Asymmetric skew Bessel processes and their applications to finance |
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| Institution: | 1. K.U. Leuven, FETEW, Naamsestraat 69, B-3000 Leuven, Belgium;2. U.v. Amsterdam, FETEW, Naamsestraat 69, B-3000 Leuven, Belgium;3. K.U. Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leuven, Belgium;1. Katholieke Universiteit LeuvenDepartment of Mathematics andUniversity Statistics Centrede Croylaan 543001 LeuvenBELGIUMK;2. Katholieke Universiteit LeuvenUniversity Statistics Centrede Croylaan 543001 LeuvenBELGIUM;3. Katholieke Universiteit LeuvenDepartment of Mathematics andUniversity Statistics Centrede Croylaan 543001 LeuvenBELGIUM;4. Katholieke Universiteit LeuvenDepartment of MathematicsCelestijnenlaan 200B3001 LeuvenBELGIUM;5. Center for Statistics Hasselt University Agoralaan - building D 3590 Diepenbeek BELGIUM |
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| Abstract: | In this paper, we extend the Harrison and Shepp's construction of the skew Brownian motion (1981) and we obtain a diffusion similar to the two-dimensional Bessel process with speed and scale densities discontinuous at one point. Natural generalizations to multi-dimensional and fractional order Bessel processes are then discussed as well as invariance properties. We call this family of diffusions asymmetric skew Bessel processes in opposition to skew Bessel processes as defined in Barlow et al. On Walsh's Brownian motions, Séminaire de Probabilitiés XXIII, Lecture Notes in Mathematics, vol. 1372, Springer, Berlin, New York, 1989, pp. 275–293]. We present factorizations involving (asymmetric skew) Bessel processes with random time. Finally, applications to the valuation of perpetuities and Asian options are proposed. |
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