Integral representations for the Hartman--Watson density

TL;DR

This paper presents new integral representations for the Hartman--Watson density, building on Yor's formula, and applies these to derive simplified laws for exponential additive functionals of Brownian motion.

Contribution

It introduces alternative integral formulas for the Hartman--Watson density and extends their application to simpler representations of related stochastic functionals.

Findings

New integral representations for Hartman--Watson density

Simplified laws for exponential additive functionals of Brownian motion

Extension of Yor's formula to broader contexts

Abstract

This paper concerns the density of the Hartman--Watson law. Yor (1980) obtained an integral formula that gives a closed-form expression of the Hartman--Watson density. In this paper, based on Yor's formula, we provide alternative integral representations for the density. As an immediate application, we recover in part a Dufresne's result (2001) that exhibits remarkably simple representations for the laws of exponential additive functionals of Brownian motion.

Equations

$$A_t^{(\mu )}:={\int }_0^t{e}^{2B_s^{(\mu )}}ds,t\ge 0;$$

$$\mathbb{P}\left(B_t\in dx,A_t\in dv\right)=\frac{1}{v}\exp \left\{-\frac{1}{2v}\left(1+e^{2x}\right)\right\}\Theta (e^x/v,t)dxdv,x\in \mathbb{R},v>0,$$

$$\mathbb{P}\left(e^{B_t}\in du,A_t\in dv\right)=\frac{1}{uv}\exp \left(-\frac{1+u^2}{2v}\right)\Theta (u/v,t)dudv,u,v>0,$$

$${\int }_0^{\infty }dt\exp \left(-\frac{{\lambda }^2}{2}t\right)\Theta (r,t)=I_{|\lambda |}(r),\lambda \in \mathbb{R}.$$

$$\frac{\mathbb{P}(A_t^{(\mu )}\in dv)}{dv}$$

$$=v^{\mu -1}\exp \left(-\frac{1}{2v}-\frac{{\mu }^2}{2}t\right){\int }_0^{\infty }dr{r}^{\mu -1}\exp \left(-\frac{v}{2}{r}^2\right)\Theta (r,t),v>0,$$

$$\Theta (r,t)=\frac{r}{\sqrt{2{\pi }^{3}t}}{\int }_0^{\infty }dy\exp \left(\frac{{\pi }^2-y^2}{2t}\right)\exp \left(-r\cosh y\right)\sinh y\sin \left(\frac{\pi y}{t}\right),$$

$$\Theta (r,t)=\frac{r}{2\pi }\exp \left(\frac{{\pi }^2}{2t}\right)\mathbb{E}\left[\exp \left(-r\cosh B_t\right)\sinh B_t\sin \left(\frac{\pi }{t}{B}_t\right)\right],$$

$${\int }_0^{\infty }\frac{dr}{r}{e}^{-r\cosh x}\Theta (r,t)=\frac{1}{\sqrt{2\pi t}}\exp \left(-\frac{x^2}{2t}\right),t>0,x\in \mathbb{R},$$

$$\Theta (r,t)$$

$$=\frac{r}{\pi }\exp \left(\frac{{\pi }^2}{8t}\right)\mathbb{E}\left[\cosh B_t\sin (r\sinh B_t)\sin \left(\frac{\pi }{2t}{B}_t\right)\right]$$

$$=\frac{r}{2\pi }\exp \left(\frac{{\pi }^2}{8t}\right)\mathbb{E}\left[\cosh B_t\cos \left(r\sinh B_t-\frac{\pi }{2t}{B}_t\right)\right].$$

$$\Theta (r,t)$$

$$\mathbb{E}\left[\cosh B_t\cos \left(r\sinh B_t+\frac{\pi }{2t}{B}_t\right)\right]=0\text{for any r>0 and t>0,}$$

$$\mathbb{E}\left[\cosh B_t\sin \left(r\sinh B_t+\frac{\pi }{2t}{B}_t-2\nu \right)\right]=0,r>0,t>0,$$

$$\mathbb{E}\left[\cosh B_t\sin \left(r\sinh B_t+\frac{\pi }{2t}{B}_t\right)\right]=0$$

$$\underset{r\to \infty }{\lim }\mathbb{E}\left[\cosh B_t\cos (r\sinh B_t)\cos \left(\frac{\pi }{2t}{B}_t\right)\right]=0,$$

$$\displaystyle\frac{1}{\sqrt{2\pi t}}\int_{\mathbb{R}}dy\,\frac{\cosh y}{\sqrt{2\pi v^{3}}}\exp\left(-\frac{\cosh^{2}y}{2v}\right)\exp\biggl{\{}-\frac{\left(y+\sqrt{-1}\pi/2\right)^{2}}{2t}\biggr{\}}$$

