On the Integral of Geometric Brownian Motion

On the Integral of Geometric Brownian MotionAuthor(s): Michael SchröderSource: Advances in Applied Probability, Vol. 35, No. 1 (Mar., 2003), pp. 159-183Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1428278 .

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Adv. Appl. Prob. 35, 159-183 (2003) Printed in Northern Ireland

? Applied Probability Trust 2003

ON THE INTEGRAL OF GEOMETRIC BROWNIAN MOTION

MICHAEL SCHRODER,* Universitiit Mannheim

Abstract

This paper studies the law of any real powers of the integral of geometric Brownian motion over finite time intervals. As its main results, an apparently new integral representation is derived and its interrelations with the integral representations for these laws originating by Yor and by Dufresne are established. In fact, our representation is found to furnish what seems to be a natural bridge between these other two representations. Our results are obtained by enhancing the Hartman-Watson Ansatz of Yor, based on Bessel processes and the Laplace transform, by complex analytic techniques. Systematizing this idea in order to overcome the limits of Yor's theory seems to be the main methodological contribution of the paper.

Keywords: Brownian motion; Laplace transform; Asian option

AMS 2000 Subject Classification: Primary 60J65; 91B28

1. Introduction

This paper works towards a synthesis of the representations derived in recent years for the law of the integral of geometric Brownian motion over a finite time interval. These processes have both a surprisingly rich theory and manifold applications ranging from the physics of random media to mathematical finance and insurance. In fact, the insurance motivated study of certain perpetuities in [3] seems to have initiated this line of research. Here the integral of geometric Brownian motion over the whole time axis is considered and is shown to be distributed as the reciprocal of a certain gamma variable. In mathematical finance, on the other hand, valuing the so-called Asian options asks for as explicit as possible an expression for a certain functional of the integral of geometric Brownian motion over a finite time interval. Drawing on his probabilistic interpretation of the Hartman-Watson identities in [13], Yor was able to determine the law of these processes in [14]. His approach, using the Laplace transform and based on Bessel processes, has proved both fertile and fundamental; the string of papers he wrote about it are now collected in [15]. In a recent development initiated in [4], however, the focus has shifted towards studying the reciprocal of the integral of geometric Brownian motion. As one of the latest results, [6] derives an integral for the latter's law in terms of Hermite functions. This is a structural advance in that Hermite functions form a well-studied class of special functions which generalize both the error function and the Hermite polynomials. While they come from boundary value problems in potential theory for domains whose surface is an infinite parabolic cylinder, they have already appeared in [9] in connection with valuing Asian options. Still, the general structure that these results suggest for a Hermite function form of the law of the integral of geometric Brownian motion itself seems to be different from Dufresne's findings in several ways.

Received 10 May 2002; revision received 4 November 2002. * Postal address: Keplerstrasse 30, D-69469 Weinheim, Germany. Email address: [email protected]

159



160 M. SCHRODER

The general point of view of this paper is that Yor's original Hartman-Watson theory can be developed into a framework for also seeing Dufresne's results and for understanding the role of Hermite functions in the law of the integral of geometric Brownian motion in particular. This seems to be different in spirit from efforts which aim at a better understanding of this circle of ideas by developing alternatives to Yor's approach, such as those of [1], [20] based on the Feynman-Kac Ansatz and now of [6] using PDE methods. We focus on the structure of the laws of the integral of geometric Brownian motion raised to any nonzero real power 8, which we term the e-power laws for short; not least to be able to explicitly pin down the effect of this parameter 8 on the structure of the laws.

As the first main findings of this paper, two forms of these e-power laws are discussed in Section 2: a first which generalizes the integral representation obtained in [14], and a second in terms of Hermite functions. The construction of the latter in Sections 5-7 is by explicitly interrelating it with Yor's form of the e-power laws. However, the relation of either of these forms of the e-power laws with those originating from [6] as recalled in Section 2.4 seem far from being obvious. So the second main result of this paper is that we are able to establish such an explicit interconnection, and this is explained in Section 8.

Thus, we are able to explicitly interconnect all three of the above representations of the

e-power law, and moreover in a way which is different from that of [18]. In fact, the latter paper explicitly interrelates Yor's integral for the e-power laws with that of Dufresne by retracing the key steps of the latter's construction in [6]. This paper's argument is built around our Hermite function form of the e-power law, which, in fact, seems to furnish some sort of missing link between the other two above forms of the e-power laws.

Conceptually, the results of this paper are obtained by further developing what we would like to call Yor's Hartman-Watson approach of [13] whose key results we have formalized in Section 3. Originally based on Bessel processes and the Laplace transform, we enhance this approach by a systematic use of complex analytic methods. This avoids the problems with Bessel processes of negative indices in particular and, in fact, seems to be the main methodological contribution of our paper.

To conclude, it remains to mention the fourth known form of the laws of the integral of geometric Brownian motion: the Barnes-type integral representations in terms of Whittaker functions derived in [21, Section 4.1]. These are proved using expansion into eigenfunctions and further Hilbert space spectral theory, and the missing link to this approach will hopefully be found in future work.

2. Statement of results

2.1. Basic concepts This paper studies the processes A(v',) for any reals 8e 0 and v defined as the eth power

ofYor's processes A(V), that is, for any nonnegative real t, we have

A VE)= (AV ) ,

where

A) = e2(vx+Bx) dx

and B is Brownian motion. The focus is on characterizing the law of A(v,E) which is denoted by a(V'e) and, at any time t, defined as given by the densities

t (w)



On the integral of geometric Brownian motion 161

log R + iO

log R

log R - iO

FIGURE 1: The contour log Co, R

..0.'' I /

FIGURE 2: The Hankel contour Co, R R.

on the nonnegative real line of the random variable AV'se). Notice that results about at(V,) imply results about the law ot(v) = ot(vl) of A(") and the law M(V) = a(v,-1) of the reciprocal of A(V) More precisely, this paper characterizes a(v,e) using integral representations. Their general structure is that as the product of the functions cv,e,t given by

I 2v/2 _

I (v t + 1/8) W(1128)(v-1)- ,,t(w)= exp (2t + w-

multiplied by integrals over the contours of integration log CO, R given in Figure 1 which are the logarithms of the Hankel contours C, R with angles 0 in [r/2, r ] and radii R > 1 (see Figure 2). We study three forms of the densities a(v,'). One in the spirit of the results of [14], the other two in the spirit of [10] or [6] using Hermite functions. Following [7, Section 10.2 ff.], Hermite functions are given in terms of the Kummer confluent hypergeometric function (c. The Hermite function H, of degree AX E C is the function on the complex plane defined by

2AF(1/2) /D A 1Z 2"F(-1/2)/ 1- 3 F((1- )/2) ' / (22 F(-/2) 2 '2;

The Hermite functions H, are holomorphic on the complex plane as functions of both their variable z and their degree A. If the real part Re(Ai) of A is greater than -1, then they have the integral representation

HgL(z) = ez2 e

e-X2 X cos 2zx - -I rdx

So 2

and so specialize to the th Hermite polynomials if A is any nonnegative integer. If the real part of i is negative, then they have the integral representation

Hg(z) = -e-u2-2zuu-(L+1) du,



162 M. SCHRODER

and so specialize via (2/~/f)H_l(z) = exp(z2)erfc(z) to the complementary error function erfc. Hermite functions are, moreover, connected with the parabolic cylinder functions and with the Kummer confluent hypergeometric function of the second kind.

