On some exponential functionals of Brownian motion

This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of L\'{e}vy's and Pitman's theorems are discussed.


Introduction
Let B = {B t , t ≧ 0} be a one-dimensional Brownian motion starting from 0 and defined on a probability space (Ω, F , P ). Denoting by B (µ) = {B (µ) t = B t + µt} the corresponding Brownian motion with constant drift µ ∈ R, we consider the exponential functional In Part I [49] of our survey, we have discussed about the probability law of A (µ) t for fixed t and about several related topics. * This is an original survey paper.
Among the results, we have shown some explicit (integral) representations for the density of A (µ) t . In particular, we have proven the following formula originally obtained in Yor [63]: θ(e x /u, t) dudx u , (1.2) where, for r > 0 and t > 0, θ(r, t) = r (2π 3 t) 1/2 e π 2 /2t ∞ 0 e −ξ 2 /2t e −r cosh(ξ) sinh(ξ) sin πξ t dξ. (1. 3) The function θ(r, t) appears in the representation for the (unnormalized) density of the so-called Hartman-Watson distribution and satisfies ∞ 0 e −α 2 t/2 θ(r, t)dt = I α (r), α > 0, (1.4) where I α is the usual modified Bessel function. For details, see Part I and the references cited therein. Another important fact, which has been used in several domains and also discussed in Part I, is the following identity in law due to Dufresne [18]. Let µ > 0. Then one has where γ µ is a gamma random variable with parameter µ, that is, x µ−1 e −x dx, x ≧ 0.
The purpose of this second part of our surveys is to present some results obtained by applying the formulae and identities mentioned in Part I to Brownian motion and some related stochastic processes.
In Section 2 we discuss about the pricing formula for the average option, so called, Asian option in the Black-Scholes model.
In Section 3 we present some formulae for the heat kernels of the semigroups generated by the Laplacians on hyperbolic spaces. By reasoning in probabilistic terms, we obtain not only the classical formulae but also new expressions.
In Section 4 we apply the results on exponential functionals to a question pertaining to a class of diffusion processes in random environments.
In Section 5 we show Dufresne's recursion relation for the probability density of A (µ) t with respect to µ which, as we have seen in Part I, plays an important role in several studies on exponential functionals.
Dufresne's relation is important in studying extensions or analogues of Lévy's and Pitman's theorems about, respectively, {M   t } are diffusion processes for any λ ∈ R. Hence, the classical Lévy and Pitman theorems may be seen as limiting results of those as λ → ∞.

Asian options
In this section we consider the Asian or average call option in the framework of the Black-Scholes model and present some identities for the pricing formula.
By the Black-Scholes model, we mean a market model which consists of a riskless bond b = {b t } with a constant interest rate and a risky asset S = {S t } with a constant appreciation rate and volatility. That is, letting r > 0, µ ∈ R and σ > 0 be constants, we let b and S be given by the stochastic differential equation where B = {B t } is a one-dimensional Brownian motion with B 0 = 0 defined on a complete probability space (Ω, F , P ). For simplicity we normalize them by setting b 0 = 1. Then we have b t = exp(rt) and S t = S 0 exp(σB t + (µ − σ 2 /2)t).
Following the standard procedure, we consider the discounted stock price S = { S t } given by S t = e −rt S t = S 0 exp(σB t + (µ − r − σ 2 /2)t).
Then, by Girsanov's theorem, there exists a unique probability measure Q which is absolutely continuous with respect to P and under which S is a martingale. Q is called the martingale measure for S and we have Let us consider the European and the Asian call options with fixed strike price k > 0 and maturity T . The payoffs are given by (S T − k) + and (A(T ) − k) + , respectively, where x + = max{x, 0} and By the Black-Scholes formula or by the non-arbitrage argument, we can show that the theoretical price C E (k, T ) and C A (k, T ) of these call options at time t = 0 are given by where E Q denotes the expectation with respect to the martingale measure Q.
By using explicit expressions for the density of A (µ) t discussed in Part I, we obtain several integral representations for C A (k, T ). However, they are complicated. Hence we omit this approach and consider instead the Laplace transform of C A (k, T ) in T .