$$\underset{r\to 0+}{\lim }\frac{{\partial }^n}{\partial r^n}\Theta (r,t)=0,n=0,1,2,\dots .$$

$$\Theta (r,t)=o(r^{\kappa })\text{for any κ>0.}$$

$${\int }_{\mathbb{R}}dx\exp \left(-\frac{x^2}{2t}\right)e^{nx}\sin \left(\frac{\pi x}{t}\right)=0$$

$$\sqrt{2\pi t}\mathbb{E}\left[e^{n{B}_t}\sin \left(\frac{\pi }{t}{B}_t\right)\right],$$

$$e^{-n^{2}t/2}\mathbb{E}\left[e^{n{B}_t}\sin \left(\frac{\pi }{t}{B}_t\right)\right]$$

$$=(-1{)}^n\mathbb{E}\left[\sin \left(\frac{\pi }{t}{B}_t\right)\right],$$

$$H_{2n}(x)$$

$$H_{2n+1}(x)$$

$$H_{\mu }(x)=\frac{2^{\mu +1}{e}^{x^2}}{\sqrt{\pi }}{\int }_0^{\infty }ds{s}^{\mu }{e}^{-s^2}\cos \left(2xs-\frac{\pi \mu }{2}\right),x\in \mathbb{R}$$

$$\frac{\mathbb{P}(A_t^{(\mu )}\in dv)}{dv}$$

$$=\frac{C_{\mu }(t)}{\sqrt{v^{3-\mu }}}\mathbb{E}\left[\exp \left(-\frac{{\cosh }^2{B}_t}{2v}\right)H_{\mu }\left(\frac{\sinh B_t}{\sqrt{2v}}\right)\cosh B_t\cos \left\{\frac{\pi }{2}\left(\frac{B_t}{t}-\mu \right)\right\}\right]$$

$$H_0(x)=1,H_1(x)=2x,x\in \mathbb{R}$$

$$\mathbb{E}\left[|F(B_t)|\right]<\infty$$

$$\displaystyle{\rm i}\$$

$$\displaystyle{\rm ii}\$$

$$\displaystyle{\rm iii}\$$

Full text

1 Introduction

Let $B=\{B_{t}\}_{t\geq 0}$ be a one-dimensional standard Brownian motion. For every $\mu\in\mathbb{R}$, we denote by $B^{(\mu)}=\{B^{(\mu)}_{t}:=B_{t}+\mu t\}_{t\geq 0}$ the Brownian motion with constant drift $\mu$ and set

[TABLE]

when $\mu=0$, we simply write $A_{t}$ for $A^{(\mu)}_{t}$. This additive functional, together with the geometric Brownian motion $e^{B^{(\mu)}_{t}},\,t\geq 0$, plays an important role in a number of areas such as option pricing in mathematical finance, diffusion processes in random environments, probabilistic study of Laplacians on hyperbolic spaces, and so on; see the detailed surveys [14, 15] by Matsumoto–Yor and references therein.

In [19], Yor proved that for every $t>0$, the joint law of $B_{t}$ and $A_{t}$ is given by

[TABLE]

or equivalently,

[TABLE]

where for every $r>0$, the function $\Theta(r,t),\,t>0$, is the (unnormalized) density of the so-called Hartman–Watson distribution ([7]) which is characterized by the Laplace transform

[TABLE]

Here for every index $\nu\in\mathbb{R}$, the function $I_{\nu}$ is the modified Bessel function of the first kind of order $\nu$; see [10, Section 5.7] for the definition. By the Cameron–Martin relation, we see from (1.2) that for every $\mu\in\mathbb{R}$ and $t>0$, the law of $A^{(\mu)}_{t}$ is expressed as

[TABLE]

where for the second line, we changed the variables with $u=vr,\,r>0$.

It is also proven by Yor [18] that the function $\Theta$ admits the following integral representation: for every $r>0$ and $t>0$,

[TABLE]

which may be rephrased conveniently as

[TABLE]

by the fact that the function $\mathbb{R}\ni y\mapsto\sinh y\sin(\pi y/t)$ is symmetric. We also refer to [3, p. 82] for (1.5), and [3, p. 175, formula 1.10.8] for the joint law (1.1).

