t i o n , w h e n o b s e r v a t i o n a l data is used for the r a n d o m p r o c e s s e s (as in [ 33] ) the task of d a t a

h a n d l i n g in s a m p l i n g the actu a l d i s t r i b u t i o n s can also r e q u i r e a s i g n i f i c a n t c o m p u t a t i o n a l effort. For the class of p r o b l e m s c o n s i d e r e d here it is u s u a l l y more e f f i c i e n t to s o lve d i r e c t l y for the r e q u i r e d s t a t i s t i c a l p r o p e r t i e s . D e p e n d i n g on the p u r p o s e of the study, it m i g h t be s u f f i c ­ ient to s o lve for the lower o r d e r m o m e n t s of the s tate v a r i a b l e s or it m i g h t be n e c e s s a r y or m o r e a p p r o p r i a t e to solve for the t r a n s i t i o n p r o b a b i l i t y d e n s i t y or some o t h e r d i s t r i b u t i o n r e l a t e d to the state v a r i a b l e s .

5

p r o b l e m is d i f f e r e n t to the m e t h o d s used by o t her a u t h o r s in r e l a t e d p r o b l e m s (as in the r e f e r e n c e s m e n t i o n e d above) in that it i n v o l v e s s o l u t i o n of an i n t e r i o r f i rst p a s s a g e p r o b l e m . An a l t e r n a t i v e a p p r o a c h to p r o b l e m s of type (1.1.2) w h e r e it is s u f f i c i e n t or a p p r o p r i a t e to solve for the lower

(typ i c a l l y first and second) o r d e r m o m e n t s of the s t ate

1, 2

The C

C

The use of m a t h e m a t i c a l m o d e l s of the type d e s c r i b e d in §1.1 leads to the d i l e m m a of c h o o s i n g b e t w e e n Ito and S t r a t o n o v i c h c a l c u l u s . The d i f f e r e n c e b e t w e e n the two a p p r o a c h e s is d i s c u s s e d in d e t a i l by Gray and C a u g h e y [22] and by M o r t e n s e n [39] but b a s i c a l l y the d i f f i c u l t y arises in the i n t e r p r e t a t i o n of s t o c h a s t i c i n t e g r a l s , that is i n t e g r a l s w i t h r e s p e c t to r a n d o m n o i s e v a r i a b l e s . The

Ito i n t e r p r e t a t i o n leads to the s t o c h a s t i c i n t e g r a l d e f i n e d so as to be c o n s i s t e n t w i t h the p r o p e r t i e s of the W i e n e r p r o c e s s , w h e r e a s the S t r a t o n o v i c h a p p r o a c h is to i n t e r p r e t the s t o c h a s t i c i n t e g r a l in a m a n n e r w h i c h is c o n s i s t e n t w i t h the rules of o r d i n a r y c a l c u l u s . The m a i n p o i n t s made by the a b o v e a u t h o r s were

(i) W h e n the e l e m e n t s of C(x,t) in (1.1.1) are

i n d e p e n d e n t of x the two i n t e r p r e t a t i o n s lead to e q u i v a l e n t m o d e I s .

(ii) Each i n t e r p r e t a t i o n is e n t i r e l y c o n s i s t e n t w i t h i n itself.

Fo r the sake of c o n s i s t e n c y we w i l l use Ito c a l c u l u s t h r o u g h o u t this t h esis, k e e p i n g in m i n d that this r e p r e s e n t s an i d e a l i s a t i o n of the r a n d o m p r o c e s s e s as d e r i v a t i v e s

of W i e n e r p r o c e s s e s . It s h o u l d be n o t e d that the ideas p r e s e n t e d in the t h e s i s are not d e p e n d e n t on this choice

7

C

2

D

M

C

S

S

2 . 1

I

In 1827 the E n g l i s h b o t a n i s t R o b e r t B r own n o t i c e d that w h e n p o l l e n is d i s p e r s e d in w a t e r the s u s p e n d e d p a r t ­ icles move a b o u t e r r a t i c a l l y in w h a t c o u l d only be d e s c r i b e d as a t h r e e - d i m e n s i o n a l r a n d o m walk. He c o r r e c t l y d e d u c e d that these e r r a t i c m o v e m e n t s w e r e the r e s u l t of the p a r t ­ icles c o l l i d i n g w i t h w a t e r m o l e c u l e s . This p h e n o m e n o n ,

s u b s e q u e n t l y n a m e d B r o w n i a n m o t i o n was s u c c e s s f u l l y e x p l a i n e d by E i n s t e i n in 1905. E i n s t e i n ' s w o r k was g e n e r a l i s e d in

a r i g o r o u s m a t h e m a t i c a l t r e a t m e n t by W i e n e r in 1923. The a p p r o a c h of E i n s t e i n and W i e n e r was to c o n s i d e r the p o s i t i o n x(t) of the p a r t i c l e as a r a n d o m p r o c e s s . This p r o c e s s

has s i n c e b e c o m e k n o w n as the W i e n e r p r o c e s s and is d e f i n e d as a r a n d o m p r o c e s s x(t) w i t h i n d e p e n d e n t and s t a t i o n a r y i n c r e m e n t s x ( t + s ) - x ( t ) (for s>0), such that the i n c r e m e n t s are G a u s s i a n d i s t r i b u t e d w i t h m e a n zero and v a r i a n c e as, for some c o n s t a n t a.

