Explicit construction of a dynamic Bessel bridge of dimension 3 (Q388880)

Let \(Z(t)= 1+\int^t_0 \sigma(s)\,dW(s)\) a deterministically time-changed Brownian motion with associated time change \(V(t)= \int^t_0\sigma^2\,ds\). Assuming that \(t< V(t)< \infty\) for all \(t>0\) and \(\exists\varepsilon> 0\,\int^\varepsilon_0(V(t)- t)^{-2}\,dt< \infty\), the authors explicitly construct a Brownian process \(X\) hitting \(0\) for the first time at \(V(\tau)\), \(\tau:= \{t> 0: Z(t)= 0\}\), and they give the semimartingale decomposition of \(X\) under the filtration jointly generated by \(X\) and \(Z\).NEWLINENEWLINE The hard part of the work is the construction up to time \(\tau\). It combines enlargement of filtration and filtering techniques. The motivation stems from an application in mathematical finance: insider trading with default risk where the insider observes the company value continuously in time, see [\textit{L. Campi} et al., Finance Stoch. 17, No. 3, 565--585 (2013; Zbl 1270.91034)].