The function $\Theta$ as well as formula (1.5) have continuously attracted researchers’ attention particularly from the view point of evaluation of Asian options in the Black–Scholes framework; see, e.g., [16, 4, 2, 11] and references therein. Yor obtained (1.5) by inverting the Laplace transform (1.3), reasoning of which is also reproduced in [14, Appendix A]. In [6, Subsection A.3], we explain (1.5) via

[TABLE]

which relation is found in, e.g., [12, Proposition 4.5(i)] and, as was observed in [14, Proposition 4.2], may be obtained by integrating both sides of (1.1) with respect to $v$. In this paper, we continue our discussion of [6, Subsection A.3] and, based on Yor’s formula (1.5) (or (1.6)), aim at providing the following alternative representations of $\Theta$:

Theorem 1.1**.**

For every $r>0$ and $t>0$, it holds that

[TABLE]

More generally, we have for every $r>0$ and $t>0$,

[TABLE]

where $\nu\in\mathbb{R}$ is arbitrary. Representations (1.8), (1.9) and (1.10) may be seen as the case $\nu=0,\pi/2,\pi/4$, respectively.

The third representation (1.10) follows readily from (1.8) and (1.9) by summing them. If we multiply (1.8) and (1.9) by $\cos^{2}\nu$ and $\sin^{2}\nu$, respectively, then taking their sum leads to the fourth representation (1.11); for details, see the proof of Theorem 1.1 given in Section 2 whose reasoning will also reveal that $\nu$ may be replaced by any complex number.

Remark 1.1*.*

1 Taking the difference of (1.8) and (1.9) leads to the following fact of interest:

[TABLE]

which will be returned to in Remark 3.1 and recalled in the proof of Proposition 4.2 below. Representation (1.10) should be considered jointly with (1.12).

2 Differentiating representation (1.11) with respect to $\nu$ yields

[TABLE]

for any $\nu\in\mathbb{R}$, which agrees with (1.12) since for any $r>0$ and $t>0$,

[TABLE]

due to the fact that the function $\sin\left(r\sinh x+\pi x/(2t)\right),\,x\in\mathbb{R}$, is an odd function.

3 In view of (1.5), we see from (1.8) that

[TABLE]

which may also be explained by the Riemann–Lebesgue lemma. The same remark is true for (1.9).

4 In a recent paper [9] by Jakubowski and Wiśniewolski, they have obtained in their Theorem 3.7 and Corollary 3.9 the third representation (1.10), appealing to the fact due to Alili and Gruet ([1, equation (1.5)]) that when $t>0$, the density of the law of $A_{t}$ is given by

[TABLE]

for $v>0$. As noted above, representation (1.10) is immediate from (1.8) and (1.9); our method used in deriving those two representations quite differs from that of [9] and hinges upon a simple observation exhibited in Lemma 2.1 below, which explains the Riemann–Lebesgue lemma in a clear fashion as to some expectations relative to $B_{t}$. In view of the injectivity of Mellin transform, representation (1.10) may also be deduced from the two representations of $\alpha^{(\nu,\varepsilon)}_{t}(w)$ given just after Theorem 2.1 of [16] by Schröder.

5 In Proposition 4.2, we also present other relevant integral representations of $\Theta$.

Theorem 1.1 has several applications. One of its immediate consequences is that for every fixed $t>0$, the derivative of $\Theta(r,t)$ of any order at $r=0+$ vanishes due to repeated occurrence of the multiple factor $\sin(r\sinh B_{t})$ in integrands when taking derivatives of (1.8) and (1.9) with respect to $r$.

Proposition 1.1**.**

Fix $t>0$. It holds that

[TABLE]

In particular, as $r\to 0+$,

[TABLE]

For the proof, see Subsection 3.1; the above fact (1.13) can also be deduced readily from (1.10) and (1.12), for which we refer the reader to Remark 3.1. As a consequence of (1.14), we see that the integral in (1.4) does converge even when $\mu\leq 0$. We recall that (1.13) is also observed in [13, Subsection 2.1] from Yor’s formula (1.5), combined with a remark by Stieltjes in 1894 stating that for any integer $n$,

[TABLE]

(see [13, equation (6.3)]). Our Theorem 1.1 allows us to obtain (1.13) more directly, without relying on Stieltjes’ remark.

Remark 1.2*.*

Since the left-hand side of (1.15) is written as

[TABLE]

(1.15) is verified by the Cameron–Martin relation:

[TABLE]

which is zero by the symmetry of Brownian motion.

