A Case of a Random Difference Equation Connected with a Filtered Binary Process

ADVANCES IN APPLIED MATHEMATICS ARTICLE NO.

17, 88]100 Ž1996.

0004

A Case of a Random Difference Equation Connected with a Filtered Binary Process Jean-Franc¸ois Chamayou Laboratoire de Statistique et Probabilites, ´ Uni¨ ersite´ Paul Sabatier, 118 route de Narbonna F31062, Toulouse, France Received June 12, 1995

Pawula and Rice have studied filtered random binary processes; the distribution of their process for special values of the parameters is computable in closed form. The same study is done for the random linear difference equation arising from triggered shot noise for parameters: 12 , 1. A closed form for the probability density of the stationary case is obtained. Q 1996 Academic Press, Inc.

I. INTRODUCTION In Refs. w3, 11x the connection between the random telegraph wave problem and the following random difference equation was established, namely, Ynq 1 s X nq1 Ž 1 y Yn . ,

n s 1, . . . ,

Ž 1.

where Y1 s X 1 and the X i are independent real random variables with the same distribution. Pawula and Rice in a series of papers w12]15x presented a study of the random telegraph signal with filtering. The complete calculations were performed for a special case of the ratio ‘‘a’’ of the Poisson process rate and the exponential decay parameter. The density of probability was given explicitly. In w5x, the triggered shot noise process was studied using the random difference equation Ž1.. The purpose of this paper is to provide a complete numerical study of the stationary distribution of the Yn in Ž1. for the distribution of the X n given by g X Ž x . s a2 Ž yLog x . x ay 1 |x0 , 1w Ž x . dx for a particular value, a s 1r2 Žsee Table I.. 88 0196-8858r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

with a ) 0

Ž 2.

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89

TABLE I a s 1r2 x

f Ž x.

gŽ x.

FŽ x.

GŽ x .

Monte-Carlo

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1

3.415 1.819 1.204 8.705 Žy1. 6.600 Žy1. 5.144 Žy1. 4.079 Žy1. 3.266 Žy1. 2.627 Žy1. 2.113 Žy1. 1.692 Žy1. 1.344 Žy1. 1.052 Žy1. 8.058 Žy2. 5.977 Žy2. 4.219 Žy2. 2.744 Žy2. 1.531 Žy2. 5.821 Žy3. 0

3.349 1.820 1.225 8.997 Žy1. 6.931 Žy1. 5.495 Žy1. 4.436 Žy1. 3.622 Žy1. 2.976 Žy1. 2.451 Žy1. 2.015 Žy1. 1.649 Žy1. 1.336 Žy1. 1.066 Žy1. 8.305 Žy2. 6.237 Žy2. 4.407 Žy2. 2.776 Žy2. 1.316 Žy2.

0.581 0.703 0.777 0.828 0.866 0.895 0.918 0.936 0.951 0.963 0.972 0.980 0.986 0.991 0.994 0.997 0.998 0.999 0.999 1

0.559 0.680 0.755 0.807 0.847 0.877 0.902 0.922 0.939 0.952 0.963 0.972 0.980 0.986 0.991 0.994 0.997 0.999 0.9997 1

2.386 1.354 1.008 7.840 Žy1. 6.000 Žy1. 4.280 Žy1. 3.680 Žy1. 2.540 Žy1. 2.220 Žy1. 1.620 Žy1. 1.520 Žy1. 1.320 Žy1. 9.000 Žy2. 7.200 Žy2. 5.200 Žy2. 3.600 Žy2. 1.800 Žy2. 1.400 Žy2. 2.000 Žy3.

II. DENSITY CALCULATIONS Introduce the following sequence of random variables: ny1

Yn s

Ž . Ý Ž y1. nymq1 exp yb Ž t 2 n y t 2 m . 4 ,

Ž 3.

ms0

where by b ) 0 and t i are the dates of a Poisson process of rate l. Hence the sign change in Ž3. is triggered by the arrival of two Poisson events. Introducing the sequence of i.i.d. random variables Zi s exp  ylŽ t i y t iy1 . 4 , we obtain Yn s

ny1

n

ms0

ismq1

Ž . Ý Ž 1. nym q1 Ł Ž Z2 i ? Z2 iy1 . brl .

