Stochastic integral representation of a Brownian bridge
= Stochastic integral representation of a Brownian bridge
{title2=$X_t=(1-t)\int_0^t(1-s)^{-1}dB_s$}
For $t<1$ this <stochastic integral> solves $dX_t=dB_t-X_t(1-t)^{-1}dt$. The <Itô product rule> also gives $X_t=B_t-(1-t)\int_0^tB_s(1-s)^{-2}ds$. The process is centered <Gaussian>, with <covariance> $\min(s,t)-st$, and has a continuous extension $X_1=0$. <Time reversal of a Brownian bridge> proves endpoint continuity without assuming it at the start.