When inserting representation (1.5) into (1.4), a double integral emerges in the description of the law of $A^{(\mu)}_{t}$. The second application of Theorem 1.1 is that, when $\mu$ is a nonnegative integer, we easily reduce that apparently complicated double integral to a single integral by Fubini’s theorem, thanks to the well-known formulae (see [10, equations (4.11.2) and (4.11.3)]) for the Hermite polynomials

[TABLE]

where $n$ is any nonnegative integer. Dealing with other values of $\mu$ as well, we put the above-mentioned reduction in Proposition 1.2 below, which recovers in part Theorem 4.2 of [5] by Dufresne. For every $\mu\in\mathbb{R}$, we denote by $H_{\mu}$ the Hermite function of degree $\mu$ and recall its integral representation when $\mu>-1$:

[TABLE]

(see Section 10.2 and equation 10.5.5 in [10] for the definition of the Hermite functions and the integral representation (1.16), respectively).

Proposition 1.2**.**

Let $\mu>-1$. For every $t>0$, the law of $A^{(\mu)}_{t}$ admits the density function expressed by

[TABLE]

for $v>0$, where C_{\mu}(t)=\bigl{(}1/\sqrt{2^{\mu+1}\pi}\bigr{)}e^{\pi^{2}/(8t)-\mu^{2}t/2}.

Recall that

[TABLE]

(see, e.g., [10, p. 60]). Expression (1.17) in the case $\mu=0$ was obtained by several authors, for which we refer the reader to [5, p. 223] as well as the beginning of [13, Subsection 2.2] (see also Remark 1.14 above); an alternative derivation of (1.17) in the case $\mu=0$ and $\mu=1$ without relying on Theorem 1.1, will be found in Section 4 (see equations (4.17) and (4.18) therein). If we consider the law of $1/(2A^{(\mu)}_{t})$, then from (1.17), we partly recover formula 4.9 in [5, Theorem 4.2] due to Dufresne, who also shows that the formula is valid for $\mu\leq-1$ as well, by developing a recurrence relation that connects the law of $1/(2A^{(\nu)}_{t})$ with that of $1/(2A^{(\mu)}_{t})$ when $\nu<\mu$. We do not pursue it here with generality, however, if we repeat integration by parts as necessary appealing to (1.13), then Theorem 1.1 enables us to reduce the computation of the case $\mu\leq-1$ to a situation where formula (1.16) applies or the function $H_{-1}$ emerges; see Remark 3.2. We also note that in [16], a contour integral representation for the density of the law of $A^{(\mu)}_{t}$, is given in terms of Hermite functions and compared with Dufresne’s representation.

We give an outline of the paper. In Section 2, we prove Theorem 1.1; we do this by preparing Lemma 2.1 which shows the equivalence of certain three relations for expectations relative to Brownian motion. Since in the proof of Theorem 1.1, one implication between two of the three relations is used, we give its proof in Section 2 and the rest of the proof of the lemma is provided in the appendix. Propositions 1.1 and 1.2 are proven in Section 3. We prove in Section 4 a family of integral identities that embraces relations in Theorem 1.1, which is then applied to the derivation of other integral representations of $\Theta$ relevant to Theorem 1.1. Finally, in the appendix, we complete the proof of Lemma 2.1.

2 Proof of Theorem 1.1

From now on, we fix $t>0$. This section is devoted to the proof of Theorem 1.1. Let two real-valued functions $F$ and $G$ on $\mathbb{R}$ be continuous for simplicity and suppose that they are even functions and satisfy

[TABLE]

Lemma 2.1**.**

Under condition (2.1), the following three relations are equivalent:

[TABLE]

Since, in the proof of Theorem 1.1, we only use the implication from i to ii, we give a proof of it below and postpone proofs of other implications to the appendix.

Proof of i $\Rightarrow$ ii in Lemma 2.1.

For every $r\geq 0$, we integrate both sides of (2.2) multiplied by $\cosh x\cos(r\sinh x)$ with respect to $x\in\mathbb{R}$. Then the right-hand side turns into that of (2.3) in view of the latter condition in (2.1). On the other hand, provided that we are allowed to use Fubini’s theorem, the left-hand side turns into

[TABLE]

as claimed. Here for the equality, we used the fact that for any $b\in\mathbb{R}$ and $r\geq 0$,

[TABLE]

which follows readily by noting $\cosh(2b)+\cosh(2x)=2\left(\cosh^{2}b+\sinh^{2}x\right)$, and changing the variables with $\sinh x=y\cosh b,\,y\in\mathbb{R}$. Usage of Fubini’s theorem mentioned above is justified by taking $r=0$ in (2.5); indeed,

[TABLE]

which is assumed to be finite in (2.1). The proof is complete. ∎

In what follows, we denote by $\mathbb{C}$ the complex plane and write $i=\sqrt{-1}$. A pair of functions $F$ and $G$ fulfilling relation (2.2) may be obtained by the residue theorem applied to a meromorphic function $f$ of the form

