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On Explicit Probability Laws for Classes of Scalar Diffusions

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Abstract

This paper uses Lie symmetry group methods to obtain transition probability densities for scalar diffusions, where the diffusion coefficient is given by a power law. We will show that if the drift of the diffusion satisfies a certain family of Riccati equations, then it is possible to compute a generalized Laplace transform of the transition density for the process. Various explicit examples are provided. We also obtain fundamental solutions of the Kolmogorov forward equation for diffusions, which do not correspond to transition probability densities.

Suggested Citation

  • Mark Craddock & Eckhard Platen, 2009. "On Explicit Probability Laws for Classes of Scalar Diffusions," Research Paper Series 246, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:246
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    File URL: https://www.uts.edu.au/sites/default/files/qfr-archive-02/QFR-rp246.pdf
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    1. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151, January.
    2. Hardy Hulley & Eckhard Platen, 2007. "Laplace Transform Identities for Diffusions, with Applications to Rebates and Barrier Options," Research Paper Series 203, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Mark Craddock & Eckhard Platen, 2003. "Symmetry Group Methods for Fundamental Solutions and Characteristic Functions," Research Paper Series 90, Quantitative Finance Research Centre, University of Technology, Sydney.
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    Cited by:

    1. Mark Craddock, 2017. "Integral Transform and Lie Symmetry Methods for Scalar and Multi-Dimensional Diffusions," Research Paper Series 380, Quantitative Finance Research Centre, University of Technology, Sydney.

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    Keywords

    Lie symmetry groups; fundamental solutions; transition probability densities; Ito diffusions;
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