2.2. Yor-type representations for a(v,e)

This section discusses representations for the law a(V'E) generalizing those which result for

at(v) from [14]. In terms of the concepts of Section 2.1, the basic result is the following contour integral representation.

Theorem 2.1. For any reals 8e 0 and v, the law ot(v,') has the representation

t(V,E)_= Y(V,E),

where the function y(v,') is given by the following double integral:

Y(Ve) (t, w) = cv,e,t(w) xeX2 12 e2/2t sinh() exp

2x

cosh(•

~ d dx o 27ri log CO, R

for any positive reals t and w and any real parameters 0 in [7r/2, 7r] and R > 1.

This theorem is proved in Section 4. We discuss two specializations of this theorem. If 0 = 7r/2 and R = 1, then

Ot (V, ) C ,E,t(w) eXr2/8t e-x2 e-y2/2t cosh(y)

2 Cos 2x sinh(y) - r dydx 2t

for any positive reals t and w. In the case 0 = 7r and R = 1 as in [14], we have

(V,E) cV,E,t(w) 2 /2t "-Xe-X2 0e

y2/2t sinh(y) sin

(x exp 2x cosh(y)) x exp -

2 dy dx

for any positive reals t and w. For 8 = 1, this last formula is seen to specialize to the double

integral for the law of A(V) resulting from [14, Section 6]. Proofs for these results are given in Section 4. They are based on what we would like to call the Hartman-Watson approach which

originates with Yor's probabilistic interpretation of the Hartman-Watson identities in [13] and whose key steps are formalized in Section 3.

2.3. Hermite-function-form representations for a (v,e)

Hermite functions, as recalled in Section 2.1, seem to be naturally connected with the analysis of the processes A(V,E). This may be surprising given Yor's form of the law &(V'i) as discussed in Section 2.2. Historically, Hermite functions thus seem to have been explicitly identified first in connection with Asian options. Here they appear in the integral representations of [9]. They were then shown to play a crucial role in representing the law f8(V) of the reciprocal of any A(V) in [6]. In fact, there seems to be a rather natural explanation for the appearance of the Hermite functions HI. The integral for a(',E) given in the following theorem, which is one of the main results of this paper and which is proved in Sections 5-7, is an expression of this.




Theorem 2.2. For any reals e s 0 and v, the law at(vl') has the representation

a(V,E) = S(V,E),

where the function S(ve) is given by the following contour integral:

S(v)(t, w) = F(v + 1) ,,t(w) e2/2t sinh(?)H_(v+l) cosh() ) d 2.ri logCo,R /2wl/

for any positive reals t and w.

Remark 2.1. If v is any negative integer, then the law is given by the product of cv,e,t multiplied by the product of the residue (-1)n/n! of the gamma function in n = -(v + 1) and the corresponding contour integral in terms of the derivative of H_A with respect to it at t =

v + 1. This last function is a new higher transcendental function which deserves further study. This statement is proved in Section 7.3.

We discuss two specializations of Theorem 2.2. If 0 = r/2 and R = 1, then

t') (w) = F(v + 1) c,e,t (W) e 2/8t eY2/2t cosh(y)

x Re e-xyi/2t H(v) -i sinh(y) ) dy

for any positive reals t and w. In the case 0 = r and R = 1, we have

t' () = F(v +1) c,e,t(w) e2/2t e-y2/2t sinh(y) 7r o

x sinm t

H-(v+l) /2cosh(y) dy

for any positive reals t and w. If v > 0, then these results visibly coincide with those of Section 2.2. But if v < 0, then we

are able to prove them only by establishing the stronger result that S(I,E) and Y(wE) considered as functions in the complex variable ft are entire and agree. This is explained in Section 7. Moreover, Theorem 2.2 seems to furnish a natural missing link between the representation of Section 2.2 for a(v'E) by y(v,E) and that originating with [6].

2.4. Dufresne-type representations for a(ve) As already mentioned, an original and intriguing perspective on a(v,E) has been developed

in [6]. Its key idea is to first compute certain double Laplace transforms of these densities using the theory developed around the hypergeometric ordinary differential equations. Analytic inversion then gives a new form of a(v,-1) in terms of integrals against Hermite functions. In terms of the concepts of Section 2.1, the latter yields, for any nonzero real e, more precisely the representation

where the function D(v'e)

is given by the following integral:

cvt( +ye 12( (cosh(y) 2

y2 D (t ) = exp + /e

exp - cosh(y) 2xc 8t 2o1s" ((io) 2t

y (r(xy)) (sinh(y) )dy



164 M. SCHRODER

for any reals t, w > 0. The overall structure of this integral for ao(v,) is similar to that of Section 2.3 but markedly different from it. In particular, notice that in Dufresne's form the factor exp(7r2/8t) is coupled with a nonoscillating Hermite function factor of the integrand, and that this integrand includes an exponential of a certain negative squared hyperbolic cosine as additional factor.

2.5. Explicit interrelatedness of the representations for c(v,e)

The laws a(ve) of A(v,e) can thus be represented in at least three ways. First, by the Yor-type integrals y(v,e) of Section 2.2, then by the Hermite function form integrals S(v,e) of Section 2.3, and finally by the Dufresne-type integrals D(v,e) of Section 2.4. These functions, however, do not appear to be explicitly related in an obvious way. In particular, it does not seem to be clear how Dufresne's function D(v,e) can be explicitly related to any of the other two.

A possible explanation for the equality

D(V,E) = y(v,e)

between Yor's and Dufresne's functions for a(v,') has been recently advanced in [18]. It retraces the key steps of [6], and its punchline is as follows. As a first step, the above identity is proved for v equal to 0 and 1 by direct computation using residue calculus. A key result of [6] expresses D(v,e) for any real drift v as, roughly speaking, the convolution of a certain D(vO,e) with integer drift vo against the density of a certain gamma variable. Conceptually, this result is established on the Laplace transform level, and it applies to lift the validity of the above identity from v equal to 0 and 1 to v ranging within a nonempty open interval of the real line. Since both functions v ? D(v,e)(t, w) and v -+ Y(v,e)(t, w) are entire for any time t and state w, they are equal as a consequence of the identity theorem of complex analysis.

To further comment on methodology, the above convolution identity has been established in [6] by analytic methods as a consequence of the functional equations satisfied by the Gauss hypergeometric function. It may thus be noteworthy that it does have a probabilistic interpretation. Indeed, [18] finds that this identity gives expression to the fact that the processes A(-v'-l) and A(v'-1) have the same law on adding to the latter process 2y,, where y, is a gamma(v) random variable independent of A(v). This last result of [17] is an extension of the main finding of [5] to the process level. In this way, the above convolution identity is seen to be part of the probabilistic development originating with [5], which is also summarized and put into perspective in [18].

However, there is a different but nevertheless natural way to explicitly relate all three of the above functions representing a(v,e), and our main result is the following

Theorem 2.3. For any reals 8 : 0 and v, there are explicit explanations for the identity

Y(v,E) = S(V,E)

between the Yor-type function y(v,e) for a(vE) of Section 2.2 and its Hermite-function-form S(vE') of Section 2.3, and for the identity

S(ve) = D(V,E)

between S(v,'E) and the Dufresne-type function D(ve) fora(v'e)

of Section 2.4 which are obtained using the Hartman-Watson approach.




