350 research outputs found
Localizing Volatilities
We propose two main applications of Gy\"{o}ngy (1986)'s construction of
inhomogeneous Markovian stochastic differential equations that mimick the
one-dimensional marginals of continuous It\^{o} processes. Firstly, we prove
Dupire (1994) and Derman and Kani (1994)'s result. We then present Bessel-based
stochastic volatility models in which this relation is used to compute
analytical formulas for the local volatility. Secondly, we use these mimicking
techniques to extend the well-known local volatility results to a stochastic
interest rates framework
The Matsumoto and Yor process and infinite dimensional hyperbolic space
The Matsumoto\,--Yor process is , where
is a Brownian motion. It is shown that it is the limit of the radial
part of the Brownian motion at the bottom of the spectrum on the hyperbolic
space of dimension , when tends to infinity. Analogous processes on
infinite series of non compact symmetric spaces and on regular trees are
described.Comment: 3
On Exceptional Times for generalized Fleming-Viot Processes with Mutations
If is a standard Fleming-Viot process with constant mutation rate
(in the infinitely many sites model) then it is well known that for each
the measure is purely atomic with infinitely many atoms. However,
Schmuland proved that there is a critical value for the mutation rate under
which almost surely there are exceptional times at which is a
finite sum of weighted Dirac masses. In the present work we discuss the
existence of such exceptional times for the generalized Fleming-Viot processes.
In the case of Beta-Fleming-Viot processes with index we
show that - irrespectively of the mutation rate and - the number of
atoms is almost surely always infinite. The proof combines a Pitman-Yor type
representation with a disintegration formula, Lamperti's transformation for
self-similar processes and covering results for Poisson point processes
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