In the following we set σ = 2 and consider under the original probability measure P to follow the same convention as in Part I and in other parts of the present article. Let T λ be an exponential random variable with parameter λ > 0 independent of B. Yor [62] (see also Part I) has shown the identity in law where a = (ν + µ)/2, b = (ν − µ)/2, ν = 2λ + µ 2 , Z 1,a is a beta variable with parameters (1, a), γ b is a gamma variable with parameter b and Z 1,a and γ b are independent. From this identity we deduce the following result.
The same formula has been proven in [24] with the help of some properties of Bessel processes.
We next present another proof of Theorem 2.2, following Donati-Martin, Ghomrasni and Yor [16], who have used, as an auxiliary tool, the stochastic process is a diffusion process with generator In fact, in [16], the authors have taken advantage of the identity in law for every fixed t > 0 and have computed the in t by using the general theory of the Sturm-Liouville operators.
We present an explicit form of the Green function for L (µ) . For this purpose we recall the confluent hypergeometric functions Φ(α, γ; z) and Ψ(α, γ; z) of the first and second kinds defined by where (α) 0 = 1 and For details about the confluent hypergeometric functions, we refer to Lebedev [38]. Φ(α, γ; z) and Ψ(α, γ; z) are linearly independent solutions for the linear differential equation We set ν = 2λ + µ 2 and define the functions u 1 and u 2 on (0, ∞) by and respectively. Then, by straightforward computations, we can check Moreover, u 2 (x) is monotone decreasing in x ∈ (0, ∞). On the other hand, recalling the integral representation for Ψ(α, γ; z): (cf. [38], p.268), we can easily show that u 1 (x) is monotone increasing. In fact, we have About the Wronskian, it is known ( [38], p.265) that which yields Here the function x −(1+µ) e 1/2x is the derivative of the scale function for Y (µ) (x). Checking the boundary conditions, we obtain the following.
Proposition 2.3. Let u 1 (x) and u 2 (x) be the functions defined by (2.2) and (2.3). Then the Green function G (µ) (x, y; λ) for L (µ) with respect to the Lebesgue measure is given by In order to proceed to a proof of Theorem 2.2, we recall the following identity presented in [16]: which may be proven by Kummer's transformation and the series expansion (2.1) of Φ. It is a special case of a general formula given on page 279, Problem 21, Lebedev [38]. See also [23]. Then, noting that u 1 (0) = 2 (µ+ν)/2 , we obtain Moreover, we recall the integral representation of Φ: Then, after some elementary computations, we arrive at Finally, by using the identity zΓ(z) = Γ(z + 1), we obtain and Theorem 2.2.

Heat kernels on hyperbolic spaces
Let H n be the upper half space in R n given by {z = (x, y) = (x 1 , ..., x n−1 , y); x ∈ R n−1 , y > 0}, endowed with the Poincaré metric ds 2 = y −2 (dx 2 + dy 2 ). The Riemannian volume element is given by dv = y −n dxdy and the distance d(z, z ′ ) between z, z ′ ∈ H n is given by the formula The Laplace-Beltrami operator ∆ n is written as We denote by p n (t, z, z ′ ) the heat kernel with respect to the volume element dv of the semigroup generated by ∆ n /2. Since p n (t, z, z ′ ) is a function of r = d(z, z ′ ) for a fixed t > 0, we occasionally write p n (t, r) for p n (t, z, z ′ ). Then, for n = 2 and 3, the following formulae are well known: Moreover, the following recursion formula due to Millson is also well known and we also have explicit expressions of p n (t, r) for every n ≧ 4: For details about the real hyperbolic space H n and the classical formulae for the heat kernels, we refer the reader to Davies [14]. Gruet [28] has considered the Brownian motion on H n , which is a diffusion process generated by ∆ n /2, and has derived a new integral representation for p n (t, r) by using the explicit expression (1.2) for the joint density of (A While the classical expressions for p n (t, r) have different forms for odd and even dimensions, Gruet's formula (3.6) below holds for every n.
Before giving a proof of (3.6), we mention its relationship to the classical formulae. First of all we note that Millson's formula (3.5) is easily obtained from (3.6) if we differentiate both hand sides of (3.6) with respect to r.