of the i n c r e m e n t s x(t+s) - x(t) = 0(s) i m p l i e s that the i n c r e m e n t s are of o r der s . H e nce the v e l o c i t y of the p a r t i c l e w o u l d have to a p p r o a c h i n f i n i t y as s t e n d e d to

zero. H o w e v e r , the W i e n e r p r o c e s s , a l t h o u g h a m a t h e m a t i c a l i d e a l i z a t i o n , is m o s t u s e f u l in m a t h e m a t i c a l m o d e l s of

s t o c h a s t i c p r o c e s s e s .

O r n s t e i n and U h l e n b e c k [43] a t t e m p t e d to o v e r c o m e this p r o b l e m w i t h the W i e n e r p r o c e s s by p r o p o s i n g a r a n d o m p r o c e s s x(t) , w h i c h has since b e c o m e k n o w n as the O r n s t e i n -U h l e n b e c k p r o c e s s , w h o s e i n c r e m e n t s x(t+s) - x(t) are s t a t i o n ­ ary and i n d e p e n d e n t and are G a u s s i a n d i s t r i b u t e d w i t h

m e a n zero and v a r i a n c e

^•Ce -1+3 I s I ] ,

for some c o n s t a n t s a and 3 w h i c h may be p h y s i c a l l y d e t e r m i n e d . T h e n for large |s| (i.e. s>>1/3) the v a r i a n c e is a p p r o x i m a t e l y a|s| c o r r e s p o n d i n g to the W i e n e r p r o c e s s , w h i l e for small

s the v a r i a n c e is a p p r o x i m a t e l y a 3 s 2/2, w h i c h i m p l i e s f i n i t e v e l o c i t y of the p a r t i c l e .

In E i n s t e i n ' s a n a l y s i s of B r o w n i a n m o t i o n he d e s c r i b e d the m o t i o n of a p a r t i c l e in terms of the t r a n s i t i o n p r o b a b i l i t y d e n s i t y f u n t i o n w ( x , t | x o rtj) of its d i s p l a c e m e n t x(t) ,

d e f i n e d such that w (x ,t | x 0 ,t o )dx is the p r o b a b i l i t y that at time t the p a r t i c l e lies b e t w e e n x and x + d x , g i v e n that it was at p o s i t i o n x Q at time t Q. E i n s t e i n s h o w e d th a t w ( x , t | x 0 ,to) s a t i s f i e s the d i f f u s i o n e q u a t i o n

9 w ( x ,t 1x p ,tq) _ a 9 2w ( x ,1 1x Q ,t 0 )

91 2 . 2

9

T h e B r o w n i a n m o t i o n p r o b l e m w a s g e n e r a l i s e d b y L a n g e v i n w h o i n c l u d e d a f r i c t i o n t e r m in h i s m o d e l , t h e L a n g e v i n

e q u a t i o n

d v

d t - b v + Z (t) (2.1.1)

w h e r e v ( t ) is t h e v e l o c i t y of t h e p a r t i c l e , - b v r e p r e s e n t s t h e d y n a m i c f r i c t i o n e x p e r i e n c e d b y t h e p a r t i c l e a n d Z(t ) is t h e r a n d o m f o r c e o n t h e p a r t i c l e . T h e d i f f u s i o n e q u a t i o n f o r L a n g e v i n 's m o d e l , d e r i v e d i n d e p e n d e n t l y b y F o k k e r in 1 9 1 4 a n d b y P l a n k in 1 9 1 7 , is

9 w ( x , v , t 1x n , v n ,t n )

8 t - v

9 w

9 x b ~ (vw)

a 2

9 v :

T h i s is t h e o r i g i n a l F o k k e r - P l a n k e q u a t i o n w h i c h w a s l a t e r g e n e r a l i s e d in a r i g o r o u s m a t h e m a t i c a l t r e a t m e n t b y

K o l m o g o r o v [ 2 7 ] .

F o r t h e s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n in t h e I to f o r m ( 1 . 1 . 2 )

d x ( t ) = a ( x , t ) d t + C ( x , t ) d W ( t ) ,

w h e r e t h e e l e m e n t s o f W ( t ) a r e m i n d e p e n d e n t W i e n e r p r o c e s s e s

w i th *

( d W (t )) = 0

a n d ( d W (t ) d W (t ) T) = D d t ,

3 w ( x , t | x 0 / t 0 ) 3 t

n

3

-

I

n 3 2 t ,

y -r-- v-- [ (CDC ) . .w (x , t x 0 , t o ) ] (2.1.2) . V-.3x.3x. 1 3

1 / 1 = 1 1 1

The terms i n v o l v i n g the f i rst d e r i v a t i v e s w i t h r e s p e c t to the e l e m e n t s of x are c a l l e d the d r i f t terms and r e p r e s e n t the d e t e r m i n i s t i c or m e a n m o t i o n of the system. The s e c o n d d e r i v a t i v e terms are the dif f u s ion terms and r e p r e s e n t

d i f f u s i o n due to r a n d o m effec t s . In terms of the t r a n s i t i o n d e n s i t y f u n c t i o n , the d i f f u s i o n terms c a use a s p r e a d i n g -out from the i n i t i a l v a l u e s (typ i c a l l y a d e l t a function, c o r r e s p o n d i n g to k n o w i n g the i n i t i a l s t a t e w i t h p r o b a b i l i t y