[TABLE]

where $x\in\mathbb{R}$ and $J(z),\,z\in\mathbb{C}$, is an odd entire function which will be taken to be either $\sinh(2z)$ or $\sinh z$ below. When $x\neq 0$, the poles $w$ of $f$ each of whose imaginary part $\mathrm{Im}\,w$ lies between [math] and $\pi$, are two points $\pm x+(\pi/2)i$. By taking a rectangular contour circling these poles and having its two sides on the two lines $\mathrm{Im}\,z=0$ and $\mathrm{Im}\,z=\pi$, residue calculus yields, at least heuristically,

[TABLE]

for $x\neq 0$. When $J(z)=\sinh(2z)$ and $\sinh z$, the above computation is justified, yielding the following lemma:

Lemma 2.2**.**

It holds that for any $x\in\mathbb{R}$,

[TABLE]

where in the latter identity, the function $S(x)\equiv S(x,t),\,x\in\mathbb{R}$, is given by

[TABLE]

Validity of (2.7) and (2.8) at $x=0$ follows by passing to the limit as $x\to 0$. We remark that these two identities are found in Lemmas 3.1 and 3.2 of [13] by Matsumoto–Yor, in which paper those two lemmas are used to show that, in the case $\mu=0$ and $1$, expression (1.4) with Yor’s formula (1.5) inserted in coincides with (1.17). What is revealed in the present paper is that coincidence for any $\mu>-1$ is also reduced to the above two identities (2.7) and (2.8) via Theorem 1.1.

We are in a position to prove the theorem.

Proof of Theorem 1.1.

First we prove (1.8). Identity (2.7) tells us that we may take

[TABLE]

in relation (2.2). It is clear that these functions fulfill the integrability condition (2.1). Therefore by Lemma 2.1, we have for every $r\geq 0$,

[TABLE]

Now representation (1.8) follows from this and Yor’s formula (1.6).

We proceed to the proof of (1.9). The second identity (2.8) in Lemma 2.2 shows that we may take in (2.2)

[TABLE]

which pair also fulfills (2.1). Therefore by Lemma 2.1, we have for every $r\geq 0$,

[TABLE]

Thanks to the fact that

[TABLE]

differentiating both sides of the last identity with respect to $r$ and appealing to formula (1.6) again, we arrive at (1.9).

Representation (1.10) is a consequence of summation of (1.8) and (1.9). To prove (1.11), fix $\nu\in\mathbb{R}$. Using the addition theorem, we develop

[TABLE]

with $R(x),\,x\in\mathbb{R}$, an odd function such that $\mathbb{E}\left[|R(B_{t})|\cosh B_{t}\right]<\infty$. Hence the right-hand side of the claimed identity (1.11) is equal to

[TABLE]

by (1.8) and (1.9), which shows (1.11) and completes the proof of the theorem. ∎

3 Proofs of Propositions 1.1 and 1.2

In the sequel, we set the function $g(r)\equiv g(r,t),\,r>0$, by

[TABLE]

As seen in the previous section, it holds that for any $r>0$,

[TABLE]

3.1 Proof of Proposition 1.1

In this subsection, we prove Proposition 1.1.

Proof of Proposition 1.1.

By relation (3.1), it suffices to show that

[TABLE]

By observing the fact that

[TABLE]

for any nonnegative integer $n$, successive differentiation of the above two representations of $g$ yields

[TABLE]

for every nonnegative integer $n$, from which (3.2) follows readily by the dominated convergence theorem. ∎

Remark 3.1*.*

Another direct way of convincing ourselves of (3.2) is to consider the mapping

[TABLE]

which determines a $C^{\infty}$-function thanks to (3.3), agrees with $2g(r)$ for $r>0$, and is identically zero on $(-\infty,0)$ in view of (1.12), and hence would yield a contradiction if (3.2) were not the case. A fact of independent interest following from the above argument is that the function given by (3.4) provides an example of a $C^{\infty}$-function on $\mathbb{R}$ that is not analytic at $r=0$; in fact,

[TABLE]

whenever $r>0$, since the left-hand side agrees with

[TABLE]

and, under the pinned measure $P(\,\cdot\mid B_{t}=0)$, $\mathrm{ess\,sup}\,A_{t}=\infty$ and $\mathrm{ess\,inf}\,A_{t}=0$. Expression (3.5) is a consequence of (1.1) together with (1.10).

3.2 Proof of Proposition 1.2

In this subsection, we prove Proposition 1.2.

Proof of Proposition 1.2.

