A parallel between Brownian bridges and gamma bridges
Let \(\left\{ \gamma _{t},t\geq 0\right\} \) be the Gamma processes with natural filtration \(G\). Then \(\{a\gamma_u/ \gamma_t,u\in[0,t]\} \) is the Gamma bridge starting at zero and ending at \(a\) with the filtration \[ G_t^*\equiv \{\sigma (\gamma_u/ \gamma _t;\;u\in[0,t])\} . \] It is proved that the Gamma filtration \(G_{t}^{\ast }\) is generalized by the Gamma process \[ \gamma _{t}^{\ast }=\sum\limits_{s\in \left( 0,t\right] }u_{s}(\Delta \gamma _{s}/\gamma _{s-}) \] where \(u_{s}\left( x\right) \) is the function implicitly defined by \[ \int\limits_{u_{s}\left( x\right) }^{\infty }\frac{e^{-z}}{z} dz=\int\limits_{x}^{\infty }\frac{dy}{y\left( 1+y\right) ^{s}},\quad x>0,~s>0. \] The harness property, the independence of \(G_{t}^{\ast }\) of \(\sigma _{t}^{+}\equiv \sigma \left\{ \gamma _{u},u\in \left[ t,\infty \right) \right\} \) and the homogenity of increments without independence of the process \( \ln \{\gamma _{1}/t\), \(t>0\} \) are proved emphasizing their similiarity to that of the Brownian bridge.
scientific article; zbMATH DE number 2151098
scientific article