350 research outputs found

    Localizing Volatilities

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    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

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    The Matsumoto\,--Yor process is _0texp(2B_sB_t)ds\int\_0^t \exp(2B\_s-B\_t)\, ds, where (B_t)(B\_t) 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 qq, when qq 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

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    If Y\mathbf Y is a standard Fleming-Viot process with constant mutation rate (in the infinitely many sites model) then it is well known that for each t>0t>0 the measure Yt\mathbf Y_t 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 Y\mathbf Y 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 α]1,2[\alpha\in\,]1,2[ we show that - irrespectively of the mutation rate and α\alpha - 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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