When n = 3, the integrand on the right hand side of (3.6) may be extended to a meromorphic function in b on C. Hence we can apply residue calculus and obtain (3.4).
In the case n = 2, which is the most interesting and important, we compute the Laplace transform in t of the right hand sides of (3.3) and (3.6). Then, using the Hankel-Lipschitz formula for the modified Bessel functions (see Watson [57], p.386), we can check the coincidence of the Laplace transforms or of the expressions for the Green function. For details, see [28], [40], [41].
We give a proof of (3.6) and see how the exponential functional A (µ) comes into the story.
, w n · ) starting from 0 with the topology of uniform convergence on compact intervals, B is the topological σ-field, B s is the sub σ-field of B generated by {w u , 0 ≦ u ≦ s} and P is the n-dimensional Wiener measure.
The Brownian motion on H n may be obtained as the unique solution of the stochastic differential equation We denote by Z z = {Z z (t, w) = (X z (t, w), Y z (t, w)), t ≧ 0} the unique strong solution satisfying Z z (0) = z = (x, y). Then we have Note that the heat kernel p n (t, z, z ′ ) may be written as where δ z ′ and δ z ′ are the Dirac delta functions concentrated at z ′ with respect to the volume element dv and the Lebesgue measure dz = dxdy, respectively, and δ z ′ (Z z (t, w)) is the composition of the distribution δ z ′ and the smooth Wiener functional Z z (t, w) in the sense of Malliavin calculus (see [32]). Therefore, we obtain where we have used the same notation for the Dirac delta functions on R n and R. Now we apply formula (1.2) for the last expression. Then we obtain Moreover, changing variables by v = y ′ /yu and using (3.1), we obtain Finally we use the integral representation (1.3) for θ(v, t). Then, changing the order of the integrations by Fubini's theorem, we obtain (3.6) after some elementary computations. Remark 3.1. Recalling formula (1.4), we can easily obtain an explicit expression of the Green function for ∆ n from formula (3.9). Remark 3.2. From the last expression of (3.8), we obtain Hence, we see that the Laplacian ∆ n on H n and the Schrödinger operator on R with the Liouville potential − 1 2 d 2 dx 2 + 1 2 |λ| 2 e 2x are unitary equivalent, which may be directly verified by Fourier analysis and has been already pointed out in Comtet [11], Debiard-Gaveau [15], Grosche [26] and so on. See also [31].
In the rest of this section, we restrict ourselves to the case n = 2 and consider two questions related to the results and formulae presented above. For other related topics, see, e.g., [2] and [29].
Let us consider the following Schrödinger operator H k , k ∈ R, on H 2 with a magnetic field: The differential 1-form α = ky −1 dx is called the vector potential and its exterior derivative dα = ky −2 dx ∧ dy represents the corresponding magnetic field. Since dα is equal to constant k times the volume element dv, we call H k a Schrödinger operator with a constant magnetic field. It is essentially the same as the Maass Laplacian which plays an important role in several domains of mathematics, e.g., number theory, representation theory and so on. For details, see [22], [31] and the references cited therein.
In [31], the authors have started their arguments from the Brownian motion on H 2 given in the above proof of Theorem 3.1 and have discussed about explicit and probabilistic expressions for the heat kernel q k (t, z, z ′ ) of the semigroup generated by H k . They have also applied the results to a study of the Selberg trace formula on compact quotient spaces, i.e., compact Riemannian surfaces, and have shown close relationship between the spectrum and the action integrals for the corresponding classical paths. It should be mentioned that some physicists have shown similar results in the context of Feynman path integrals prior to [31]. See, e.g., [12], [25], [27].
Explicit formulae for several quantities related to the operator H k , e.g., the Green functions, the heat kernels, have been obtained by Fay [22] by harmonic analysis. On the other hand, starting from computations by Feynman path integrals, Comtet [11] and Grosche [26] have obtained explicit forms of the Green functions.