JEAN-FRANC ¸OIS CHAMAYOU

90

Note that the distribution of Zi is uniform on w0, 1x and if a s lrb , then the distribution of X i s Ž Z2 i ? Z2 iy1 .

br l

is Ž2.. The random variables Ž Yn .`ns1 clearly satisfy the difference equation Ž1., with the above independent random variables X i . Let us consider the stationary distribution F Ž n ª `. of the Markov chain Ž Yn .`ns1 defined by Ž1. Žthe proof of the existence of F is easy.: clearly the alternating series Ss

`

m Ý Ž y1. X1 . . . X m

ms0

converges, and the hypothesis of Proposition 1 in w4x is thus fulfilled; see also Wervaat w16x. The distribution function FY satisfies the functional equation FY Ž y . s

1

HY Ž 1 y F

Y

ž

1y

y x

/

g Ž x . dx,

y g x 0, 1 w .

Ž 4.

Introducing the variable s s yrx and replacing g Ž x . by Ž2. we get FY Ž y . s a2 y a

1

Hy

Ž 1 y FY Ž 1 y s . .

ž

yLog

y

ds

s

aq1

ž //

s

.

Clearly this implies that F is infinitely differentiable on x0, 1w. Differentiating, we get F9 Ž y . s a

FŽ y. y

y a2 y ay1

1

Hy

Ž1 y F Ž1 y s. .

ds s

aq 1

,

and differentiating again we have y 2 f 0 Ž y . q Ž 3 y 2 a . yf 9 Ž y . q Ž 1 y a2 . f Ž y . s a2 f Ž 1 y y . ,

Ž 5.

where f Y is the probability density of y. It is easy to verify that this delayed differential equation of order 2 can be transformed by two differentiations into an ordinary fourth-order differential equation. Only two values of ‘‘a’’ give rise to real simplifications. For a s 1, we obtain a third-order ordinary differential equation for the probability

A RANDOM DIFFERENCE EQUATION AND FILTERED BINARY PROCESS

91

density, 2 Ž y y 1. y 2 f - Ž y . q Ž 2 y y 3. y Ž y y 1. f 0 Ž y .

q Ž y 2 y y q 1 . 9 Ž y . y yf Ž y . s 0 which can be solved by a series method. The numerical results are presented in Table II. Let us now consider the other simple case a s 1r2. THEOREM 1. For a s 1r2, if f is the density of the stationary distribution of the Marko¨ chain Ž Yn .`ns1 of Ž1., and if uŽ y . s

y g x 0, 1 w ,

'y r Ž 1 y y . f 9 Ž y . , 3

Ž 6.

then u is a solution of the following hypergeometric differential equation 4 y Ž y y 1 . u0 Ž y . q 4 Ž 2 y y 1 . u9 Ž y . q 2 u Ž y . s 0.

Ž 7.

Proof. We get from Ž5. for a s 1r2 4 y 2 f 0 Ž y . q 8 yf 9 Ž y . q f Ž y . s f Ž 1 y y . ,

TABLE II as1 x *0.1

f2Ž x.

f Ž x.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

3.776 2.687 2.072 1.651 1.337 1.091 0.892 0.728 0.591 0.475 0.377 0.294 0.224 0.165 0.116 0.076 0.044 0.021 0.006 0

3.864 3.077 2.464 1.969 1.565 1.233 0.960 0.736 0.554 0.408 0.292 0.201 0.133 0.082 0.047 0.024 0.010 0.003 0.0004 0

y g x 0, 1 w .

Ž 8a .

JEAN-FRANC ¸OIS CHAMAYOU

92

Replacing y by Ž1 y y ., we get 2

4Ž Ž 1 y y . f 9Ž 1 y y . .9 q f Ž 1 y y . s f Ž y . .

