Abstract: In Neuenschwander [3] it was
shown that in asset price models suchas the Hull-White [1] one (logarithm of a stock price as sum ofBrownian
motionanditeratedBrownianmotionsthedistribution of the common process consisting of the asset price
and the Brownian motions constituting the iterated integral at a fixed time is a
sufficient statistic for the whole model within the framework of models with
sums of Lévy processes and iterated Lévy processes. (Note that this
distribution at a fixed time can, for example, be determined by observations of
i.i.d. models.) In the same paper, Neuenschwander [3], a similar result was also
shown for compound Poisson processes with determinate Lévy measure and without
Brownian part. In the present note, we will have a look at models with both
Brownian part and compound Poisson part. We will prove that the above-mentioned
result remains true if, in addition to the Brownian motions, also compound
Poisson processes the distributions of whose jumps have a tail decrease which is
at least as fast as that of a Gaussian distribution in the first two coordinates
and at least exponential in the third one are included additionally. Another
sufficient condition is that there is a direct sum decomposition of such that the probability laws of the underlying Lévy processes are given
by the product measures of a Brownian part on the one subspace and a compound
Poisson process with determinate Lévy measure on the other subspace. In the
general model (without any moment assumptions or other hypotheses), the law of
the process on any (arbitrary short) interval is a sufficient statistic for the
law of the whole process.
Keywords and phrases: compound Poisson process, Hull-White model, retrieval of law of process, Heisenberg group.