From the point of view of probability theory along the line of [31], another explicit representation for the heat kernel q k (t, z, z ′ ) has been shown in [1] by using an extension of formula (1.2) and Gruet's formula (3.6). We introduce the result in [1] together with some arguments taken from [31].
To show an explicit representation for q k (t, z, z ′ ), we recall from Proposition 2.2 in [31] (see also the references therein) that q k (t, z, z ′ ) may be written in the form for some positive function g k (t, r). This is a consequence of the group action of SL(2; Theorem 3.2. The function g k (t, r) on the right hand side of (3.10) is given by Proof. We show (3.11) when |k| < 1/2. Formula (3.11) for a general value of k follows from this result on the special case by analytic continuation. We use the same notations as those in the proof of Theorem 3.1.
In fact, it is easy to show By using the Itô formula, we have As in the proof of Theorem 3.1, we consider the conditional distribution of (w 1 t , Then it is easy to see that this conditional distribution is a two-dimensional Gaussian distribution with mean 0 and covariance matrix Taking the conditional expectation and using the Cameron-Martin theorem, we obtain In the same way as is mentioned in Remark 3.2, we may write where q λ,k (t, ξ, η) denotes the heat kernel of the semigroup generated by the Schrödinger operator H λ,k on R with the Morse potential given by In [1] and Part I, we have shown an explicit representation for q λ,k (t, ξ, η): where the function θ(r, t) is given by (1.3) and φ = 2λe (ξ+η)/2 / sinh(u). For λ < 0, we have q λ,k (t, ξ, η) = q −λ,−k (t, ξ, η). We now recall the remark following the statement of Theorem 3.2 and consider the case x ′ = x. Then, combining (3.12) and (3.13), we obtain Note that the integral is convergent if |k| < 1/2. Then, using the integral representation (1.3) for θ(r, t) and carrying out the integration in λ first, we obtain By Gruet's formula (3.6), we have It is now easy to show (3.11) from these formulae.
Similar arguments to those in the proofs of Theorems 3.1 and 3.2 are available to study the Laplace-Beltrami operators on the complex and quaternion hyperbolic spaces. Also on these symmetric spaces of rank one, we have explicit expressions of Brownian motions as Wiener functionals and we can show explicit representations for the heat kernels and for the Green functions. For details, see [40].
Next we consider the diffusion process on H 2 associated to the infinitesimal generator where ν ≧ 0 and µ > 0. The operator L ν,µ is invariant under the special transforms on H 2 of the form z → az+b, a > 0 and b ∈ R, while the operator H k and, in particular, the Laplacian ∆ 2 are invariant under the action of SL(2; R). The diffusion process starting from z = (x, y) associated to L ν,µ may be realized as the unique strong solution {Z defined on a two-dimensional Wiener space. As in the case of Brownian motion on H 2 , Z (ν,µ) is also represented as a Wiener functional by Since µ is assumed to be positive, Y (ν,µ) t converges to 0 as t tends to ∞. Following [4], we show that the distribution of X (ν,µ) t converges weakly as t → ∞ and that we can specify the limiting distribution. It is enough to consider the special case x = 0 and y = 1.
For details on the normalizing constant C ν,µ , see [4]. Note that, if ν = 0 and µ = 1/2, that is, if we consider the Brownian motion on H 2 , the limiting distribution is the Cauchy distribution as in the case of the hitting distribution on lines of standard Brownian motion on R 2 . In general, the limiting distribution belongs to the type IV family of Pearson distributions (cf. [34]).
It should be mentioned that the functional has been much studied in the context of risk theory. See Paulsen [50] and the references cited therein about this. In [50] the density is derived when ν > 1. See also [2] and [3] about some results in special cases. For further related discussions, see [47] and [64].
We present a probabilistic proof taken from [4], where we also find an analytic proof.
Proof. The limiting distribution coincides with that for the stochastic process {X We also consider the diffusion process { X (ν,µ) t } given by By the invariance of the law of Brownian motion under time reversal from a fixed time,X (ν,µ) t and X (ν,µ) t are identical in law for any fixed t > 0. Therefore, to prove the theorem, we only have to check L (ν,µ) * f = 0 for the adjoint operator L (ν,µ) * to L (ν,µ) . ) has been studied in [1] (see also Part I).