1) w h i l e the d r i f t terms cause a t r a n s l a t i o n of the d e n s i t y f u n c t i o n . T h e s e ideas are i l l u s t r a t e d in §2.8 w h e r e we look at an e x a m p l e of a s t o c h a s t i c c ubic o s c i l l a t o r and p r e s e n t r e s u l t s in terms of c o n t o u r p l o t s of the d e n s i t y

f u n c t i o n .

E q u a t i o n (2.1.2) is u s u a l l y c a l l e d the F o k k e r - P l a n k or the f o r w a r d K o l m o g o r o v e q u a t i o n . In this the s i s we w i l l use the latt e r name s i nce the te r m " F o k k e r - P l a n k

e q u a t i o n " m o r e c o r r e c t l y r e f e r s to the p a r t i c u l a r case of the s y s t e m d e s c r i b e d by the L a n g e v i n e q u a t i o n (2.1.1).

S i n c e those e a r l y s t u d i e s of B r o w n i a n mot i o n , d i f ­ f u s i o n m o d e l s have b e e n us e d in a w i d e v a r i e t y of a p p l i ­ cations. P r o b a b l y the m o s t s i g n i f i c a n t of t h ese was the study by C h a n d r a s e k h a r on s t o c h a s t i c p r o b l e m s in P h y s i c s and A s t r o n o m y [15]. In that a r t i c l e C h a n d r a s e k h a r c o n ­

s t a t i s t i c s and p r o b a b i l i s t i c p r o b l e m s in s t e l l a r d y n a m i c s . R a n d o m v i b r a t i o n s have also b e e n s t u d i e d by M o y a l [ 41 ] , Liu [ 35] and C a u g h e y [ 14] . S a g i r o w [ 5 0] has p r o c e e d e d f u r t h e r in this area in his study of s t o c h a s t i c m e t h o d s in the d y n a m i c s of s a t e l l i t e s . O t h e r a p p l i c a t i o n s i n c l u d e s t u d i e s of s i g n a l s w i t h r a n d o m n o i s e (S t r a t o n o v i c h [ 55] ) , p o p u l a t i o n g r o w t h (Tuckwell [57], C a p o c e l l i and R i c c i a r d i [13] and B r a d f o r d and P h i l l i p [ l l ] ) and a s t o c h a s t i c m o d e l of i n t e r a c t i n g s p e c i e s (Goel, M a i t r a and M o n t r o l l [ 20] ) -We i n v e s t i g a t e the last of these in d e t a i l in A p p e n d i x A.

In C h a p t e r 3 we p r e s e n t a d i f f u s i o n m o d e l of o c e a n - d r i f t m i g r a t i o n . This has p a r t i c u l a r r e l e v a n c e to the q u e s t i o n of the o r i g i n s of the s e t t l e r s of the P o l y n e s i a n i slands in the P a c i f i c Ocean. In §2.2 we look at the a s s u m p t i o n s and i m p l i c a t i o n s of the use of d i f f u s i o n m o d e l s and in §2.4 we s h o w that t h e r e is a c l ose c o r r e s p o n d e n c e b e t w e e n a s i m p l e r a n d o m w a l k and the e x p l i c i t f i n i t e d i f f e r e n c e s o l u t i o n of the r e l e v a n t d i f f u s i o n e q u a t i o n .

U s u a l l y d i f f u s i o n mod e l s m a k e use of the f o r w a r d K o l m o g o r o v e q u a t i o n . However, in §2.3 we c o n s i d e r the a p p l i c a t i o n of the b a c k w a r d K o l m o g o r o v e q u a t i o n (defined in §2.2) to the i n t e r i o r f i r s t p a s s a g e p r o b l e m , w h i c h is r e l e v a n t to our s t u d y of o c e a n - d r i f t m i g r a t i o n in C h a p t e r 3 .

S i n c e the r e c e n t a p p l i c a t i o n s of d i f f u s i o n m o d e l s are far m o r e c o m p l i c a t e d than the o r i g i n a l B r o w n i a n m o t i o n studies, the d i f f u s i o n e q u a t i o n s c o n c e r n e d are rar e l y

K o l m o g o r o v e q u a t i o n s a n d t h e i r n u m e r i c a l s o l u t i o n . T h e K o l m o g o r o v e q u a t i o n s a r e m e m b e r s o f a c l a s s of d e g e n e r a t e e l l i p t i c - p a r a b o l i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s c o n s i d e r ­ e d b y F i c h e r a [ X 7]

>

a n d N i r e n b e r g [ 2 6 ] • F i c h e r a p r e s e n t e d a u n i f i e d t h e o r y f o r e q u a t i o n s i n v o l v i n g t h e s e c o n d o r d e r l i n e a r d i f f e r e n t ­ i a l o p e r a t o r

L (w ) V , , , 92w

l

3 = 1 l 3

v 8w

I

1=1 l

( 2 . 1 . 3 )

d e f i n e d o n a n o p e n c o n n e c t e d s e t w i t h b . . = b . . f o r i , j = 1 , 2 , . .