Ž 8b .

Adding Ž8a. and Ž8b. yields 4

½Ž y

2

2

f 9 Ž y . . 9 q Ž Ž 1 y y . f 9 Ž 1 y y . . 9 s 0.

5

Integrating we get 2

y 2 f 9Ž y . y Ž 1 y y . f 9Ž 1 y y . s K .

Ž 9.

Evaluating at y s 1r2, shows that K s 0. Differentiating Ž9. we obtain 4 Ž y 2 f 9 Ž y . . 0 q f 9 Ž y . s yf 9 Ž 1 y y . ; multiplying by Ž1 y y . 2 and using Ž8a. and Ž8b., we have 4Ž 1 y y .

2

Ž y 2 f 9 Ž y . . 0 q Ž Ž 1 y y . 2 q y 2 . f 9 Ž y . s 0.

Setting pŽ y . s y 2 f 9Ž y . we finally get 2

4 Ž 1 y y . y 2 p0 Ž y . q Ž 1 y 2 y q 2 y 2 . p Ž y . s 0.

Ž 10 .

The solution of Ž10. is Žcf. w8, -2-393x. pŽ y . s

'y Ž 1 y y . u Ž y . ,

where u is the solution of the hypergeometric differential equation Ž7.. We now rely on the classical theory of the hypergeometrical differential equation, as described in w6, p. 1045ffx. With the standard notation Ž a , b , g . the coefficients of Ž7. yield ab s 12 , a q b s g s 1. Therefore the solutions of Ž7. are linear combinations of the Gauss hypergeometric functions s1 Ž y . s2 F1 Ž a , 1 y a ; 1; y . ,

s2 Ž y . s2 F1 Ž a , 1 y a ; 1; 1 y y . ,

where a s Ž1 q i .r2. Our next task is to find C1 and C2 such that u Ž y . s C1 s1 Ž y . q C 2 s 2 Ž y . corresponds to the f in Theorem 1. We now prove the following.

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93

THEOREM 2. If the solution of Ž7. is such that Ž6. holds, where f is the density of the stationary distribution of Ž1., then u Ž y . s py2 G

ž

3qi 4

4

Ž s1 Ž y . q s 2 Ž y . . ,

/

and where G is the gamma function. Proof. The density of probability is fY Ž y. s

'1 y z

1

Hy

 C1 2 F1Ž a , 1 y a ; 1; z .

z 3r2

qC2 2 F1 Ž a , 1 y a ; 1; 1 y z . 4 dz,

Ž 11 .

where C1 and C2 are unknown constants. Since for a q b s g , Ž . Ž . Ž Ž . Ž .. 2 F1 a , b ; g ; 1 y z , yG a q b r G a G b Log z when z ª 0, then one deduces easily that

ž

f Y Ž y . , y 2C2

G

ž

1qi 2

2

/

Log y

/'

y

when y ª 0.

The distribution function is FY Ž y . s

y

1

H0 Hj

'1 y z z 3r2

 C1 2 F1Ž a , 1 y a ; 1; z . qC2 2 F1 Ž a , 1 y a ;1; 1 y z 4 dz d j .

Ž 12 .

The moments are M Ž s. s

1

1

H0 j Hj s

'1 y z z 3r2

 C1 2 F1Ž a , 1 y a ; 1; z . qC2 2 F1 Ž a , 1 y a ; 1; 1 y z . 4 dz d j . Ž 13 .

Interchanging the order of integration, we get M Ž s. s

1 sq1

1

H0

z sq1r2y1 Ž 1 y z .

3r2y1

=  C1 2 F1 Ž a , 1 y a ; 1; z . q C2 2 F1 Ž a , 1 y a ; 1; 1 y z . 4 Ž 14 .

JEAN-FRANC ¸OIS CHAMAYOU

94

and Žcf. w6, Section 7.5, p. 848x.

