Stochastic filtering theory studies the problem of estimating an unobservable 'signal' process <I>X</I> given the information obtained by observing an associated process <I>Y</I> (a 'noisy' observation) within a certain time window [0, <I>t</I>]. It is possible to explicitly describe the distribution of <I>X</I> given <I>Y</I> in the setting of linear/guassain systems. Outside the realm of the linear theory, it is known that only a few very exceptional examples have explicitly described posterior distributions. We present in detail a class of nonlinear filters (Bene filters) which allow explicit formulae. Using the explicit expression of the Laplace transform of a functional of Brownian motion we give a direct computation of the unnormalized conditional density of the signal of the Bene filter and obtain the formula for the normalized conditional density of <I>X</I> for two particular filters. In the case in which <I>X </I>is a diffusion process and <I>Y</I> is given by the equation <I>dY<SUB>t</SUB> </I>=<I> dh</I>(<I>s</I>,<I>X<SUB>s</SUB></I>)<I>ds </I>+ <I>dW<SUB>t</SUB></I>, where <I>W</I> is a Brownian motion independent of <I>X, Y</I><SUB>0</SUB> = 0 and <I>h </I>satisfies certain conditions, the evolution of the conditional distribution of <I>X</I> is described by 2 stochastic partial differential equations: a linear equation - the <I>Zakai</I> equation - which describes the evolution of an unnormalised version of the condition distribution of <I>X</I> and a nonlinear equation - the <I>Kushner</I> - <I>Stratonovitch </I>equation - which describes the evolution of the conditional distribution of <I>X</I> itself. We construct several measure valued processes, associated with the two equations, whose values give the conditional distribution of <I>X</I> (in the first case unnormalised). We do this by means of converging sequences of branching particle systems. The particles evolve independently, moving with the same law as <I>X</I>, and branch according to a mechanism that depends on the their locations and the observation <I>Y</I>. The result is a cloud of paths, with those surviving to the current time providing an estimate for the conditional distribution of <I>X</I>.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:645151 |
| Date | January 1996 |
| Creators | Crisan, Dan Ovidiu |
| Publisher | University of Edinburgh |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/1842/13489 |
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