13 3 i

n

l

i / j = 1 ij

> 0

S o f n - d i m e n s i o n a 1 , n a n d

in S .

s p a c e ,

T h e l a t t e r c o n d i t i o n d o e s n o t e x c l u d e t h e p o s s i b i l i t y t h a t f o r s o m e x e S a n d n o n - z e r o v e c t o r £

n I b i / j = 1 ij

(x) £ . £ .

~ i 3

0

w h i c h c o i n c i d e s w i t h t h e d e g e n e r a t i o n o f L to a n o p e r a t o r of f i r s t o r z e r o o r d e r . A n i m p o r t a n t c o n s e q u e n c e of t h i s is t h a t i n s o l u t i o n o f e q u a t i o n s o f t h i s t y p e , b o u n d a r y c o n d i t i o n s a r e n o t n e c e s s a r i l y s p e c i f i e d o n t h e e n t i r e b o u n d a r y of S. W e c o n s i d e r t h i s a s p e c t of t h e e q u a t i o n s in § 2 . 5 .

s o l u t i o n of (2.1.3) when b . .=0 for i*j. The latter

con-i]

s i d e r e d in some deta i l the K o l m o g o r o v e q u a t i o n a r i s i n g from the f i r s t p a s s a g e p r o b l e m in s i m p l e B r o w n i a n m o t i o n and w e r e able to c o m p a r e the a n a l y t i c a l s o l u t i o n for the r e l e v a n t p r o b a b i l i t y d e n s i t y f u n c t i o n w i t h s o l u t i o n s from a M o n t e C a r l o s i m u l a t i o n and s e v e r a l f i n i t e d i f f e r e n c e s c h e m e s . Hill [25] e x t e n d e d the r e s u l t s of C a n n o n and Hill to the more g e n e r a l p r o b l e m (2.1.3) . He s h o w e d that his f i n i t e d i f f e r e n c e s c h e m e is w e l l - d e f i n e d in that it leads to a set of linear e q u a t i o n s w i t h the same n u m b e r of e q u a t i o n s as u n k n o w n s and that the s o l u t i o n s to the f i n i t e d i f f e r e n c e e q u a t i o n s c o n v e r g e to a u n i q u e s o l u t i o n as the size of the fin i t e d i f f e r e n c e m e s h tends to zero. K u s h n e r [ 3 0] , in a far more g e n e r a l a p p r o a c h than our

§2.4, has s h own that the fin i t e d i f f e r e n c e s o l u t i o n of e q u a t i o n (2.1.3) c o r r e s p o n d s to c e r t a i n f u n c t i o n a l s of M a r k o v c h a i n s .

In §2.6 we look at H i l l ' s f i n i t e d i f f e r e n c e s c h e m e in some d e t a i l and sh o w that it leads to a set of linear e q u a t i o n s for w h i c h the c o e f f i c i e n t m a t r i x is large and s p a r s e and has some very d e s i r a b l e p r o p e r t i e s w h i c h make it m o s t a m e n a b l e to i t e r a t i v e r a t h e r than d i r e c t solu t i o n . In §2.7 we d i s c u s s the s t a b i l i t y p r o p e r t i e s of Hill's

2,2

A

I

U

se

D

M

T h e t e r m d i f f u s i o n , a p p l i e d to a s t o c h a s t i c p r o c e s s , h a s b e e n e x p l i c i t l y d e f i n e d b y G i h m a n a n d S k o r o h o d [19 ]

(pp 6 4 - 6 5 ) as a p r o c e s s f o r w h i c h t h e t r a n s i t i o n d e n s i t y f u n c t i o n w ( x , t | y , s ) p o s s e s s e s t h e f o l l o w i n g p r o p e r t i e s :

(i) f o r x , y , ze (-°° , °°) a n d s , t , v e C o , T ] S . T . 0 ^ s < v < t

I i'00 I I

w ( x , t I y , s ) = J ^ w ( x , t | z , v ) w ( z , v | y , s ) d z ( 2 . 2 . 1 )

T h i s is t h e S m o l u c h o w s k i o r C h a p m a n - K o l m o g o r o v e q u a t i o n

-t-h-e- o-f,

(ii) f o r a n y e > 0 a n d s , t e [ 0 , T ] w i t h s < t a n d x , ye (-0 0,0 0)

t t s ^ ^ , W ( x , t | y , s ) d y = 0 ( 2 . 2 . 2 )

I y - x I > e

T h i s c o n d i t i o n i m p l i e s t h a t it is i m p r o b a b l e t h a t l a r g e c h a n g e s in t h e p r o c e s s x ( t ) w i l l o c c u r d u r i n g s h o r t p e r i o d s o f t i m e .