We insert representation (1.11) into (1.4) putting $\nu=\pi\mu/2$. Then for $\mu>-1$, Fubini’s theorem entails that (1.4) is rewritten as

[TABLE]

where we set the function $h(x),\,x\in\mathbb{R}$, by

[TABLE]

which is equal, by changing the variables with $r=\sqrt{(2/v)}s$, to

[TABLE]

by the integral formula (1.16) of $H_{\mu}$ with $\mu>-1$. Inserting the last expression of $h$ into (3.6) and rearranging terms lead to (1.17) and conclude the proof. ∎

We end this section with a remark on the case $\mu\leq-1$.

Remark 3.2*.*

We take $\mu=-3/2$ and $-2$ as an illustration. Noting relation (3.1), we use the function $g$ to rewrite the integral in (1.4) with respect to $r$ as

[TABLE]

The proof of Proposition 1.2 shows that $I(\mu)$ may be expressed in terms of $H_{\mu}$ when $\mu>-1$. If we take $\mu=-3/2$, then integration by parts yields

[TABLE]

owing to (1.13). Recalling (1.11), we have

[TABLE]

for every $r>0$ and $\nu\in\mathbb{R}$. Therefore choosing $\nu=-(3/4)\pi$ and appealing to Fubini’s theorem and (1.16), we may express the second term on the right-hand side of (3.7) in terms of $H_{-1/2}$. In the case $\mu=-2$, we have in the same way as above,

[TABLE]

Noting that

[TABLE]

and that $|\sin(r\sinh B_{t})/r|\leq|\sinh B_{t}|$ for any $r>0$, Fubini’s theorem entails that the second term on the right-hand side of (3.8) is written as

[TABLE]

By the formulae

[TABLE]

the integrand in the last expectation may be expressed in terms of $H_{-1}$. As for formula (3.10), see [10, equation (10.5.3)]. Formula (3.9) may easily be verified by writing

[TABLE]

and using Fubini’s theorem. The above illustration in the two cases indicates that, when $-(m+1)\leq\mu<-m$ for some positive integer $m$, the density (1.4) of the law of $A^{(\mu)}_{t}$ admits an expression in terms of $H_{\mu+m}$ and $H_{\mu+m+1}$ (while in the case $\mu=-1$, only $H_{-1}$ emerges).

4 A further discussion

In this section, we develop further the discussion used in deriving Lemma 2.2 to obtain a family of integral identities that includes relations (1.10) and (1.12) as its special cases, which, in turn, yields (1.8) and (1.9); another set of integral representations of $\Theta$ relevant to Theorem 1.1, is also provided.

We keep $t>0$ fixed. We prove

Proposition 4.1**.**

For every $r,\lambda\geq 0$ and $\gamma\in\mathbb{R}$, it holds that

[TABLE]

Before giving a proof of the above proposition, we explain in the remark below how to obtain several relations in Section 1 from (4.1): note that, thanks to (2.9), differentiating both sides of (4.1) at $\lambda=0$ yields the relation that for every $r\geq 0$ and $\gamma\in\mathbb{R}$,

[TABLE]

Remark 4.1*.*

1 Taking $\gamma=-\pi/(2t)$ in (4.2), we have relation (1.10). If we take $\gamma=\pi/(2t)$, then relation (1.12) also follows, which, together with (1.10), entails (1.8) and (1.9).

2 Moreover, if we take $\gamma=0$ in (4.2), then in view of the implication from ii to i in Lemma 2.1, we have

[TABLE]

(see also Remark 4.2 below), which may be restated, by replacing $x$ and $t$ by $x/2$ and $t/4$, respectively, and by using the scaling property of Brownian motion, as

[TABLE]

Note that the left-hand side is rewritten as

[TABLE]

by Fubini’s theorem. Therefore once the characterization (1.7) of $\Theta$, namely the Laplace transform of $\Theta(r,t)/r$ in the variable $r>0$, is at our disposal, the integral representation (1.5) follows by the injectivity of Laplace transform. We also note that in view of Lemma 2.1, relation (4.3) is equivalent to

[TABLE]

which is a relation observed in [6, Subsection A.3].