This result is proved in Section 8 and gives another explicit explanation for the equality y(v,e) = D(v,e) in particular.

With regard to the Hartman-Watson approach, recall that [13, Section 5] develops a probabilistic characterization of the Hartman-Watson measures defined for any positive real a as the probability measures rla on the positive real line with Laplace transforms

o 7a(dw) = (a) Re(z) > 0,

where I,,(a) is the modified Bessel function. As summarized in [14, p. 527], this characterization is in terms of the law of A(0) conditional on the Brownian motion B. Elaborating on this point of view, the present paper develops the Hartman-Watson approach as the Laplace transform analysis of the law of A(v,e) in its broader sense. As such, it has two key steps: the computation of these Laplace transforms, which is based on probabilistic insights, and their analytic inversion, which needs complex analytic ideas. The results at the base of these two steps are formalized in Section 3 as we have distilled them from [13], and their direct application in Section 4 in particular re-establishes the results of [13, Section 5] about 7a mentioned above. The main novel feature of this paper's approach, however, is to enhance the originally purely probabilistic methods, based on Bessel processes in particular, by a more systematic use of complex analytic ideas. This technique is primarily developed in the crucial Sections 6-8. And in this way, the paper seeks to plead the case for the Hartman-Watson approach as a powerful method for obtaining the most incisive insights into both the probabilistic and the analytic structure of the processes A(V,e)

Still, at present none of the approaches seems to furnish a way to integrate the Comtet and Monthus form of a(v'e) reported in [6, Equation (1.3)] into the picture. This representation is obtained by expanding

ao(v) with respect to the eigenfunctions of the infinitesimal generator of

A(V). It is based on an analysis of the spectral theory of this operator, and hopefully the missing link to this approach can be found in future work.

3. Three basic auxiliary results

3.1. A basic Laplace transform

A key result of the Laplace transform approach to analyzing problems involving the processes A(',e) from Section 2.1 given by

Ak't,) = (A(t))E,

where

A = e2(Bs+/w) dw

has been the factorization of geometric Brownian motion afforded by the Lamperti identity

eB,+pt = R(-)(A l ),

proved for instance in [10, Chapter II, Section 5]. Here R(") is the Bessel process of index jI started at 1 at time t = 0. We have distilled from [14] the following consequence of this factorization for computing Laplace transforms of expectations in the processes A(We), which is a basic tool for computing the above forms of their laws a(Je).



166 M. SCHRODER

Proposition 3.1. Let 8 be any nonzero real. If i 0, then for any measurable functions f and g on the nonnegative real line,

e-zt E[f

(A}t'e))g(eBt+gt)]

dt

= 0

f(y)g(p)s-' P exp -~I 2

A2 Pe)dpd '0' ZO

Y 2y1/EI/2 zz y21/

d

for any sufficiently large positive real z.

This is based on the following time-change result, which is of independent interest.

Lemma 3.1. If /t > 0, then for any measurable function F on the nonnegative real line,

E e-ztF(A ) dt = E e-zr(L)(dw) (R

where r(A)(w) = inf{u I A(A)(u) > w}.

Proof Notice that, by construction, w F-+ r(A")(w) is the inverse map of t ~- A(A)(t). Time changing thus gives

E[ e-zt F(A))dt = E[ e-zr(w)F(w) dr((w)].

In fact, A(") starts at zero at time t = 0 by construction, and the strong law of large numbers for Brownian motion implies that A(U) (t) goes to infinity as t goes to infinity. To interpret the differential, apply the inverse function theorem to get that

dw dr(A) (w) =dw

exp(2(/gx + Bx))Ix=r(,)(w)

Using the Lamperti identity recalled above, the denominator is equal to the square of the Bessel process of index /t at time w, as was to be shown for the lemma.

Proof of Proposition 3.1. There are two main ideas for establishing the proposition by explicitly computing the Laplace transform

L(z) = e-zt E[f (At

"e))g(eBt+t)] dt.

The first is to make the simpler process exp(Bt + ut) more complicated in a controlled fashion using Lamperti's identity. Using Tonelli's theorem in addition, this gives

L(z) = E[ e-zt f (Az'e))g(R() "(A ")) dt .

Applying the time-change result of Lemma 3.1, also based on Lamperti's identity, we have

L(z) = Ef e-Zr()(w)f(we)g(R ) ]. ?R'>




If we apply the change variables w = yl/e, inside the expectation, reverse the Tonelli theorem and integrate against the semigroup densities of the resulting index-j/ Bessel processes, then we obtain that

f

io Ee•(I

) f(y)g(p) ( Yl/e- 1 L(z) =

00 00

E[e-zr()(y/-) R

yl

= p]f] ()2

Pyl)(1, p) dpP dy.

Here p ? pw) (1, p) is the time-w semigroup of the Bessel process of index /t starting at 1 at time t = 0. Since /i is nonnegative, this semigroup is explicitly known. For instance, recall from [19, Chapter X, Section 1] its representation in terms of I-Bessel functions:

p0) (1, p)P=/.Iexp -

I+ .P ' w 2w) w"

The second idea is to handle the conditional expectation factor in the integrand of L (z) using [13, Theoreme 4.7, p. 80] which asserts that, for nonnegative real /u, it is given by the following quotient of I-Bessel functions:

l,/2z+2pI E[e-z'w IR R) = p] =

for any positive reals z, w and p. On substitution, the proof of the proposition is complete.

3.2. A basic Laplace inverse

This section explicitly reviews how the Bessel function factors of the integrands of the identity of Proposition 3.1 are obtained as Laplace transforms. Working with continuous functions on the nonnegative real line of at most exponential growth, the Laplace transform ?(f) of any such function f is defined by

?(f)(z) = j e-Z x f(x) dx

for any complex number z in a half plane contained sufficiently deep within the complex right half plane. This Laplace transform perspective on the Bessel factors is already at the base of the inversion argument of [13, Section 5], and, using the concepts of Section 2.1, the precise result is the following.

Lemma 3.2. For any complex numbers qj with positive real part and [t,

X-I(I (7))(t W= e-

2(12 i o)g C,2t

el

2/2t+qcosh(4) sinh(4) d4 -l /2z+2t))(t) --

27ri figCo

for any reals t > 0, 0 in [r/2, r1] and R> 1.