(iii) t h e r e e x i s t f u n c t i o n s a ( x , t ) a n d b ( x , t ) s u c h t h a t f o r a n y e > 0 , s , t £ [ 0 , T ] a n d x , y e ( - 00,00 ) w i t h s < t

1 im 1

1 4 s t - s

a n d 1 i m 1

t ^ s t - s

/ ( y - x ) w ( x ,t I y , s ) dy y - x I < e

/ ( y - x ) 2w ( x , t | y , s ) d y y - x I < e

a ( x , t )

b ( x , t ) .

( 2 . 2 . 3 )

( 2 . 2 . 4 )

15

F r o m t h o s e a s s u m p t i o n s s e v e r a l a u t h o r s

(f

9 w ( x , t 1 y , s )

9t 9 x[ a (x ,t )w] + 9 x— 2- [ b ( x , t ) w ] (2.2.5)

(the f o r w a r d K o l m o g o r o v e q u a t i o n ) a n d

9 w (x ,t 1y ,s) 9 s

/ x 9 w , , , x 9 2 w

'a ( Y 's)ä7 ‘ ^b ( Y 's)ä ^ (2.2.6)

(the b a c k w a r d K o l m o g o r o v e q u a t i o n ) .

F o r a s t a t i o n a r y p r o c e s s th e t r a n s i t i o n d e n s i t y f u n c t i o n

9 9

is a f u n c t i o n of t-s. H e n c e r— = --r— a n d the b a c k w a r d 9 s 9 1

K o l m o g o r o v e q u a t i o n m a y be w r i t t e n as

9 w (x , 1 1 y , s )

91

a(y's)ä7 + *

5b(y's)ä7

T h i s is t h e m o r e c o n v e n i e n t f o r m f or u s e in t he f i r s t p a s s a g e p r o b l e m (e.g. §2. 3 a n d r e f e r e n c e s [16 ] , [53] ) .

T h e M a r k o v p r o p e r t y c a n be l o o s e l y d e f i n e d as m e a n i n g t h a t t h e f u t u r e s t a t e of t he s y s t e m d e p e n d s o n l y on the p r e s e n t s t a t e a n d n o t on t he p a s t . In p r a c t i c e o n e w o u l d

e x p e c t t h i s to a p p l y to a w i d e r a n g e of p h y s i c a l s y s t e m s as it is t he p r o b a b i l i s t i c i n t e r p r e t a t i o n of N e w t o n i a n d y n a m i c s . O n e t y p e of p r o b l e m f o r w h i c h t h e M a r k o v a s s u m p ­ t i o n is n o t a p p l i c a b l e is t h a t of a s y s t e m w i t h " m e m o r y " ,

for e x a m p l e

dx (t )

d t a ( x , t ) + t

f

+ b ( x , t ) Z (t ) , (2.2.8)

i n t e g r a b l e f u n c t i o n r e p r e s e n t i n g a r a n d o m noise process. This p r o b l e m has been s t u d i e d by Mo r i I 3 8

J

taken by P a w u l a [4 5] wh o s h o w e d that w h e r e a s for a M a r k o v p r o c e s s one c o uld d e r i v e the K o l m o g o r o v e q u a t i o n s from the S m o l u c h o w s k i - C h a p m a n - K o l m o g o r o v e q u a t i o n (2.2.1), for a g e n e r a l s t o c h a s t i c p r o c e s s one can d e r i v e g e n e r a l i s e d K o l m o g o r o v e q u a t i o n s from the total p r o b a b i l i t y rule

w(x, t I T (t) ) = (x , 11 y , s ;

T

T

Here T(t) is an a r b i t r a r y set of v a l u e s x ( t i ) , x (t2) » . . . A d i f f e r e n t a p p r o a c h to the n o n - M a r k o v p r o c e s s was

00

x(t )e[0,t]. The g e n e r a l i s e d K o l m o g o r o v e q u a t i o n is .K

9 w(x ,

1 1r

00 j j

1

17

by

A . (x, t

I

I

Note that the generalised Kolmogorov equations may be of infinite order. An example of this phenomenon was given by Ramakrishnan [47] in his study of randomly-fluctuating fields .

Another of Pawula's results is that

if A <°° for all positive integers n and if A „ =0 for some -- n --- 2 n ---positive integer n then A n:=0 for n> 3 . (2.2.12)

This means that if the generalised Kolmogorov equation is of finite order it must be of order two or less. Thi-s— -r e s u l t io of— g^e-a-b-i-mp-e-rbarft e^-~r ■ -e-e-p-e-eia-i-by-.i - a - p ^ d r :

» a t i a n s , .it..caa m ean that th-e a^s4ompLtixui--xxf—a - M a r k o v. p-r o OQ sc— i s- no t ae-&&s&ary f or t he—K-e-t-nte-^o-rov- eqat-ie-ns of—

S QG-on-d— or d o r -to. -bo - app-li c-abl e . To see this consider a process for which the conditional moments Aj and A 2 exist and are finite. Then if the general conditional moment is given by

n .

a

1 im n

At4-0 At , for n = 1,2,...

w (A x , t ) 1__ (A x - a l ) 2 / 2 a 2

/ 2tt a ,

I q ( A x - a 2)k / (2.2.13)

k = 0

w h e r e the c o e f f i c i e n t s q 2 , q 2 , . . . ,q all r e m a i n f i n i t e as