Remark 3.4. Set
We have, using some obvious notation, Hence we obtain However, we have not succeeded in obtaining Theorem 3.3 from this expression. Remark 3.5. The limiting distribution with density f (ξ) belongs to the domain of attraction of a stable distribution, whose characteristic function φ is of the form where c > 0, −1 < γ < 1 and z ∈ R.
It is also the case if we consider the hitting distribution on {Im(z) = a}, that is, the distribution of X

Maximum of a diffusion process in random environment
The purpose of this section is to survey the work by Kawazu-Tanaka [35] on the maximum of a diffusion process in a drifted random environment. In [35], several equalities and inequalities for the exponential functionals of Brownian motion are used.

A scale function S (c) (x) = S (c)
W (x) for X(W ) is given by By the general theory of the one-dimensional diffusion processes, {S (c) (X t (W ))} may be represented as a random time change of another Brownian motion and, based on this representation, several interesting results have been obtained. For these results, see, e.g., Brox [8], Hu-Shi-Yor [30], Kawazu-Tanaka [36].
The question we discuss in the present section is how the tail probability P(max t≧0 X(t) > x) decays as x → ∞. We have and several results on the exponential functional given by (1.1) are quite useful in this study. The random variables S (c) (x) and S (c) (−∞) are independent. We also note (cf. (1.5)) that −S (c) (−∞) is distributed as 2γ −1 2c , where γ 2c is a gamma random variable with parameter 2c.
Before proceeding to a proof for each assertion, we rewrite the right hand side of (4.1) into different forms. We set A (c) = −S (c) (−∞) and Moreover, considering the time reversal of W , we easily obtain By the Cameron-Martin theorem, we also obtain (4.4) By considering the time reversal again, we may write

Proof of (i). From (4.2) and (4.4), we have
where γ µ is a gamma random variable with parameter µ > 0. Easy evaluation of the right hand side yields the assertion. Before proceeding to a proof of (ii), we prepare two lemmas. Proof. The first assertion is easily shown by time reversal. We can show the second assertion from the identity which has been shown in Part I, Proposition 5.9. However, we give another direct proof. Set Then we have ϕ ′ (x) = ψ(x) and, if we show we obtain (4.5) by L'Hospital's theorem. By the scaling property of Brownian motion, we have and, by the Laplace principle, we also have Hence, applying the dominated convergence theorem, we obtain Hence, using the independence of increments of Brownian motion, we obtain Proof of (ii). Set A (1) = −S (1) (−∞). Then, since A (1) and B (1) x are independent, we have where γ 2 is a gamma variable with parameter 2 and we have used the Cameron-Martin theorem for the second equality. Therefore we obtain from (4.5). We next prove for some absolute constant C. Combining this with (4.6) above, we obtain the assertion. For this purpose we note the elementary inequality Then we obtain For the first term on the right hand side, we have For the second term, we use the Cauchy-Schwarz inequality to show Then, using Lemmas 4.1 and 4.2, we obtain (4.7) and the result of (ii).
Proof of (iii). We prove this case by using formula (1.2). To do this in a direct way, we note that and that the latter is 4A is defined by (1.1). Then, by using (4.1) and (1.2), we obtain x .
From this identity we see that the order of decay is x −3/2 e −c 2 x/2 and, by using the dominated convergence theorem and changing the variables in the integration, we obtain the assertion. For details, see the original paper [35].

Exponential functionals with different drifts
The purpose of this section is to show a relationship between the laws of the exponential functionals of Brownian motions with different drifts. In this and the next sections, we consider several stochastic processes or transforms on path space related to the exponential functional {A (µ) t }. In particular, the following transform Z plays an important role. For a continuous function Let ν < µ and consider two exponential functionals A (ν) = A(B (ν) ) and A (µ) = A(B (µ) ): where B (ν) s = B s + νs and B = {B s } is a one-dimensional Brownian motion with B 0 = 0 as in the previous sections.
We first consider the case where ν = −µ and µ > 0 and, using the result in this special case, we will give the general result in Theorem 5.4 below.