Proof The proof is based on the following classical Hankel contour integral for the modified Bessel function I which can be found in [12, Section 17.231, p. 362] for instance:

1 ( iog

=R e-pg+r cosh(4) d' 2I( log

COR



168 M. SCHRODER

for any complex numbers qj and p with Re(q) > 0. To identify the first exponential factor of this last integral's integrand as a Laplace transform if p = (2z + /2)1/2, recall from [2, Beispiel 8, p. 50 ff.] the standard Laplace transform

-?2/4t e-2/ ?(z ()(z), where ~r(t)= e- _ 2 ert

for any complex numbers z with positive real part and such that both ? and its square ?2 have positive real parts. For ? in the contour log CO,R, this last condition can be checked to be satisfied if the radius R is large enough. First considering this case, we then have

e- 2 = e-zte-(1/2)12t e-2/2t dt Jo N/27r t3

for any complex z with sufficiently big positive real part, whence on substitution

I 2z ( ecosh(,)

f e-zt e_(1/2 /2t dt d4. 2Z+i2og CO2,R rt3

The point now is to justify interchanging the order of the Laplace transform with the integration over the Hankel contour in this double integral. Moreover, choosing 0 in (7r/2, 7r] for this, the

hyperbolic cosine in the second exponent has a negative real part. If Re(q) > 0 in addition, then the integrand of the above double integral can be shown to go to zero in absolute value as (z, ?) goes to infinity. Thus, under these restrictions, Fubini's theorem gives

e-(1/2)/2t G ? 21(I 2z+2(7))

(t) = G27r

(t)

setting

G(t) =

o eqcosh(?)e-2/2t d?.

G 2(t i lOgCo,, With its integrand entire, using Cauchy's theorem G, is independent of any chosen contour

log Co,R with e E [r/2, 7r] and R > 1, and partial integration is then seen to give

G,(t) = Iqt e-,2/2t+7cosh(k) sinh(?) d?. 27ri Jlog co,R

This completes the proof of the lemma.

3.3. A basic analyticity criterion

This section addresses the construction of analytic functions by integration from complex- valued functions f on product spaces of the form V x X. Here V is any subset of the complex plane with a nonempty interior and X is any measurable subset of a measure space with measure jix. The precise result to be proved in this set-up is the following analyticity criterion.

Lemma 3.3. For any integrable complex-valued map f on V x X, the map on V defined by

F(v) = f (v, x)

d/x(x) is analytic on V if for almost all x in X, the map v a- f(v, x) is analytic on V.




Remark 3.1. Integrability of any complex-valued function f on V x X is assured if there is an integrable map g on X such that If (v, x)I

_< g (x) for any x in X.

Proof With the remark being clear, the idea for the proof of the lemma is to establish analyticity of F on V by applying Morera's theorem; see [8, Section 10.17, p. 208] for instance. For this we have to show that

A F(v) dv = 0

for any triangle A contained in the interior of V. Writing out F in terms of its defining integral, notice that the integrand f of the resulting double integral is integrable by assumption. An application of Fubini's theorem, thus gives

fA F(v) dv = A f(v, x) dv

dutx(x). By assumption, the integrands of almost all inner integrals are analytic on V. Cauchy's theorem as in [8, Section 10.13, p. 205] applies to these and shows that they are equal to zero. Thus, the integral of F over A is equal to zero, as was to be shown.

4. Proof of the Yor-type integral for the densities a(v,e)

4.1. Conditional form of the densities a(v,e)

Following [14], this section reviews the construction of certain conditional densities ae,t (x, w) subordinated to the law a(v,E) of A(v,e). This is based on applying the Girsanov transformation to the given measure Q to give one for which Wt = vt + Bt becomes a standard Brownian motion. Dropping reference to this new measure, we then have, for any measurable function f on the nonnegative real line into itself,

EQ [f(A(v'e))] = E[f (A (,e))gv(eWt)]e-(t/2)v2,

defining gv(x) = xv. Computing this expectation becomes a special case of computing expectations of the form

E[f (A(O' ))g(e ')] with Borel maps f and g from the nonnegative reals into themselves. Obtaining the laws a(V'e) is reduced to obtaining the conditional densities which, for any reals t > 0 and x, are given by the maps

wt - ae,t(x, w) := Q(A(O'e) E dw I Bt = x)

and which as densities of powers of A(0) (t) conditional on Bt = x are independent of v.

4.2. Laplace transforms of the conditional densities This section adapts the computations of [14] to computing the Laplace transform with respect

to time of the conditional densities w -* a,t, (x, w) defined in Section 4.1. The precise result in terms of modified Bessel functions I, is the following.

Lemma 4.1. For any nonzero real e and any reals w > 0 and x,

1 x2\ l-1 1 + e2X 1 eX ) 12 exp ---- ae,t(x, w) = 8 exp 2

-1 I) (t)

for any positive real t.



170 M. SCHRODER

Proof The idea of the proof is to compute in two ways the Laplace transform

L(z) = e-zt E[f (A oE))g(eBt)] dt

for any Borel functions f and g on the nonnegative real line into itself and with z > zo for some sufficiently large positive real zo. First, Proposition 3.1 applied with i = 0 gives

L(z) = f (w)g(eX) ex1 + e2 e d dw

on applying the change of variables p = exp(x). On the other hand, conditioning on the Brownian motion B, we have tautologically

L(z) = f (w)g(ex) e-zt /2t

(x, w) dt dx dw.

In fact, first express the expectations defining this Laplace transform as obtained by integrating against the respective densities a(v,') at time t. Then express these latter densities as conditional densities ae,t (x, w) integrated against the semigroups of the conditions Bt = x. Since f and g are arbitrary maps, on comparison we obtain that

x2 e-1 + e2x ex t exp a( , t (xw)(z) =

,.x e(-,x2)()=)exp T22t 7 w 2wl1/Ew

I /2

for any real z > zo. We extend the validity of this identity to the whole complex half plane H = {z I Re(z) > zol using analytic continuation. For this we have to establish analyticity on H of both sides of the identity considered as functions in z. Indeed, granting this for the moment, the identity theorem then applies and extends the validity of the above Laplace transform identity to the whole complex half plane H as desired.

To establish the required analyticity properties, first consider the right-hand side of the above identity. It is analytic on the whole right complex half plane in particular since the Bessel function is entire in its degree and the square root is analytic on the complex plane with the nonpositive reals deleted.

With regards to analyticity in z of the Laplace transform of the left-hand side of the above identity, notice that an application of the analyticity criterion Lemma 3.3 reduces this to the integrability of its integrands f(z, t) = exp(-zt)(27rt)-1/2 exp(-x2/2t)ae,t(x, w) if z is in any compact subset of H and t ranges over the positive real line. On taking absolute values, this is implied by Proposition 3.1, and the proof of the lemma is complete.

4.3. Analytic Laplace inversion for the conditional densities

Analytic Laplace inversion of the transforms for the conditional densities ae,t(x, w) of Lemma 4.1 is along the lines of [13, Section 5]. In fact, it reduces to the explicit Laplace inversion of Lemma 3.2, and the precise result is found to be as follows.

Lemma 4.2. For any reals e f 0 and t > 0, the conditional density ae,t at any x, w > 0 is given by the following contour integral:

1 x 2 b

t(

w 2 exp(x) N e

x aE, 2i 1, C exp +

/e cosh() sinh(4) 1d,

2V-•Txp --2et~xw - 2zi lgCo,R

2t W1




with real parameters 0 in [7r/2, 7r] and R > 1 and with

bEtw e +1/e)xp 1 + e2x

2" t ex= we +1/8 201

4.4. Proof of the Yor-type integral for ac(v,e) For the proof of the representation of Theorem 2.1 of the laws a1V',) by the Yor-type integrals

y(V,E), again compute in two ways the expectation

E[f] = E[f(A•')]

for any measurable function f of the nonnegative real line into itself. For the first of these, successively apply the Girsanov identity of Section 4.1 and formally express the resulting expectation by double integration against the appropriate densities. This gives

E[f] =

e-(1/2)vt

vx f (w) 2/2tae,t(x, w) dw dx.

tfRfeXf(w) 27rtt

On substitution for the density from Lemma 4.2 and the change of variables y = exp(x),

E[f] = yY Vf(w)c* exp 2i - •I)

g C expR ( - + cosh(4)) sinh(4) d~ dw dy,

setting

c* = c,,t(W) V(2w1/0)v+1

A Tonelli argument permits the interchange of the first two integrations in this triple integral. Changing variables (2wl/e) 1/2x = y in the resulting middle integral and recalling the defining double integral for y(v,E)(t, w) of Section 2.2, on comparison we obtain that

E[f] = f (w) Y(v,')(t, w) dw.

