The Brownian bridge is a stochastic process defined as a Brownian motion with
the condition that it comes back to the origin at time $t=1$. A Brownian bridge $X$ can be obtained from a Brownian motion using
$$X_t=B_t-tB_1.\tag1$$
This definition extends to any final time $T$ and any final point $Y$ in any dimension $d$ using the same definition
$$X^{d,Y}_t=B^{d}_t-\frac tT(B^d_T-Y),$$
where $B^d$ is a $d$ dimensional Brownian motion.
I was wondering if the probability density of $X$ was known in the literature and if so, if someone could provide me with a reference.
Best Answer
Edited to remove useless material
So silly of me, the answer is straightforward. Consider a Brownian bridge $X$ in $d$ dimensions, starting at $x=0$ at time $t=0$ and reaching its final point $R$ at time $t>0$. Then the caracteristic function is determined by the equation (1) in the Question : $$ \mathbb{E}[X_u]=\frac utR, \qquad \mathbb{E}[X_u^2]-\mathbb{E}[X_u]^2=\frac{u(t-u)}t.$$ It follows that the density probability of $X_u$ is a Gaussian distribution centered on $\frac utR$ with variance $\frac{u(t-u)}t$ or explicitely $$p_{X_u}(x)=\left(\frac{t}{2\pi u(t-u)}\right)^{d/2}\exp\left(-\frac{t\left(\frac utR-x\right)^2}{2u(t-u)}\right).$$