On the Law of a Triplet Associated with the Pseudo-Brownian Bridge

Abstract

We identify the distribution of a natural triplet associated with the pseudo-Brownian bridge. In particular, for B a Brownian motion and T ₁ its first hitting time of the level one, this remarkable law allows us to understand some properties of the process $(B_{\mathit{uT}_{1}}/\sqrt{T_{1}},\ u \leq 1)$ under uniform random sampling, a study started in (Elie, Rosenbaum, and Yor, On the expectation of normalized Brownian functionals up to first hitting times, Preprint, arXiv:1310.1181, 2013).

A Appendix

1.1 A.1 A Simple Proof for the Joint Law of $(1/\sqrt{\tau _{1}},L_{U\tau _{1}})$

The fact that

$$\displaystyle{( \frac{1} {\sqrt{\tau _{1}}},L_{U\tau _{1}})\mathop{ =}\limits_{ \mathcal{L}}(L_{1},\varLambda )}$$

can obviously be deduced from Theorem 1. However, interestingly, we can give a simple proof for this equality in law. Indeed, for λ ≥ 0 and l < 1, we have

$$\displaystyle\begin{array}{rcl} \mathbb{E}[\text{e}^{-\lambda \tau _{1} }\mathrm{1}_{\{L_{U\tau _{ 1}}\leq l\}}]& =& \mathbb{E}[\frac{1} {\tau _{1}} \int _{0}^{\tau _{1} }\mathit{ds}\mathrm{1}_{\{L_{s}\leq l\}}\text{e}^{-\lambda \tau _{1} }] {}\\ & =& \mathbb{E}[ \frac{\tau _{l}} {\tau _{1}}\text{e}^{-\lambda \tau _{1} }]. {}\\ \end{array}$$

Now, consider in general (τ _l ) a subordinator and denote by ψ its Laplace exponent. Thus, we have

$$\displaystyle\begin{array}{rcl} \mathbb{E}[ \frac{\tau _{l}} {\tau _{1}}\text{e}^{-\lambda \tau _{1} }]& =& \mathbb{E}[\tau _{l}\int _{0}^{+\infty }\mathit{dt}\text{e}^{-(t+\lambda )\tau _{1} }] {}\\ & =& \int _{0}^{+\infty }\mathit{dt}\mathbb{E}[\tau _{ l}\text{e}^{-(t+\lambda )\tau _{l} }]\text{e}^{-(1-l)\psi (t+\lambda )}. {}\\ \end{array}$$

Using the fact that the Laplace exponent is differentiable on $\mathbb{R}^{+{\ast}}$, we get

$$\displaystyle\begin{array}{rcl} \mathbb{E}[ \frac{\tau _{l}} {\tau _{1}}\text{e}^{-\lambda \tau _{1} }]& =& \int _{0}^{+\infty }\mathit{dt}l\psi ^{{\prime}}(t+\lambda )\text{e}^{-l\psi (t+\lambda )}\text{e}^{-(1-l)\psi (t+\lambda )} {}\\ & =& l\text{e}^{-\psi (\lambda )}. {}\\ \end{array}$$

This proves the independence of $1/\sqrt{\tau _{1}}$ and $L_{U\tau _{1}}$ and the fact that $L_{U\tau _{1}}$ is uniformly distributed. The equality in law

$$\displaystyle{ \frac{1} {\sqrt{\tau _{1}}}\mathop{ =}\limits_{ \mathcal{L}}L_{1}}$$

is easily obtained by scaling.

1.2 A.2 On a One Parameter Family of Random Variables Including α

In this section of the appendix, we consider the family of variables defined for 0 < c ≤ 1 by

$$\displaystyle{\alpha _{c} =\varLambda L_{1} - c\vert B_{1}\vert,}$$

as an extension of our study of

$$\displaystyle{\alpha \mathop{=}\limits_{ \mathcal{L}}\alpha _{1/2}\mathop{ =}\limits_{ \mathcal{L}}B_{\mathit{UT}_{1}}/\sqrt{T_{1}}.}$$

The variables α _c , although less natural than α _1∕2, enjoy some similar remarkable properties. Indeed, Propositions 3 and 2 admit the following extensions.

Proposition 5

Let 0 < c ≤ 1 and $C = 1/c$ . Let Λ and U be two independent uniform variables on [0,1] and

$$\displaystyle{A_{c} =\varLambda U - c(1 - U).}$$

We have

$$\displaystyle\begin{array}{rcl} & & (A_{c}\vert A_{c} > 0)\mathop{ =}\limits_{ \mathcal{L}}\mathit{VZ}_{C} {}\\ & & (-A_{c}\vert A_{c} < 0)\mathop{ =}\limits_{ \mathcal{L}}\mathit{cV } {}\\ & & \mathbb{P}[A_{c} > 0] = 1 - c log (1 + C), {}\\ \end{array}$$

where V and Z _C are independent, with V uniform on [0,1] and Z _C a random variable with density given by

$$\displaystyle{ \frac{C} {1 - c log (1 + C)} \frac{\mathit{d\,zz}} {(1 + \mathit{C\,z})}\mathrm{1}_{\{0<z<1\}}.}$$

Proposition 6

Let 0 < c ≤ 1. The following equalities in law hold.

$$\displaystyle\begin{array}{rcl} & & (\alpha _{c}\vert \alpha _{c} > 0)\mathop{ =}\limits_{ \mathcal{L}}\vert N\vert Z_{C} {}\\ & & (-\alpha _{c}\vert \alpha _{c} < 0)\mathop{ =}\limits_{ \mathcal{L}}c\vert N\vert {}\\ & & \mathbb{P}[\alpha _{c} > 0] = 1 - c log (1 + C). {}\\ \end{array}$$

Proof

To establish Proposition 5, we simply compute the density of A _c . Proposition 6 ensues since

$$\displaystyle{\alpha _{c}\mathop{ =}\limits_{ \mathcal{L}}R_{1}A_{c}}$$

and

$$\displaystyle{R_{1}V \mathop{ =}\limits_{ \mathcal{L}}\vert N\vert,}$$

with the same notation as previously. □