On the other hand, expressing E[f] as obtained by integration against the density tX(v,E) at time t, we have tautologically

E[ff] = (w) )(w) dw.

The identity of Theorem 2.1 follows on comparison, and the proof is complete.

5. Probabilistic part of the proof of the Hermite-function-form representation for alt(v,")

5.1. Laplace transforms of a(v,e)

The proof of the Hermite-function-form representation for a(',•)

of Theorem 2.2 is by a Laplace transform analysis. As its first step, this section computes a second integral representation for the Laplace transform of these densities. The precise result is the following.



172 M. SCHRODER

Lemma 5.1. Let e be any nonzero real. If v > 0, then

(t a')(w))(Z) ))= (2wl

)v/2exp ( E

I 2z+v 2w/

) 1ex2 dx ?(t •, o (W))(z)

[2w12 for any positive real w and any complex z with a sufficiently big positive real part.

Proof Let f be any measurable function on the nonnegative real line and consider the map F on the positive real line given by

F(t) = E[f (Av'))].

The idea is to compute in two ways its Laplace transform

Z(F)(z) = j e-zt F(t) dt

at real arguments z > zo first, where zo is any sufficiently large real. First, Proposition 3.1 applied with g - 1 gives, on changing variables with

(2w'•E)1/2x = p,

? ) f ) (2wl/e)v/2 ( -1 j

)Io' 2x(

S~(F))(z) f(w) elw 2 1/ e-x2 dx dw

o exp

)

I/2zw2•/2w1E for z > zo any sufficiently large positive real. On the other hand, F(t) is by construction obtained by integrating f over the positive real line against the e-power densities a(,'E) at time t. In this way, Z(F)(z) becomes a double integral and applying Tonelli's theorem we obtain

?,(F)(z) = f f(w)e-zt v's) (w) dt dw

= f(w)J(aO ') '(w))(z) dw

as a second representation for it. On comparison of these two representations for ? (F)(z), the identity of the lemma thus holds for any reals z > zo since f is arbitrary. The validity of this identity then extends to the whole complex half plane H = {z I Re(z) > zol by an analytic extension argument similar to that for Lemma 4.1. The additional fact to be shown for this is that the analyticity of the I-Bessel function factors of the maps

(Z, x) -*I 2

•21/e)XV-

e-x2

are preserved on integration with respect to the variable x over the positive real line when z is in H. To see this, use the asymptotic behaviour of the Bessel functions near zero (see [7, Section 5.7]) and towards infinity (see [7, Section 5.11]) to construct as a first step functions majorizing the absolute values of these integrands which depend continuously on the parameter z. The desired analyticity is then a consequence of the analyticity criterion Lemma 3.3, and the proof of the lemma is complete.




5.2. Analytic Laplace inversion for a(v,e)

As a second step in the Laplace transform analysis leading to the Hermite function form for

a(V&l) of Theorem 2.2, this section analytically inverts the Laplace transform of Lemma 5.1. The precise result is the following .

Lemma 5.2. The assertions of Theorem 2.2 are valid if v > 0.

Proof The idea for the inversion of the Laplace transform of Lemma 5.1 is to justify Laplace inversion under the integral sign and apply the inversion result of Lemma 3.2. Interpreting Laplace inversion as given by the contour integral of Riemann's inversion formula, this reduces to justifying the interchange of the order of integration in the following double integral:

1 zo+ioo foo

Sezt Iu(z) (?(w)x)xV- e-x2 dx dz, 27ri z0-ioo 0

where zo > 0, t is any positive real, g(z) = (2z + v2)1/2 and 1r(w) = 2(2w l/ )-1/2. Applying Fubini's theorem, this further reduces to proving the integrability of this double integral's integrand. For this, express the Bessel function factor as the Hankel contour integral from [12, Section 17.231, p. 362]:

I(z)@((W)x) =

ilo exp(-It(z)4 + i7(w)x cosh(4)) d4.

Then study the asymptotic behaviour of the resulting double integral's integrand as (z, x, 4) goes to infinity. Hereby z varies on the line through zo parallel to the imaginary axis, x varies in the positive real line, and 4 varies in the contour log CO,iR. First restrict to contours with 0 in (7r/2, 7r] and R so large that log COR and its image under squaring are contained in the right half plane. As in the proof of Lemma 3.2, the above integrand is then seen to be exponentially decreasing to zero as (z, x, )) goes to infinity. Laplace inversion under the integral sign is thus justified in particular. Using Lemma 3.2,

(V,e) 2 2 2x cosh(4)) Ct (w) = C,e,t(w) e-xv exp -+ )sinh(4) d4 dx o 2xil 2t

on collecting constants. The asymptotic behaviour established above implies, moreover, that the order of the two integrations can be interchanged. Since v is positive, the resulting inner integral is a value of a certain Hermite function of negative degree recalling the latter's integral representation for negative degrees from Section 2.1. As a precise result we obtain

(Ve) (w) VE,t(W)f e2/2t sinh()F(v

+ 1)H(v) cosh()d• Ut (W)= 2ri

log CO,R(v H/2Wl/ whose right-hand side is equal to S(Ve) (t, w), as required.

To lift the restrictions on 0 and R, notice that the integrand of this last integral is entire as a function of 4. Using the asymptotic expansion for Hermite functions

Hz nl (--1 _)2k (--1)k ( 1 )

Hk(z)

= (2z) k! (2z)2k

+ O (12nRe(,)

k=O



174 M. SCHRODER

of [7, Equation (10.6.7), p. 292], valid for any complex z with larg(z)I < 37r/4, the integrals are finite if 0 is in [nr/2, 7r] and R > 1. Cauchy's theorem applies and shows that the integral is independent of 0 and R > 1. This extends to constellations with R = 1 by continuity, and the proof of the lemma is complete.

6. Analyticity preliminaries

6.1. Analyticity properties of the Yor-type integrals Y(v,e)

This section considers the contour integral factors of the Yor-type integrals y(v,') of Theo- rem 2.1 for the law a(V',) as functions in the index v. For any nonzero real e and any positive reals t and w, consider the functions which, for any complex i, are given by

Pe,t,w(it) = j x0e-x2g e,t,w(x) dx,

where fe,t,w are the functions on the real line given by

e,1t,w(x) =

e-/2t sinh() exp ( 2x cosh() ) 27i lo JgCO,R /21//

over any logarithmic Hankel contour log CO,R. By construction, we have

Y'A'")(t, w) = cL,,e,t(w)Pe,t,w(iL)

if the integrals of either side are finite. The result to be proved is the following.