A t ^ O a n d m u s t s a t i s f y

N (i) (ii) (iii) /2 Tra. /27ra, /27Ta

- ( z - a , ) /2a,

l q k (z-otj) dz = 1,

k = 0 ,oo

r oo

o N

- (z - a ,) / 2 a „ r , . k

z e 1 2 z q, ( z - a i ) dz = a x ,

k = 0

N

2 - ( z - a ) / 2 a 0 v / v k

e 1 2 l q, (z - a x ) dz = a,

k = 0

U s i n g i n t e g r a t i o n by p a r t s w e f i n d t h a t

n - 2 - z 2 / 2 a

Zn e ‘ z 2 / 2 a 2 dz = ( n - l ) a z e 2 dz /

f o r n £2 a n d by r e p e a t e d i n t e g r a t i o n by p a r t s w e g e t

n -z / 2 a , ,

z e 2 dz

0, if n is o d d

n ! a 2 7 2

2 n / 2 ( f ) :

(2.2.14) , if n is e v e n

H e n c e w e c a n e v a l u a t e the n - o r d e r m o m e n t of the i n c r e m e n t s w i t h d e n s i t y f u n c t i o n (2.2;13) as

N

/2 7Ta2

n - ( z - a .) / 2 a , v , .

■ e i 2 Z q k (z" a i }

k = 0

/2ira

•oo 2 N

, , \n -z / 2 a , r k n

(z + a l ) e 2 I q z dz

-«> k = 0 k

/2rra -0° k = 0

I

r n- r - z 2/2 a

z a j e

2

I qk z dz

19

n N

- I

l

is e v e n

(k + r) ! n - r ( k + r ) / 2

2 (k + r )/2 , q k a * ^ 2 u s i n g ( 2 . 2 . 1 4 ) .

B u t s i n c e o^, a 2 ~ 0 ( A t )

^ m a x . n - r /A . ( k + r ) / 2

a £ . 0 (At) At

n k , r

<; 0 (At) n/2

H e n c e t he n - o r d e r c o n d i t i o n a l m o m e n t is

l i m n ^ l i m ^ . . , n / 2 - l A n = At + 0 Ä I S A t + 0 ° (At)

T h e r e f o r e f o r n >2 A =0 a n d t he g e n e r a l i s e d K o l m o g o r o v e q u a t i o n n

will be of o r d e r t wo or les s in e a c h of t he i n d e p e n d e n t v a r i a b l e s , r e g a-r d l o s e— ©-£— w h o t h o r— feb-e— p»o-e e & & ■■^)s-&e&-se s-— -t h e — M a r k o v p r o p e r t y — (-3 .3 .1) .

A t t h i s p o i n t w e s h o u l d n o t e t h a t a l t h o u g h the d i s c u s s i o n so f a r h a s b e e n l i m i t e d to th e o n e - d i m e n s i o n a 1 ca s e , thi s

h a s b e e n p u r e l y f or the s a k e of n o t a t i o n a l s i m p l i c i t y a n d

t h a t , as s h o w n by P a w u l a [45] a n d b y S o o n g [54] (pp 1 8 3 - 1 8 5 ) , th e t h e o r y is e q u a l l y a p p l i c a b l e to t h e n - d i m e n s i o n a l c a s e . T h e n - d i m e n s i o n a l g e n e r a l i s e d K o l m o g o r o v e q u a t i o n is

9w 91

'i k

#■ k

n n ■ j = l

( -1) __9 x .Y (k . )

D

k , k l 2

(2. 2 . 1 5 )

w h e r e t he c o n d i t i o n a l m o m e n t s are d e f i n e d by

. n k

k l' k 2... k n At4-0 ^ < . Y X _{1 = 1}

A typical problem for which one would want to use a diffusion model is in the study of an otherwise d et er ­ ministic system which is subjected to random disturbances.

Then in order to take advantage of the computational fea s i b ­ ility of a diffusion model, the usual assumption is that the random disturbance may be approximated by a white noise process, that is with a zero-mean, stationary, Gaussian process with constant spectral density. It must be noted that this

is a mathematical idealization and is physically impossible since the constant spectral density (i.e. all frequencies participate with equal intensity) would require infinite

energy. The Fourier transform of the spectral density is the a utocorrelation which in this case is a delta function. Hence another description of white noise as a de 1t a- co r r e 1ated process, meaning that there is zero correlation between

values of the process at different times.

The use of a physical impossibility to model a physical reality provides a dilemma in the study of stochastic processes. It lead Goel, Maitra and Montroll to attempt a compromise

in their stochastic model of interacting species [20].

They proposed a random process whose fluctuations are "not too rapid" and for which the autocorrelation function is "not a delta function, but is more spread out". From this rather loose definition they derive a Kolmogorov equation with no diffusion term, which implies that their assumption has lead them back to a deterministic model. However,

careful-21

investigation. In Appendix A we make a detailed study of the stochastic model of interacting species, with the aim of demonstrating the implications of fitting a diffusion model to a physical system. We find that sufficient c o n ­ ditions for the use of a diffusion model are analogous to the standard assumptions made in the study of Brownian motion (for example see Chandrasekhar [15]), namely

(i) The random fluctuations are very rapid compared with other changes in the system. In a practical situation this becomes largely a matter of scaling in that during a time interval of suitable length we would expect a random noise variable to undergo many changes of value of the

order of magnitude of its range of values while increments in all other variables in the system are very small compared with their ranges of values.