It would also be of interest to note that by taking $\gamma=\pi/(4t)$ in (4.1), there holds the following symmetry with respect to the variables $r,\lambda\geq 0$:

[TABLE]

In order to prove Proposition 4.1, we consider a contour integral of a meromorphic function $f(z),\,z\in\mathbb{C}$, of the form

[TABLE]

where $x\in\mathbb{R}$ is fixed and $G$ denotes an even entire function. If we take the same contour as used in Section 2 supposing $x\neq 0$, then we have for a suitable choice of $G$,

[TABLE]

by residue calculus applied to $f$. As will be seen, we are allowed to take $G(z)=\exp(-\lambda\cosh z)\cos(\gamma z)$ and $\exp(-\lambda\cosh z)\sin(\gamma z)\sinh z$ for $\lambda\geq 0$ and $\gamma\in\mathbb{R}$, obtaining the following lemma: set

[TABLE]

Lemma 4.1**.**

For any $\lambda\geq 0$ and $\gamma\in\mathbb{R}$, relation (2.2) holds for the following pairs of functions $F(x)$ and $G(x)$, $x\in\mathbb{R}$:

[TABLE]

Proof.

In the case $G(z)=\exp(-\lambda\cosh z)\cos(\gamma z),\,z\in\mathbb{C}$, because of the fact that

[TABLE]

for any $L\in\mathbb{R}$ and $0\leq\eta\leq\pi$ when $\lambda\geq 0$, it follows readily that

[TABLE]

which justifies (4.4), yielding pair i. The same justification is also true in the case $G(z)=\exp(-\lambda\cosh z)\sin(\gamma z)\sinh z,\,z\in\mathbb{C}$, and leads to ii. As it is clear that $\gamma$ may be any complex number in the above argument, it is also possible to obtain ii from i by using the following relations:

[TABLE]

for any $x,\gamma\in\mathbb{R}$. We conclude the proof of the lemma. ∎

Remark 4.2*.*

Relation (4.3) is nothing but the case $\lambda=\gamma=0$ in i; in other words, it is obtained simply by taking $G\equiv 1$ in (4.4).

Using the above lemma, we prove Proposition 4.1.

Proof of Proposition 4.1.

By Lemma 2.1 and i of Lemma 4.1, we have for any $r,\lambda\geq 0$ and $\gamma\in\mathbb{R}$,

[TABLE]

which shows that the first derivatives with respect to $\lambda$ of the following two expressions agree:

[TABLE]

where we set

[TABLE]

(We are allowed to interchange the order of differentiation and expectation thanks to (2.9).) Moreover, by the Riemann–Lebesgue lemma, the former expression (4.5) converges to [math] as $\lambda\to\infty$, and by the bounded convergence theorem, the latter expression (4.6) does as well. Therefore the two expressions (4.5) and (4.6) agree. Similarly, by Lemma 2.1 and ii of Lemma 4.1, we have

[TABLE]

If we consider the following two expressions

[TABLE]

then by differentiating them with respect to $\lambda$ and $r$ successively, and by using the same reasoning as above, the last identity entails that those two expressions also agree. Consequently, the difference of (4.5) and (4.7) coincides with that of (4.6) and (4.8), which proves the proposition. ∎

Identity (4.2) enables us to obtain yet another set of integral representations of $\Theta$, which we put in the next proposition.

Proposition 4.2**.**

For every $r>0$ and $t>0$, it holds that

[TABLE]

More generally, we have for every $r>0$ and $t>0$,

[TABLE]

where $\nu\in\mathbb{R}$ is arbitrary.

Proof.

The third representation (4.11) follows by taking $\gamma=3\pi/(2t)$ in (4.2) and noting (1.6). Then the first two representations (4.9) and (4.10) are obtained by observing that their arithmetic mean agrees with (4.11) and their difference vanishes because of (1.12). The last representation (4.12) is proven in the same way as in the proof of (1.11). ∎

Remark 4.3*.*

By combining Theorem 1.1 and Proposition 4.2, it is also possible to derive the following representations: for every $r>0$ and $t>0$,

[TABLE]

In fact, replacing $\gamma$ in (4.2) by $\pi/(2t)\pm\gamma$, one may deduce that for every $r>0$, $t>0$ and $\gamma\in\mathbb{R}$,

[TABLE]

We recall from [6, Proposition 3.3] that for any $\lambda,r\geq 0$,

[TABLE]

Hence, taking $\gamma=0$ in (4.1), we have the following relation:

[TABLE]

As for the former relation (4.13), we also refer to [8, Proposition 2.4], which may be regarded as the case where $r$ in (4.13) is replaced by a purely imaginary number with modulus not exceeding $\lambda$. To our knowledge, the latter relation (4.14) has not been noticed before. We conclude this section by pointing out that one can easily derive from (4.14) simple explicit expressions of the laws of $A_{t}$ and $A^{(1)}_{t}$; notice that in [6], relation (4.13) is obtained independently of formula (1.1), by using so-called Bougerol’s identity and a certain invariance formula for Cauchy random variable.