M Ž s. s

G sq

ž

1 sq1

1

3

G

/ ž /½ .

2 2 GŽ s q 2

C 1 3 F2 a , 1 y a , s q

ž

qC2 3 F2 a , 1 y a ,

ž

3

1

; 1, s q 2; 1

2

; 1, s q 2; 1

2

/5

/

. Ž 15 .

It is known that 3 F2

Ž ?, ? , ? ; ? , ? ; 1 .

the generalized hypergeometric function of argument 1 can be reduced for s s 0, 1, 2 to simpler expressions Žgamma functions., in the Watson case w8x, 3 F2

Ž a , 1 y a , d; 1; 2 d; 1 .

Ž s s 1. ,

and in the contiguous cases studied by Lavoie w9, 2, 10x 3 F2

Ž a , 1 y a , d; 1, 2 d q 1; 1 . ,

s s 0, 2,

3 F2

Ž a , 1 y a , d; 1, 2 d y 1; 1 . ,

s s 0, 2.

Since the moments are also directly given by the difference equation Ž1., using 1 s

1

H0 y f Ž y . dy s H0

x s g Ž x . dx ?

1

H0

s

Ž 1 y y . f Ž y . dy,

Ž 16 .

we get

m1 s

1r4

Ž 1 q 1r2.

2

Ž 1 y m1 . « m1 s

m2 s

1

Ž s s 1.

10

Ž s s 2.

1 30

and so on for higher s. Using the Watson result w9x we get for s s 1

Ž C1 q C 2 .

G 2 Ž 3r2 . 2 G Ž 3.

3 F2

ž

a,1 y a,

3 2

; 1, 3; 1 s m 1 s

/

1 10

,

Ž 18 .

where G

3 F2 Ž a , 1 y a , d; 1, 2 d; 1 . s

G

ž

aq1 2

1

G 1 G dq

G

2 2ya 2

1

GŽ d. 2 1ya a G dq G dq 2 2

ž / Ž. ž /ž /ž

/

/ž

.

/

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95

After straightforward calculations using d s 32 ,

ž

C1 q C 2 2

/ž

p2

G

ž

3qi 4

4

/

/

s 1.

Ž 19 .

Now, for the contiguous cases Žcf. w9x. for s s 0, we get

Ž C1 q C2 . A 0 q Ž C2 y C1 . B0 s 1 s m 0 , where 1

A 0 q B0 s

2

1

G2

G2 A 0 y B0 s

ž / ž ž / ž 3 F2

2

2

a

ž

G 1y

ž

2

a 2

a Ž1 y a . 4

1

a,1 ya,

3 F2

G 1q

B0 s K ?

; 1, 2; 1

2

/

1

2

A0 s K

3

a,1 y a,

G 1q

/ž

G 1y

/ž ž G G

ž ž

1qa 2 3ya 2

2

; 1, 2; 1

1ya 2 1ya 2

/

/ //

G 1y

/ž /ž

G 1q

a 2 a 2

/ /

and Ks

G 2 Ž 1r2 . 2 a Ž1 y a . GŽ a . GŽ1 y a .

.

Then A0 s p

2

2 G

ž

3qi 4

4

/

,

using the Gauss duplication formula for the gamma function. But Ž C1 q C2 . A 0 s 1. Hence, C1 s C2 s C Ž B0 / 0.. The same result could have been obtained using s s 2, a very useful tool for computing the 3 F2 Ž1. for s ) 2 is the algorithm given by Wimp in w17x.

JEAN-FRANC ¸OIS CHAMAYOU

96

III. NUMERICAL CALCULATIONS < G ŽŽ3 q i .r4.< 2 has been computed by Pawula and Rice using an integral representation w12x. We prefer to use the fast converging series representation of 2 F1 Žfor argument 1r2; see w1x., 2 F1

ž

1qi 1yi , ; 1; 1r2 s 'p 2 2

G

/

ž

3qi 4

2

/

Ž 20 .

and G

3qi

ž

4

2

s 1.288788405.

