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Integral Transform and Lie Symmetry Methods for Scalar and Multi-Dimensional Diffusions

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  • Mark Craddock

    (School of Mathematical and Physical Sciences, University of Technology Sydney)

Abstract

In this paper we demonstrate the various ways in which Lie symmetry group and integral transform methods can be combined and applied to solve some important types of problems in the theory of diffusion processes. We may compute various kinds of transition probability densities, as well as densities for diffusions which are conditioned to be either reflecting or absorbed at some boundary. Reflection and absorption for squared Bessel processes on the line x = a is studied. We also show how various first hitting times may be computed. We study some higher dimensional diffusions and show how transform methods can be used to extend some one dimensional results to higher dimensions. We also produce a general formula for the sums of certain one dimensional processes. Finally, we introduce what seems to be a new class of processes which have nearly exact densities.

Suggested Citation

  • 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.
    • Handle: RePEc:uts:rpaper:380
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    File URL: https://www.uts.edu.au/sites/default/files/QFR-2017-rp380.pdf
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. Mark Craddock & Martino Grasselli, 2016. "Lie Symmetry Methods for Local Volatility Models," Research Paper Series 377, Quantitative Finance Research Centre, University of Technology, Sydney.
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    Keywords

    Lie symmetry groups; fundamental solutions; transition densities;
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