Taking $\lambda=0$ in (4.14), we have

[TABLE]

Moreover, if we differentiate both sides of (4.14) at $\lambda=0$, we also have

[TABLE]

where on the left-hand side, we used the Cameron–Martin relation. Inserting the rewriting

[TABLE]

into the right-hand sides of (4.15) and (4.16), and using Fubini’s theorem, we see that for $v>0$,

[TABLE]

thanks to the injectivity of Laplace transform. These two expressions agree with (1.17) when $\mu=0$ and $\mu=1$, respectively.

Remark 4.4*.*

1 On the other hand, by taking $\lambda=0$ in the former relation (4.13), we have

[TABLE]

for any $r\geq 0$, which relation is explained by Bougerol’s original identity

[TABLE]

Here and below, $\{\beta(s)\}_{s\geq 0}$ denotes a one-dimensional standard Brownian motion independent of $B$. Moreover, if we differentiate both sides of (4.13) at $\lambda=0$, then by the Cameron–Martin relation, we have

[TABLE]

for any $r\geq 0$, where on the right-hand side, $\varepsilon$ is a Rademacher (or symmetric Bernoulli) random variable taking values $\pm 1$ with probability $1/2$, independently of $B$. By rewriting the left-hand side of the last identity as

[TABLE]

the injectivity of Fourier transform entails the following variant of Bougerol’s identity:

[TABLE]

For Bougerol’s identity and its variants including the above one, see the survey [17] by Vakeroudis; different kinds of extensions of Bougerol’s identity may be found in [6].

2 Relation (4.14) also enables us to derive the expression of the joint density (1.1) with the integral representation (1.8) of $\Theta$ inserted in, but we omit details here.

Acknowledgements. The author would like to thank anonymous referees for their valuable comments.

Appendix

We complete the proof of Lemma 2.1.

Proof of ii $\Rightarrow$ iii in Lemma 2.1.

Given $x\in\mathbb{R}$, we integrate both sides of (2.3) multiplied by $e^{-r\cosh x}$ with respect to $r\geq 0$. Then by Fubini’s theorem, the left-hand side turns into that of (2.4). Therefore it suffices to show that

[TABLE]

By the latter condition in (2.1), we may also use Fubini’s theorem to rewrite the left-hand side of the claimed identity (A.1) as

[TABLE]

On the other hand, by the symmetry of $G$, the right-hand side of (A.1) is equal to

[TABLE]

Noting the fact that

[TABLE]

for any $x,y\in\mathbb{R}$, we compare the last expression with (A.2) to conclude identity (A.1). ∎

We turn to the proof of the implication from iii to i. To this end, we prepare the following lemma:

Lemma A.1**.**

For every $x,b\in\mathbb{R}$, it holds that

[TABLE]

Proof.

We may assume $|x|\neq|b|$; validity in the case $|x|=|b|$ is verified by passing to the limit. By symmetrization and by the relation $\cosh(2b)+\cosh(2y)=2\left(\cosh^{2}b+\sinh^{2}y\right)$, the left-hand side of the claimed identity is equal to

[TABLE]

which is rewritten, due to relation (A.3), as

[TABLE]

where we changed the variables with $\sinh y=z$ in the second line. Now the claimed identity follows. ∎

We are prepared to finish the proof of Lemma 2.1.

Proof of iii $\Rightarrow$ i in Lemma 2.1.

We appeal to the injectivity of Fourier transform. For this purpose, we first observe that

[TABLE]

Indeed, the former observation is immediate from (2.6) and the former condition in (2.1) while the latter is clear by the latter condition in (2.1). For an arbitrarily fixed $\xi\in\mathbb{R}$, we integrate both sides of (2.4) multiplied by $\cos(\xi x)$ with respect to $x\in\mathbb{R}$. Then by the latter finiteness in (A.4) and Fubini’s theorem, the right-hand side turns into

[TABLE]

where we used the fact that

[TABLE]

which is verified by standard residue calculus. On the other hand, as for the left-hand side of (2.4), we have

[TABLE]

where we used Lemma A.1 for the second line, Fubini’s theorem for the third thanks to the former finiteness in (A.4), and fact (A.6) for the fourth. Since the last expression agrees with (A.5) for any $\xi\in\mathbb{R}$ and the function $G$ is assumed to be symmetric, the injectivity of Fourier transform entails relation (2.2). The proof completes. ∎

Remark A.1*.*

By using Lemma A.1, implication i $\Rightarrow$ iii may also be proven in the same manner as in the proof of i $\Rightarrow$ ii given in Section 2.