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

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)].

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)

Citations:

Zbl 1270.91034
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