(ii) The correlation time of the random fluctuations should be small compared with the interval in (i). This implies that the expectation of the net result of the random variable over the time interval should be the mean value of the random variable.

■A 1 o o wo ha-v-e -seea- .that ■-£-e-r— a— n o n - M a r k o v— pr o-o-e-o o— a-«---a dd i tirQ-n-a-«---a-l— &a-«---amp; c m 4ition f or. the use of a- •dif-f-e-s-i-e-n— — -that t h e in-er-emen-ts— of t h e fifce-t a— v a r i a b l e s — be n e ar— Ga-ue s-ia-n-% "N o te— that- -thi-s is - an ass-u-mp t-i-o-n— r e g a-r-d-i-n-g— t-h e— d e ns i ty— f-u-n e t i o-«

a h — each— p ax-ticula r i a s t a a t — of— -time r- w h e-re as —t-h e o t h e r — t w o a s s u m p t i o n s are— -G-o-n-cerne d— w-i-t-h-- -t-he — c o r r e l a t i o n— s tree tur-e-a-s— a— f-u-n.of i o n of tim e T-he--p-a-r-ti-Gu-la-r— a d v a n t-age— t h e wh i t c

2,3 T

he

I

n t e r i o r

F

irs t

P

a s s a g e

P

r o b l e m

A n a l y s i s of the f i r s t p a s s a g e p r o b l e m has in the p a s t b e e n c o n c e r n e d w i t h d e t e r m i n i n g the p r o b a b i l i t y d i s t r i b u t i o n or m o m e n t s of the first p a s s a g e time for a s t o c h a s t i c p r o c e s s

to the e x t e r i o r b o u n d a r i e s of some r e g i o n c o n t a i n i n g it. Fo r such p r o b l e m s , as s h o w n by S i e g e r t [53] and by D a r l i n g and S i e g e r t [16], the p r o b a b i l i t y d e n s i t y f u n c t i o n of f i r s t p a s s a g e times is a s o l u t i o n of the b a c k w a r d K o l m o g o r o v

e q u a t i o n . An e x a m p l e of this is the s t u d y of v i b r a t i o n a l s y s t e m s s u b j e c t e d to r a n d o m d i s t u r b a n c e s (see for e x a m p l e [ 4] , [ 32] or [ 5 l] ) w h e r e the aim is to find the p r o b a b i l i t y d e n s i t y f u n c t i o n or m o m e n t s of the time t a k e n for the a m p l i ­ tude of the v i b r a t i o n to e x c e e d p r e s c r i b e d limits for the f i r s t time. This is p a r t i c u l a r l y a p p l i c a b l e in the study of the r e l i a b i l i t y of s y s t e m s .

C o n s i d e r a s t a t i o n a r y M a r k o v p r o c e s s r e p r e s e n t e d by its p o s i t i o n x(t) in a clo s e d r e g i o n S of n - d i m e n s i o n a l C a r t e s i a n space. For some tar g e t r e g i o n RcS the i n t e r i o r f i r s t p a s s a g e p r o b l e m is c o n c e r n e d w i t h d e t e r m i n i n g the p r o b a b i l i t y d e n s i t y f u n c t i o n f (0 ,t|y,0) of the first p a s s -age times t from a s t a r t i n g p o i n t yeS = S\R to some p o i n t

R

on a , the b o u n d a r y of R (see F i g u r e 2.3.1). We assume that R

the t r a n s i t i o n d e n s i t y f u n c t i o n w Q (x,t|y,0) of the p o s i t i o n xeS s a t i s f i e s the b a c k w a r d K o l m o g o r o v e q u a t i o n

3w (x,t|y,0) n 3w (x,t|y,0)

V

■

t'l'v"

n

+ *3 I b . (y , t) i f j - 1 J ~

3 2w q (x ,t I y /0)

9 y j

respectively, as discussed in §2.1. The initial condition for (2.3.1) is

w 0 (x ,0 Iy,0) = S ( x - y ) m (2.3.2) In general the boundary condition on 0^ could allow for partial reflection and partial absorption. This would

take the form, for y e a the outer boundary of S, ~ s s

9 w 0 (x , 11 y , 0 )

p (^s } W ° I ~S ' 0) + [1- P (y s )]---d v ~(’y )---- = °' ( 2 • 3 • 3 )

where p (y ) is the probability of total absorption across represents spatial different-the boundary at y and 77—

-~S

iation in the direction of the outward normal to the boundary a t y .

We define Vo (y(t) = / wo( x , t | y , 0 ) d x (2.3.4)

~ O ~ ~

and note that this is the probability of x(t)eS conditional on x (0 ) = y e S .

