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Markovian bridges: Construction, Palm interpretation, and splicing. (English) Zbl 0844.60054

Cinlar, E. (ed.) et al., Seminar on stochastic processes, 1992. Held at the Univ. of Washington, DC, USA, March 26-28, 1992. Basel: Birkhäuser. Prog. Probab. 33, 101-134 (1992).
Take a standard Brownian bridge \(B\) from 0 to 0 on the unit interval, and an independent uniform random variable \(U\). Remove the excursion of \(B\) in which \(U\) falls, close up the gap, and rescale the path to have length 1, producing a process \(Y\). The law of this path is absolutely continuous with respect to the law of the Brownian bridge, and has density proportional to \(L_1\), where \(L\) is the local time process at zero of \(B\). This result is an application of the general results developed in this paper. The main material is concerned with formulating and proving rigorously various intuitively plausible integral identities, which would have to be true “under suitable conditions”. The suitable framework taken in this paper is that where one has a right process \(X\) on a Luzin state space \(E\), whose semigroup maps Borel functions to Borel functions, and whose paths are right continuous with left limits. One also needs to suppose the existence of transition densities with respect to some measure \(m\), and the existence of a right process \(\widehat {X}\) in duality with \(X\).
Now fix \(x, y \in E\), and \(l > 0\) such that \(p_l(x,y) > 0\). The splicing of two bridges is going to be achieved by taking a probability measure of the form \(\rho (z,t) \nu (dz)dt\) on \(E \times (0,l)\) (where \(z\) will be the place where splicing occurs, and \(t\) will be the time where splicing occurs), and then choosing \((Z,T)\) according to this law and stringing a bridge from \((x,0)\) to \((Z,T)\), followed by a bridge from \((Z,T)\) to \((y,l)\). The measure \(\nu\) must be the Revuz measure of a continuous additive functional \(A\). The main result (Proposition 4) establishes the law of the spliced process, by describing its density with respect to the law of the bridge from \((x,0)\) to \((y,l)\). This density is \(\int^l_0 f (X_t, t) dA_t\), where \(f(z,t) = \rho(z, t) p_l (x,y)/\{p_t (x,z) p_{l - t} (z,y)\}\).
For the entire collection see [Zbl 0780.00020].

MSC:

60J65 Brownian motion
60J60 Diffusion processes
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