/

The series representation for a hypergeometric function can be used, namely, 2 F1

Ž a , 1 y a ; 1; y . s

Ž a . mŽ1 y a . m

`

Ý

Ž m! .

ms0

2

ym,

where Ž a. m is the Pochhammer symbol: Ž a . m s a Ž a q 1. . . . Ž a q m y 1. for 0 F y F 0.85; see w7 a x. Even better is the method with acceleration of convergence presented in w7 b x Žwhich can be extended to complex parameters.. for 0.85 - y - 1. The integral representation for the hypergeometric function was used. We get for the real part: 2 F1

ž

1qi 1yi , ; 1; y 2 2

ž

s 1

G

ž

1qi 2

/ 2

/

1

/H

cos Ž 12 Log Ž Ž 1 y t . r Ž t Ž 1 y yt . . . . dt

't Ž 1 y t . Ž 1 y yt .

0

,

noting that G

ž

1qi 2

2

/

s

p Ch Ž pr2 .

,

Ž 21 .

Žsee w1x., the imaginary part being zero since a and Ž1 y a . are complex conjugates. The quadrature formula Ž25-4-39 in w1x. makes use of Tchebichev polynomials. The integral Ž11. giving the density is computed

A RANDOM DIFFERENCE EQUATION AND FILTERED BINARY PROCESS

97

using

f Y Ž y . s f Ž 1r2 . q C

1r2

Hy

'Z Ž 1 y Z . Z2

Ž 2 F1Ž a , 1 y a ; 1; Z . yg x 0, 1 x

q2 F1 Ž Ž a , 1 y a ; 1; 1 y Z . . dZ,

Žsee Appendix II for f Ž1r2. calculation . and Ž12. for the distribution function is

FY Ž y . s yf Ž y . y yC

1 2

1r2

Hy

f Ž 1r2 . q F Ž 1r2 .

'Z Ž 1 y Z . Z

Ž 2 F1Ž a , 1 y a ; 1; Z .

q2 F1 Ž a , 1 y a ; 1; Z . . dZ,

y g x 0, 1 x Ž 23 .

Žsee Appendix II for FY Ž1r2. calculation .. Tables I and II provide a comparison between f Y Ž x . and the density g X Ž x . and between FY Ž x . and the distribution function GX Ž x .. f 2 Ž x . is the exact density of probability of X 2 Ž1 y X 1 ., L2 Ž 1 . ? Log Ž x . q L 3 Ž x . y L 3 Ž 1 . , yL

x g x 0, 1 x ,

Ž 24 .

where L 2 and L 3 are respectively the dilogarithm and the trilogarithm functions; see w1x.

V. CONCLUSION Motivated by a paper of Pawula and Rice we study a special case of a random linear difference equation Ž1.. The cases where the random variable X i Žin Ž1.. is beta-distributed have been widely studied w4x since closed forms can be easily derived Ževen for non-linear difference equations.. Other cases of X i distributions, like the one studied here, give rise very rarely to a closed form for the density of the stationary case. However, one can notice that the logarithmic distribution used here is in fact the product of two special beta distributions.

JEAN-FRANC ¸OIS CHAMAYOU

98

APPENDIX I: AN INVESTMENT MODEL Starting capital invested: 1 First year: Y1 s X 1 ? 1. X i random rates g w0, 1x. The X i are independent. Second year: Y2 s X 2 Ž1 y Y1 .. We use 2Y1 s 2 X 1 and continue with the capital Ž1 y Y1 . invested. Ž n q 1. th year: Ynq1 s X nq1Ž1 y Yn ., n s 1, 2, . . . the random difference equation is obtained. We reinvest Yny 1 and use 2Yn . We must repay the deficit, if 2Yn y Yny 1 - 0; to avoid this, we can spend Yn each year and make a short term investment with the remaining Yn . The aim of the model is to keep the long term capital stable, just swapping < Yn y Yny1 < each year, using the fact that random reduction of the rate by half is highly improbable. The advantage of the model is the use of double the funds, instead of a simple expectation of the year dividends.

