We start by recalling some basic results on Brownian bridges and properties of conditional expectations that will be used in the sequel. The interested reader may find further discussions on Brownian bridges, among others, in the books [@key-1] of Karatzas and Shreve or [@key-3] of Revuz and Yor .

As usual, the set of natural numbers is denoted by $\mathbb{N}$ and the set of real numbers by $\mathbb{R}$. If $A\subseteq\mathbb{R}$, then the notation $A_{+}$ stands for $A_{+}:=A\cap\{ x\in\mathbb{R}:x\geq0\}$. If $E$ is a topological space, then the Borel $\sigma$-algebra over $E$ will be denoted by $\mathcal{B}(E)$. If $A$ is a set, its indicator function will be denoted by $\mathbb{I}_{A}$.

Let $(\Omega,\,\mathcal{F},\,\mathbf{P})$ be a complete probability space. By $\mathcal{N}_{P}$ we denote the collection of $\mathbf{P}$-null sets. If $X$ is a random variable, the symbol $\sigma(X)$ will denote the $\sigma$-algebra generated by $X$.

Let $W:(\Omega,\mathcal{F})\rightarrow(C,\mathcal{C})$ be a map from $\Omega$ into the space of continuous real-valued functions defined on $\mathbb{R}_{+}$ endowed with the $\sigma$-algebra generated by the canonical process, so that with each $\omega\in\Omega$ we associate the continuous function $W(\omega)=(W_{t}(\omega),\, t\geq0)$.

Given a strictly positive real number $r$, the function $\beta^{r}:\Omega\rightarrow C$ defined by $$\beta_{t}^{r}\left(\omega\right):=W_{t}\left(\omega\right)-\frac{t}{r\vee t}W_{r\vee t}\left(\omega\right), \ t\geq0, \ \omega\in\Omega\,,$$ is called 