Theorem 5.1. Let µ > 0 and let γ µ be a gamma random variable with density Γ(µ) −1 x µ−1 e −x , x > 0, independent of B. Then one has the identity in law The identity in law for a fixed t > 0 has been obtained by Dufresne [19] and the extension (5.2) at the process level has been given in [47].
Proof. We sketch a proof based on the theory of initial enlargements of filtrations (cf. Yor [59]). Another proof based on some properties of Bessel processes has been given in [47].
Let B

As a consequence, it follows that
∞ , we obtain the theorem.
Next we consider the stochastic processes Z (−µ) = Z(B (−µ) ) and Z (µ) = Z( B (µ) ): Then we have d dt Hence, from the identity (5.2), we obtain Z(B (−µ) ) (law) = Z(B (µ) ) for any µ > 0. Moreover, by (5.3), we have the pathwise identity Z The study of the stochastic process Z(B (µ) ) is in fact the main object of the next section. We will show that, for any µ ∈ R, Z(B (µ) ) = {Z (µ) t } is a diffusion process with respect to its natural filtration {Z s , s ≦ t}, and that this result gives rise to an extension of Pitman's theorem ( [51], [54]).
A key fact in the proof of the above mentioned result is the following Proposition 5.2, which also plays an important role in the rest of this section.
Before mentioning the proposition, we note another important fact. By (5.4), we easily obtain the following: for every t > 0, In particular, Z (µ) t is strictly smaller than the original filtration B (µ) t of the Brownian motion B, as is also shown very clearly in the next proposition.

is a generalized inverse Gaussian distribution and is given by
and K µ is the modified Bessel (Macdonald) function.
Proof of the corollary. The first assertion is easily obtained. For the second assertion, we consider random variables I (±δ) z whose densities are given by the right hand side of (5.6), replacing ±δ for µ. Then, assuming that I (−δ) z and γ z are independent, we have I = I (−δ) z + 2zγ δ . In fact, more general identities in law for generalized inverse Gaussian and gamma random variables are well known. See Seshadri [56], [46] and the references therein.
Hence, from the identity (5.6) considered for µ = 0, we obtain We postpone a proof of Proposition 5.2 to the next section and, admitting this proposition as proven, we show a general relationship between the probability laws of the exponential functionals of Brownian motions with different drifts.
where γ δ is a gamma random variable with parameter δ independent of {B (µ) s }. Proof. We start from (5.7). Then, for any non-negative function ψ on R + , we have E ψ e Bt z e δBt Z t = E ψ e Bt z + 2γ δ e −δBt Z t .
We also deduce from the first assertion of the corollary Multiplying by (Z t ) −ν on both hand sides, we rewrite the last identity into Then we obtain, for any non-negative functional F , By the Cameron-Martin theorem, we now obtain This identity is equivalent to (5.8) because of (5.4) or (5.5).

Some exponential analogues of Lévy's and Pitman's theorems
In this section we consider the two stochastic processes where B (µ) t = B t + µt and B = {B t } is a one-dimensional Brownian motion starting from 0.
Our purpose here is to show that both ξ (µ) and Z (µ) are diffusion processes, that is, they give representations for some diffusion processes starting from 0, with respect to their natural filtrations and that this result gives rise to analogues or extensions of the celebrated Lévy and Pitman theorems.
We start by recalling these classical theorems. Set Then the Lévy and Pitman theorems may be stated in the following general form with any µ ∈ R.
t } be the bang-bang process with X (µ) 0 = 0 and with parameter µ, that is, the diffusion process with infinitesimal generator every λ > 0 and we can recover the Lévy and Pitman theorems as the limiting cases by letting λ → ∞. Furthermore, there is an exponential analogue of the second assertion of Theorem 6.2 since we have shown in the previous section (see (5.5) holds for every t > 0. We can easily show that ξ (µ) is a diffusion process. In fact, from the Itô formula, we deduce which implies the following.
and its infinitesimal generator is given by (ii) For every fixed t > 0, one has On the other hand, it is not as easy to show that Z (µ) is a diffusion process. By the Itô formula, we have and we need to take care of the third term on the right hand side.