Proposition 6.1. All functions Pe,t,w are entire.

Corollary 6.1. The maps A t-+ y(/,') (t, w) are entire for any reals e : 0 and t, w > 0.

Proof With the corollary clear by construction, notice that the proof of the proposition reduces to a local question. In fact, dropping references to indices e, t, w and k to simplify the notation, it is sufficient to prove entireness of P for i ranging within any compact subset V of the complex plane with nonempty interior. As an application of the analyticity criterion Lemma 3.3 this further reduces to construct an integrable function g on the positive real line such that

Pp(it, x) < g(x)

for any x > 0 and i in V setting p(iL, x) = exp(-x2)xAu*(x). For this construction first apply Cauchy's theorem to see that *r is independent of the contours log CO,R chosen. The function *r that results from taking 0 = 7r and R = 1 can be shown to coincide with the one considered in [14, Equation (6.g)], whence

lim x"~ (x) = 0 x40

in particular for any u in V. For any c > 0, sending / in V and x in (0, c] to x # (x) thus extends to a continuous map on V x [0, c]. Let Cv be the maximum that its absolute value achieves. On the interval [c, oc) majorize I" by a positive constant Dc depending on c. For instance, choosing 0 = xr and R = 1, take

Dc e2/2

e-y2/2t sinh(y)exp( 2c cosh() dy.




To continue, let 7r, denote the map on the positive real line sending any x to its IXvth power,

where I v is the maximum of the real parts of all elements in V. Moreover, choosing c > 1, the function g on the positive real line defined by

g(x) = exp(-x2)(Cvl[o,c) + DcAVl[c,oo))(x)

then has the desired majorizing properties, and the proof of the proposition is complete.

6.2. A connection with Hermite functions

This section gives a first explanation for the relation between the two forms of the law a (v,e) of Sections 2.2 and 2.3. This is by interpreting the integrals P,,t,,w (/) of Section 6.1 in terms of the contour integral factors of the functions S(v,~) of Section 2.3. More precisely, for any reals e : 0 and t, w > 0, consider the functions

F,1

•()=

f,t,w (it, ?) d? 2.7ri,(•) gCOR

setting

fe,t,w(, ) = e-/2t sinh())/H-2t( ) dcosh(

whence

S(',e)(t, w) = ci,e,rt(w)F(ix

+ 1)Fe,t,w(iL)

when either side is finite. Then the precise result is the following.

Lemma 6.1. For any complex number it with Re(it) > -1,

Pe,t,w (Ai) = F(Ai + 1)Fe,t,w( (A)

for any positive reals t and w and nonnegative integer k.

Proof To see this, revert to definitions and write out the integrals Pe,t,w(it) as double integrals. The lemma's identity follows on interchanging the order of integration as soon as we know the integrability of this double integral's integrand. This, however, is shown by an argument analogous to that for Lemma 3.2 or Lemma 5.2, and the proof is complete.

6.3. Analyticity results

This section addresses analyticity of the functions Fe,t,w,k of Section 6.2. The result to be proved is the following.

Proposition 6.2. All functions Fe,t,w are entire.

Proof The proof is similar to that of Proposition 6.1. First, establishing this result is a local question. In fact, dropping reference to indices E, t and w to simplify the notation, it is sufficient to prove entireness of F for &Y

ranging within any compact subset V of the complex plane with nonempty interior. Since in particular Hermite functions are entire functions in their degree, an application of the analyticity criterion Lemma 3.3, again reduces this to showing that there is an integrable function g on log CO,R with If (,, ()I < g(4) for any ( in log CO,R and

It in V.

The construction of such a function is based on certain properties of Hermite functions. Using the recurrence rule

H,1+l(z) - 2zH,(z) + 2lHH,_l (z) = 0



176 M. SCHRODER

for Hermite functions of [7, Equation (10.4.6), p. 289], first reduce to V being contained in the left complex half plane. Using [7, Equation (10.6.7), p. 292], the Hermite function factors

H_(,+l) in F have the asymptotic behaviour

H_(+1)/t(z) = (2z)-` + O IZ3_-Re(1)

for any complex z with larg(z)l < 37r/4. Hence, the factors exp(-_2/2t) dominate the asymptotic behaviour of the integrands of F as |I1 goes to infinity. Moreover, using the compactness of V, construct a majorizing function g for the integrands depending on V but independent of the single elements it in V. The proof of the proposition is complete.

7. Analytic part of the proof of the Hermite-function-form representation for a(v'e)

7.1. Coincidence of the integrals This section provides the key for relating the two integrals of Sections 2.2 and 2.3 for the

laws a(',e)

for general complex /t. To recall the current state of the argument, for any reals E 0 0 and t, w > 0, we have on the one hand the contour integral factors P,,t,w of the Yor-type functions Y(0',) discussed in Section 6.1, and on the other hand the contour integral factors Fe,t,w of Section 6.2 of the functions S(',e) of Section 2.3. The main result is then the following generalization of Lemma 6.1.

Proposition 7.1. For any positive reals t and w, the functions itt '(it + 1)Fe,t,w (it) and it -+ PE,t,w(it) are entire, and we have

Pe,t,w(It) = F'(i + 1)Fe,t,w(it)

for any complex Ai.

Proof For any fixed positive reals t and w, the function ~t F,t,w (it) is entire from Proposition 6.2. The function it P,t,w(it) was proved entire in Proposition 6.1. From Lemma 6.1 we have the equality of the proposition for any positive A. Now apply the identity theorem. This equality then holds for any complex A first in the sense of meromorphic functions since its gamma function factor has simple poles in the negative integers. With its left-hand side an entire function, the right-hand side of this identity cannot have any poles, and so this equality holds for any complex number u between entire functions. This completes the proof of the proposition.

7.2. Proof of the Hermite-function-form representation for a(v,')

The analysis of the relatedness of the functions Y(O',) of Section 2.2 and S(I~',) of Section 2.3 now makes possible a proof of Theorem 2.2. Recall that the latter asserts that S(v,e) represents a(v',) for any real v. Using the concepts of Section 2.1, recall that Theorem 2.1 gives, for any real v, the representation

(V,E) = y(v,E).

By construction of the functions P~,t,w

in Section 6.1 we have the following representation for y(v,e) .

Y(v'e) (t, w) = ce, v,t(w)Pe,t,w(v)

for any positive reals t and w. In particular, for any real v, we have from Proposition 7.1

Pe,t,w(v) =

F(v- + 1)Fe,t,w(V).




Recalling how the functions Fe,t,w were defined in Section 6.2 as contour integral factors of S(v,e) (t, w), this identity gives, on multiplication with the appropriate constants c of Section 2.1,

Y(V,e)(t, w) = cv,e,t(w)Pe,t,w(V) =

cv,e, t(w)F(v + 1)Fe,t,w(V)

= S(,E) (t, w).

On substitution into the identity of Theorem 2.1, we thus obtain the identity

a (VE) = S(V,E)

for any real e # 0 and v, and the proof of Theorem 2.2 is complete.

