Explicit construction of a dynamic Bessel bridge of dimension 3. (English) Zbl 1290.60075

Campi, Luciano; Cetin, Umut; Danilova, Albina

Electron. J. Probab. 18, Paper No. 30, 25 p. (2013).

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 \(Z\).
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 [L. Campi et al., Finance Stoch. 17, No. 3, 565–585 (2013; Zbl 1270.91034)].

Reviewer: Heinrich Hering (Rockenberg)

MSC:

60J60	Diffusion processes
60G44	Martingales with continuous parameter
60G05	Foundations of stochastic processes
93E11	Filtering in stochastic control theory
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

dynamic Bessel bridge; enlargement of filtration; filtering; martingale problem; insider trading

Citations:

Zbl 1270.91034