APPENDIX II: f Y Ž1r2. AND FY Ž1r2. CALCULATION 1

f Y Ž 1r2 . s C

H1r2

'Z Ž 1 y Z . Z2

Ž 2 F1Ž a , 1 y a ; 1; Z . q2 F1 Ž a , 1 y a ; 1; 1 y Z . . dZ

sC

1

½H

0

y

'Z Ž 1 y Z .

1r2

H0

Z2

'Z Ž 1 y Z .

1r2

q

H0

Ž1 y Z .

2

Ž 1.

2 F1

Ž a , 1 y a ; 1; Z . dZ

2 F1

Ž a , 1 y a ; 1; Z . dZ . Ž 2 .

5

Let I0 s

1

H0

'Z Ž 1 y Z . Z2

2 F1

Ž a , 1 y a ; 1; Z . dZ

s G Ž 1r2 . G Ž 3r2 . ?3 F2 Ž a , 1 y a ; y1r2; 1, 1; 1 .

ž

s Ž yp . ? 2 ch Ž pr2 . G 2

ž

3qi 4

4

/

p3

/

Ž 3.

Žsee w10x for F32 calculation . and I1 s y

1r2

H0

Ž1 y 2 Z .

Ž ZŽ1 y Z . .

F 3r2 2 1

ž

a 1ya , ; 1; 4Z Ž 1 y Z . dZ, Ž 4 . 2 2

/

A RANDOM DIFFERENCE EQUATION AND FILTERED BINARY PROCESS

99

using a Goursat quadratic transformation for the Gauss hypergeometric function Žsee w1, p. 561; valid for 0 F Z F 1r2.. and the change of variable v s 4ZŽ1 y Z ., 1

I1 s 2

vy1 r2y12 F1

H0

s y2

G Ž y1r2.

a 1ya , ; 1; v d v 2 2

ž

/

?3 F2

G Ž 1r2 .

a 1ya , , y1r2; 1, 1r2; 1 . . 2 2

ž

/

Ž 5.

Use is made of the algorithm in w17x to compute 3 F2 : Cs G FY Ž 1r2 . s

ž

1 2

4

3qi

p 2;

/

4

ž / 2

s C Ž I1 q I0 . s 0.2112966;

'j Ž 1 y j .

1r2

f Ž 1r2 . q C

1

f

H0

j

Ž 2 F1Ž a , 1 y a ; 1; j .

q2 F1 Ž a , 1 y a ; 1; 1 y j . . d j . Ž 6 . Using the same method, FY Ž 1r2 . s

1 2

f Y Ž 1r2 . q C Ž I2 q I3 . ,

where I2 s

'j Ž 1 y j .

1

H0

1yj

2 F1

Ž a , 1 y a ; 1; j . d j

s G Ž 3r2 . G Ž 1r2 . 3 F2 Ž a , 1 y a , 3r2; 1, 2; 1 .

s

p

?

2



ch Ž pr2 . G 2

p G

ž

3qi 4

4

q2

p

/

ž

4

3qi

/

4

0

3

Ž 7.

See w9x for the calculation in terms of gamma functions and I3 s s s

1r2

Ž1 y 2j .

2 F1

H0 'j Ž 1 y j . 1

1

H 2 0

ž

v 1r2y12 F1

1 G Ž 1r2 . 2 G Ž 3r2 .

ž

ž

a 1ya , ; 1; 4j Ž 1 y j . d j 2 2

/

a 1ya , ; 1; v d v 2 2

/ ž ?3 F2

/

a 1ya , , 1r2; 1, 3r2; 1 . 2 2

/

Ž 8.

100

JEAN-FRANC ¸OIS CHAMAYOU

Use is made of the algorithm in w17x to compute 0.9628974.

3 F2

and F Ž1r2. s

ACKNOWLEDGMENT I am deeply indebted to Professor G. Letac for reading the manuscript and making very helpful comments.

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