7.3. Proof of Remark 2.1

The proof of Remark 2.1 is based on the following observation. Since v - ce,v,t (w) x

F(v + 1)Fe,t,w(v) is entire from Proposition 7.1, the poles of the gamma function factor are cancelled by zeros of the remaining factors. On inspection, only the integral factor v - Fe,t,w (v) can develop such zeros, and the remark follows.

8. Relating the integrals for the laws

8.1. Principal ideas of Dufresne's approach As a preliminary to the proof of Theorem 2.3, this section reviews key ideas of the proof

in [6] that the function D(v,e) represents a(ve) in the way described in Section 2.4. The case E = -1 considered in [6] is sufficient for this because of the standard relation

c(VE) (w) i(1-)-(llE) -l/E)

which is valid for any real v and any nonzero real e, and recalling that ,f(v) = c(v,-1) from Section 2.1. To take care of the different normalizations, moreover, notice that

(v) (V) t (w) =

2fit (2w),

where w - 83 v)(w) is the density of the reciprocal of 2A(v) considered in [6]. Working with this last normalization, the idea now is to apply two Laplace transforms to

P(v). As is explained in [6, Section 2], first take the Laplace transform with respect to the state variable, then take the Laplace transform with respect to time. Formalizing this gives, for any real a, the double Laplace transforms

H(v) (s, z) = ?(t (w C( w a W(v ()w)) (s))(z)

or, equivalently,

H(V)(s, z) = e-zt E[Xte-sx']

dt

with X = (2A(V))-1 the reciprocal of 2A(v). These double transforms are shown to exist for any complex numbers s and z sufficiently deep within the right half plane, whence 8(V) can be obtained by the following two-step procedure. As a first step, explicitly compute the double Laplace transforms Ha. Then, as a second step, invert them analytically:

we (V ) _? ?-11

(dz1 (H(V))(t))(w)



178 M. SCHRODER

for any positive reals w and t. This program has been carried out in [6] yielding the representation of a(v,e) by the function D(v'e) recorded in Section 2.4. In the sequel, we establish alternative methods for both explicitly computing the double Laplace transforms Ha and effecting the above analytic double inversion.

8.2. A second way of computing Dufresne's double Laplace transform

This section discusses our way of explicitly computing the double Laplace transforms

H(v) (s, z) = e-zt E[Xte-sx']

dt

defined in Section 8.1. Here, a is any real, X is the reciprocal of the the process 2A(V), and s, z are any complex numbers sufficiently deep within the right half plane. The precise result to be proved about this function is the following integral representation in terms of I-Bessel and K-Bessel functions.

Proposition 8.1. For any reals a and v such that a - v is positive,

H(•)(s, z) = 21- -I-aj -l 2z; ( )(s + 1)(1/2)(va)Kav_( 5s + 1) d

for any complex numbers s, z sufficiently deep within the right half plane.

Corollary 8.1. For any real v,

+(V) (SZ)=--7f V-11e/ d? HvQ112(s, z)2= 2 00 ,/ z d7

for any complex numbers s, z sufficiently deep within the right half plane.

Remark 8.1. For any reals a and v such that a - v is positive, we have in terms of the Gauss hypergeometric function 2 F1 the representation

2-1 ['(a)F(b

+ a) ( 1 )(s, z)= I a,b+a;a+b+1; a ) (s + 1)b+a F(a + b + 1) s+1)

for any complex numbers s and z sufficiently deep within the right half plane, where a = a(z) = (B(z) + v)/2 with B(z) = (2z + v2)1/2 and b = b(z) = a(z) - v = (B(z) - v)/2.

This last identity is a consequence of the integral representation of Proposition 8.1. It was called the modified Weber-Schafheitlin integral in [11, Section 13.45, p. 400] where it is proved by expanding the I-Bessel function factor of the identity in the proposition into its series and integrating term by term. Notice that [6, Theorem 2.1] develops an alternative approach to establishing this identity by solving boundary-value problems for the hypergeometric differential equation.

Proof of the Proposition 8.1. The proof relies in particular on the basic Laplace transform of Section 3.1. Indeed, first apply Girsanov's theorem to obtain that

( z) = 2 e-(z+v2/2)t

E[fs/2(A01-))g(eB')] dt,




defining f,(x) = xaI exp(-wx) and g(x) = xV. An application of Proposition 3.1 with it = 0 and e = -1 then gives

1

0

p0'+P2 H(v)(s, z) fs/2(y)g(p) exp - I 2z (py) dpdy. Jy 2y-'

Further analytic steps are needed to bring this integral into the required form. Using the asymptotic behaviour of the I-Bessel function of [7, Section 5.16], this double integral's integrand is seen to be integrable for Re(z) >> 0 and Re(s) > 0. Thus, apply Fubini's theorem, interchange the order of integration and change variables ? = py to obtain that

H(v) (s, z) = Ia f 0Pv-a-1f0 -1/-A(p+q2/4p H) 2s z) = a0J- 0 2z+v2(?)e-a

P) d dp,

defining A = A(4) = /2 and q = q(s) = 2/s + 1. Now reverse Fubini's theorem. Recall from [7, Equation (5.10.25)] the integral representation for the K-Bessel function

K(z) = ( t f W+2/4w) -(it+1)

dw

valid for Iarg(z)I < r/4, then change variables y = Ap to obtain that

H(v) (s, z)= 2a y /II2z+2()2

2 1/s + 1a-v

This completes the proof of the proposition.

Proof of Corollary 8.1. The corollary follows from Proposition 8.1 using the fact that the K-Bessel function factor of degree a - v = is given by

K12(x)- = e-x (2x)

for any positive real x; see for instance [7, Equation (5.8.5)].

8.3. A second way of inverting Dufresne's double Laplace transform

From Section 2.3 recall the functions S(v,e) defined by contour integration against Hermite functions as

S(v)(t, w) = (v + 1) t(w) e-2/2t sinh(4)Hi(v+l) -cosh(-

) d 27ri log

CO,R /2w/

for any positive reals t and w. This section shows how they can be obtained by analytical inversion of certain integrals of Section 8.2 for the double Laplace transforms HT of Section 8.1. Our precise result is as follows.

Theorem 8.1. If v > -1 anda = v + ?, then

)iHt) ((t)) ve r2 S(Vl') (t, w)

for any positive reals w and t.



180 M. SCHRODER

Proof The key idea is to show that the double Laplace inversion for

L(t, w) =

:Is z

(C-H ) (V ) - ( Hv+ l/2(t) (0)

can be effected by Laplace inversion under the integral sign of the integral representation for

H, of Corollary 8.1, where it becomes separated into two single inversions, as follows:

fOw./7 - / 2 t

e-wVr1)( L(t, w) = 2v+1/2 j 1

zl(I/ +2+2())(t)s (2-

d.

To see this, interpret the inversions as being effected by the contour integrals of the Riemann inversion formula and get a triple integral for L(t, w) on substitution. Laplace inversion under the integral sign is then a consequence of Fubini's theorem if the integrand of this triple integral can be shown to be integrable near zero and towards infinity. On inspection, integrability near the origin reduces to showing that the factor ~V-1lI(2z+v2)1/2 () remains bounded for 4 near zero. Considering the zeroth term of the series [7, Equation (5.7.1)] for the I-Bessel function, this will hold on taking the inversion contour sufficiently deep within the right complex half plane. To establish integrability towards infinity, express the I-Bessel function factor using the integral representation of [12, Section 17.231, p. 362] as

S V 27 og CO,R exp(-v/2z+v2 + 4 cosh(qr)) dr.