Here we recall Proposition 5.2, which implies by the integral representation for K µ (cf. [38], p.119) Hence, admitting Proposition 5.2 as proven, we have obtained the following.
t } whose infinitesimal generator is given by (ii) For every t > 0, Z The diffusion processes Z (−µ) and Z (µ) have the same probability law.
To compare with the original Lévy and Pitman theorems, we present the following, which can be obtained from Theorems 6.3 and 6.4 by the scaling property of Brownian motion. We call a diffusion process a Brownian motion with drift b if its generator is given by 1 2 d 2 dx . Theorem 6.5. (i) For any λ > 0, the stochastic process is a Brownian motion with drift (ii) The stochastic process Remark 6.1. By using the integral representation for K µ (x) [38], p.140), we can show The rest of this section is devoted to a proof of Proposition 5.2, which has played an important role not only in this section but also in the previous section and in Part I of our survey. To show the proposition, we prove an identity for an anticipative transform on path space, which may be regarded as an example of the Ramer-Kusuoka formula. For the Ramer-Kusuoka formula, see [9], [37], [52] and [58].
Another proof for the proposition which uses several properties of Bessel processes has been given in [45] and a proof based on Theorem 5.1, featuring the generalized Gaussian inverse distributions, has been given in [46].
For our purpose we consider one more transform on path space. For an Rvalued continuous function φ on [0, ∞), We now summarize some properties of these transforms on path space which are easy to prove but play important roles in the following. Proposition 6.6. Letting A and Z be the transforms on path space defined by (5.1) and T be defined by (6.3), one obtains The next theorem gives an example of the Ramer-Kusuoka formula. On the left hand side of (6.4) below, the transform T α/e (µ) t depends on e (µ) t and it is natural to call it anticipative. For more discussions about this transform, related topics and references, see [17].
Step 3. We prove (6.4) for general values of µ. We start from Theorem 5.1, which says that holds for any µ > 0 and for any non-negative functional F on C([0, t] → R), where γ µ is a gamma random variable with parameter µ independent of B (µ) . We rewrite the left hand side of (6.7) into the following way: where we have used the Cameron-Martin theorem for the second identity. For the right hand side of (6.7), we rewrite Now, comparing (6.8) and (6.9), we get ∞ 0 e −η E F 1 A s + 2η, s ≦ t (ηe t ) µ dη η = ∞ 0 e −η E F 1 A s , s ≦ t (η/e t ) µ dη η .
Since this identity holds for any µ > 0, we obtain we obtain (6.5). For Step 2, we replace F exp(ηe t ) by F in (6.6). Then, since exp(T α (B) s ) = e s /(1 + αA s ), we obtain With this aim in mind, we replace F by ϕF in (6.10). Then we have Hence, for our purpose, it is sufficient that exp α 2 e t 1 + αZ t − 1 e t ϕ B t − log(1 + αZ t ), A t 1 + αZ t = 1 (6.11) and, if we take we get (6.11). Therefore, we obtain We note Z(T α/et (B)) = Z(B) (Proposition 6.6) and use (6.12). Then we obtain The proof is completed.
We are now in a position to give a proof of the key proposition.
Proof of Proposition 5.2. At first we consider the case µ = 0. We set Q ω,z t (·) = P (·|Z t , Z t = z), the regular conditional distribution given Z t . Then, taking F in (6.4) as ϕ(1/A t )G(Z s , s ≦ t) for a non-negative Borel function ϕ on (0, ∞) and for a non-negative functional G in view of Proposition 6.6, we obtain Now we assume for simplicity that the distribution of e t under Q ω,z t has a density g z (x) with respect to the Lebesgue measure. Then we have Since the function ϕ is arbitrary, we obtain z −1 g z (v/z) exp(−ηv/z) = (2η + 1/z) exp − η (2η + 1/z)v g z ((2η + 1/z)v), where we have set v = 1/x. From the last identity, we obtain g z (x) = const. x −1 exp − 1 2z x + 1 x by simple algebra and, by using the integral representation (6.2) for the Macdonald function, we obtain (5.6) when µ = 0. For a general value of µ, a standard argument with the Cameron-Martin theorem leads us to the result.