We now consider the problem with the region R as

a sink. That is, we impose the condition of total absorption across the boundary Ö . Then w ( x , t | y , 0 ) , the transition

• R ~ ~

p r o b ability density of x(t) in S will satisfy the backward

~ R

Kolmogorov equation 3 w (x ,t I y ,0)

--- 21— g-jy22--- = Lw ( x , t I y , 0 ) (2.3.5)

with initial condition

w ( x, 0 Iy / 0) = 6 (x-y)

and boundary conditions on O that S

25

p (y ) w (x , t I y , 0) +

[l-p(y )]

** o ~ ~ o ^ o

9w ( x ,11y ,0)

(y

s)

f ° r y g€ O g a n d t> 0 .

We a l s o i m p o s e the b o u n d a r y c o n d i t i o n on

o

a b s o r p t i o n )

w ( x ,t | y , 0)

~ ~ R (2.3.8)

f or y e O a n d t> 0 . ~ R R

D e f i n e V (y , t) w (x ,t y ,0) dx (2.3.9)

f or y eS a n d n o t e t h a t t h i s is the p r o b a b i l i t y of ~ R

x ( t ) e S c o n d i t i o n a l on x(0) = y e S a n d x ( v ) ^ a fo r a ny v e t o ,t]

~ R ~ ~ R ~ R

T h e p r o b a b i l i t y d e n s i t y f u n c t i o n f ( x , t | y , 0 ) of f i r s t p a s s a g e t i m e s f r o m y to x is d e f i n e d s u c h t h a t f ( x , t | y , 0 ) d t

is the p r o b a b i l i t y of the f i r s t p a s s a g e t i m e f r o m y to x b e i n g in the i n t e r v a l [ t , t + dt] . Hence the p r o b a b i l i t y

of x (t ) e O f or t^O is

~ R

w h e r e

F ( R , t | y , 0 )

f

(a

K ~

ft

f

(a

f (x,t y , 0 ) d x

(2.3.10)

(2.3.11)

T h e d e n s i t y f u n c t i o n f ( a R /t|y,0) m a y be f o u n d f r o m V Q (y,t) a n d V ( y ;t) a c c o r d i n g to the f o l l o w i n g t h e o r e m :

T

For y eSR and

v 0

(y-t)»

v(y,t) and f(aR ,t|y,o) as

d e f i n e d in ( 2 . 3 . 4 ) , (2.3.9) a n d (2.3.11) r e s p e c t i v e l y ,

P R O O F C o n s i d e r the b e h a v i o u r of x ( t ) e S w h e n the regi o n R exi s t s in S but is not a sink. T h e n there are three p o s s i b l e e v e n t s w h i c h we may o b s e r v e for the p r o c e s s w h i c h

y e S R at t=0.

x(t)eS, w i t h p r o b a b i l i t y

Pi = Vo (y /1 )

x ( t ) e S but x ( v ) ^ G for any time ve[0, t l .

~ ~ R

Th e p r o b a b i l i t y of this e v e n t is P2 = V (y , t) .

x ( t ) e S and x ( v ) e G for some time v e C o , t ] .

~ ~ R

T h a t is, x(t) has r e a c h e d some p o i n t on G

~ R

for the f i r s t time at v^t and then m o v e d on to some o t h e r p o i n t in S d u r i n g the r e m a i n i n g i n t e r v a l Cv,t]. The p r o b a b i l i t y of this e v ent i s

P 3 = / t f(G ,v I y ,0)Vo (G ,t - v ) d v .

0 R ~ R

C l e a r l y , e v e n t (i) can be r e a l i s e d only by one or o t h e r of even t s (ii) or (iii) . Then, since e v e n t s (ii) and (iii) are m u t u a l l y e x c l u s i v e and e x h a u s t i v e ,

Pi = P 2 + P 3

w h i c h p r o v e s the theorem. □

C o m b i n i n g r e s u l t (2.3.12) w i t h the d e f i n i t i o n (2.3.10) of F(R, t | ^ , 0 ) we see that

F (R ,t I y ,0) = Vo(y,t) - V (y ,t )

+ / t f (G ,v I y ,0) [ 1-Vo (G ,t - v ) ]dv

o R ~ R

star t s from

(i)

(ii)

(iii)

27

Now, if w g integrate equation (2.3.1) , the backward Kolmogorov equation for w Q (x,t|y,0) , with respect to x over the region S we find that

- L ) V o (y, t) = 0

Similarly, integration of (2.3.5) over S gives R

(— - L)V(y,t)

Therefore, on applying the operator (-^— - L) to equation a t

(2.3.13) we get

(— - L) F (R, t | y , 0) = ( A

3 t

-

If we take the Laplace transforms of both sides notation

F * ( R , A ,y ) = 00

/

t

★ roo -At I

f (a , A , y ) =

K

f(aR ,t | y ,0)d t ,

★ oo

r “At

Vq (y , A ) =

i e

and noting that

F (R ,0 I y ,0) = f (a ,0 y,0) = 0, for yeS

~ K ~ K

we find that

(A-l )f (R/A,y) = (A-L)[^- - v Q (aR ,A)]f (o ,A,y)

* *

But, since f (ö ,A,y) = X F (R,A,y) (from the Laplace transform of (2.3.10)), this becomes

^ ^ ^