Thus, a fourfold integral for L(t, w) results. Taking the z-inversion contour deeper within the right half plane if necessary, this fourfold integral's integrand,

e(w/2)s etz

IV-1e

-I h(s, z, , q) =

exp(-v/2z+v2 + 4 cosh(r)) 2tri 2ti 2r i s-+ '

is then seen to decay exponentially as its arguments go to infinity. This implies the integrability of h towards infinity, and thus the integrability towards infinity of the triple integral defining L(t, w). Notice that the factor 4v-1 does obstruct integrability of h near zero. At this stage, interchanging the order of all four integrations is thus not justified in general and for v in (- 1, 0] in particular.

To proceed, notice that, for the s-inversion factor in the above integral for L (t, w),

•sl

(e -/?-+li w

(e -• w

? e( s+

) () = e-w/2 (-

4/i ()

e-w/2 `12 e- 2/2w

using [2, Beispiel 8, p. 50 f.] for the last equality. Moreover, using Lemma 3.2 to compute the z-inversion factor in L(t, w) gives on substitution that

2v+1/2L(t, w) = e1/2

)w t

x f-

e-2/2t sinh(:)

x "excosh()e-x2/2w dx d4. 2K1

og CO,R0




With the integrand of this double integral now seen to be integrable for v > -1, notice that the reversal of the order of integration is justified on applying Fubini's theorem. To complete the proof of the theorem change variables y = (2w)-1/2x in the inner integral of the above double integral and express the result as a Hermite function.

8.4. Relating the different representations for a(Ve) This section focuses on the three different functions for the laws a(v,e) of Section 2.1. It

establishes Theorem 2.3 by providing explicit explanations for the identity between the functions Y(v,e) of Yor type of Section 2.2 and the Hermite-function-form maps S(v,e) of Section 2.3, and for the identity between the latter and the functions D(v,e) of Dufresne type of Section 2.4.

Notice that the identity Y(v,e) = S(V,E)

was explicitly shown in Section 7 for establishing Theorem 2.2. In fact, recall that we have

y(v,E) = ce,v,t(w)PE,t,w(v) from Section 6.1 and S(v,E) = ce,v,t(w) F(v + 1)Fe,t,w(V) from Section 6.2, and that Section 7.2 explains why these two functions are equal.

To explicitly establish S(v,E) = D(v,E)

reduces to the case of e = -1, recalling from Section 8.1 how a(vE) is determined in terms of

a(v'-'). The punchline of the argument here is as follows. Setting a = v + 1, consider the function L which, for any positive reals t and w, is defined by

L(t, w) - l, l HV)()) x-H(.

From Section 8.1 it follows that L(t, w) equals (w/2)aY'v)(w/2). However, one of the key points of the arguments is not to make use of this interpretation. Instead, using Riemann's inversion formula, proceed by interpreting L as given by the following double integrals:

L(t, w) = f es(w/2)eztH(V)(s, z) dz ds 2L(t,

w)Re(s)=so 2i Re(z)=zo

for any positive reals t and w, where so and zo are sufficiently large positive reals. The key point of the argument now is to compute this double integral in two ways, the first of which essentially gives D(v'-l) and the second S(v'-'). This will then complete the proof. However, it seems to be both appropriate and advisable to discuss the fine points of the argument too.

On the one hand, under the restriction v > -1, Theorem 8.1 explicitly computes L(t, w) for any t, w > 0 as follows:

L(t, w) = 2 S(V'-l) (t, w).

Recall that this computation is eventually based on the Hartman-Watson approach and that it essentially proceeds by separating the Laplace inversions under the integral sign of the function

On the other hand, the integrals L(t, w) have been explicitly computed in [6, Theorem 4.2] in the way already indicated in Section 2.5. Recall that, roughly speaking, this proceeds by successively interchanging the order of integration in L(t, w), transforming the outer integral that results, explicitly computing the inner integrals that result for v equal to 0 and 1, and extending the validity of the computations to v ranging within certain subintervals of the



182 M. SCHRODER

real line by means of a certain integral representation in terms of integer drift densities. In fact, Dufresne [6] thus directly computes two formulae for L(t, w) in terms of the confluent hypergeometric function QD which combine to give the function D. More precisely, if -1 < v < 0, then

L(t, w) = -= o,t

from [6, Equation (4.15), p. 238], where

8t(w) =

2 dvt(w) ey2/2t cosh(y) cos ( ~ 2 2 wsinh2(y)) dy,

with

dv,t (w) =

2 cv,-1,t(2w)e2

2/8t 7r

If -2 < v < 1, then [6, Equation (4.17), p. 239] asserts that

L (t, w) = 0

-3(v ( where

S ,,(w) =2( 2 + 1

xf e-y2/2t cosh(y) sinh(y) sin( X> Dj + 1, ; -w sinh2(y) dy.

On suitably weighting them, these two formulae for L(t, w) add up to give D(v,e). More precisely, [6, p. 239] shows that

L(t,w)= ?1+X)8.,, +

1

-X)8,,) = 2 D(v"'-) (t, w)

for -1 < v < 0, setting A = exp(-inrv). On comparison, we thus obtain natural explicit equalities between S(v,-1) and the two densities 3kv) in the pertinent ranges of v. The desired equality,

is then implied by these two identities if -1 < v < 0. To lift this restriction on v, recall from [6, Theorem 4.2] that v -+ Dv,-'1) (t, w) is entire for any positive reals t and w. Since, from Section 7.2, v F-+ S(v,-1)(t, w) is also entire for any positive reals t and w, the identity theorem shows the above equality between S(v,-') and D(v,-') to be valid for all complex v. This completes the proof of Theorem 2.3.

8.5. Epilogue

Reflecting on the nature of the Hartman-Watson approach of this paper, it appears to be a most effective method for uncovering the general structures governing the processes A(v,E) Moreover, it is at the base of most concrete results about these processes and their laws, in particular a(v,8). As the reader may have become aware, there is by now an ample supply of explicit representations for ac(vE). So it may appear to be natural to try to work with them




directly as analytical objects, and I am grateful to the referee for an observation in this direction. Indeed, computing with the specialization of Theorem 2.2 to 0 = 7r/2 and R = 1 given in Section 2.3, our function S(v',) can be seen to be a convex combination of those of Dufresne as follows:

S (v,-) - D (V,-) + D (v,-) 2 +2 1 Here, the Dk) are the functions obtained from the densities 8(V) of Section 8.4 by

D (V,) w) 1 (1/

) -1/E)

for any positive reals t and w. The author would be more than gratified if the reader took this paper as a starting point for

developing his or her own ideas about understanding the processes A(v,).

Acknowledgements

Support by the Deutsche Forschungsgemeinschaft via a project with Professor Hipp at Universitat Karlsruhe and the hospitality of the Institut fiir Mathematik, Universitat Mannheim, are gratefully acknowledged. I would like to thank Professors Yor and Matsumoto for sending preprints of their work and for useful comments on earlier versions of this paper. While the usual disclaimer applies, I also wish to thank the referee for a careful reading of the manuscript.

References

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