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Squared radial Ornstein-Uhlenbeck processes and inverse Laplace transforms of products of confluent hypergeometric functions

Year 2017, Volume: 46 Issue: 3, 409 - 417, 01.06.2017

Abstract

In this paper, we consider the squared radial Ornstein-Uhlenbeck process and associated Kolmogorov backward equation (the Laguerre heat
equation). For this process, we obtain the Green function of the Laplace transform of the transition density function in terms of the confluent hypergeometric functions and present new representations for the inverse Laplace transform of the products of confluent hypergeometric functions.

References

  • Abramowitz, M. and Stegun, I. A. Handbook of mathematical functions with formulas, graphs, and mathematical tables, 9th printing, Dover, New York, 1972.
  • Alili, L., Patie, P. and Pedersen, J. L. Representations of the first hitting time density of an Ornstein-Uhlenbeck process, Stochastic Models 21(4), 967-980, 2005.
  • Ansari, A. Remarks on the green function of space-fractional biharmonic heat equation using Ramanujan's master theorem, Kuwait Journal of Science Accepted, 2016.
  • Ansari, A. Fractional exponential operators and nonlinear partial fractional differential equations in the Weyl fractional derivatives, Applied Mathematics and Computation 220, 149-154, 2013.
  • Ansari, A., Refahi Sheikhani, A. and Saberi Naja, H. Solution to system of partial fractional differential equation using the fractional exponential operators, Mathematical Methods in the Applied Sciences 35, 119-123, 2012.
  • Ansari, A., Ahamadi Darani, M. and Moradi, M. On fractional Mittag-Leer operators, Reports on Mathematical Physics 70(1), 119-131, 2012.
  • Ansari, A. Fractional exponential operators and time-fractional telegraph equation, Boundary Value Problems 125, 2012.
  • Ansari, A., Refahi Sheikhani, A. and Kordrostami, S. On the generating function $e^{xt+y\phi(t)}$ and its fractional calculus, Central European Journal of Physics 11(10), 1457-1462, 2013.
  • Bennati, E., Rosa-Clot, M. and Taddei, S. A path integral approach to derivative security pricing: 1. Formalism and analytical results, International Journal of Theoretical and Applied Finance 2(4), 381-407, 1999.
  • Chandrasekhar, S. Dynamic friction. I. General considerations: The cofficient of dynamical friction, Astrophysical Journal 97, 255-262, 1943.
  • Dattoli, G., Srivastava, H.M. and Zhukovsky, K. Operational methods and differential equations with applications to initial value problems, Applied Mathematics and Computation 184, 979-1001, 2007.
  • Dattoli, G., Ottaviani, P.L. and Vazquez, L. Evolution operators equations: integration with algebric and nite difference methods. Applications to physical problems in classical and quantum mechanics and quantum eld theory, Rivista del Nuovo Cimento 20, 1-133, 1997.
  • Dattoli, G., Ricci, P.E. and Khomasuridze, I. Operational methods, special polynomial and functions and solution of partial differential equations, Integral Transforms and Special Functions 15, 309-321, 2004.
  • Dattoli, G., Ricci, P.E. and Sacchetti, D. Generalized shift operators and pseudo-polynomials of fractional order, Applied Mathematics and Computation 141, 215-224, 2003.
  • Gao, F. and Jiang, H. Moderate deviations for squared radial Ornstein-Uhlenbeck process, Statistics and Probability Letters 79, 1378-1386, 2009.
  • Gluss, B. A model for neuron ring with exponential decay of potential resulting in diffusion equations for probability density, Bulletin of Mathematical Biophysics 29, 233-243, 1967.
  • Going-Jaeschke, A. and Yor, M. A survey and some generalizations of Bessel processes, Bernoulli 9(2), 313-349, 2003.
  • Graczyk, P. and Jakubowski, T. Exit time and poisson kernels of the Ornstein-Uhlenbeck diffusion, Stochastic Models 24, 314-337, 2008.
  • Jun, Y. Quasi-Stationary distributions for the radial Ornstein-Uhlenbeck processes, Acta Mathematica Scientia 28B(3), 513-522, 2008.
  • Lamberton, D. and Lapeyre, B. Introduction au calcul stochastique appliqué à la nance, Model Assisted Statistics and Applications, Ellipses:, Paris, 1997.
  • Leonenkoa, N.N., Meerschaert, M.M. and Sikorskii, A. Fractional Pearson diffusions, Jour- nal of Mathematical Analysis and Applications 403, 532-546, 2013.
  • Melnikov, Y.A. and Melnikov, M.Y. Green's functions construction and applications, Model Assisted Statistics and Applications, De Gruyter, Berlin, 2012.
  • Moslehi, L. and Ansari, A. On M-Wright transforms and time-fractional diffusion equations, Integral Transforms and Special Functions 28(2) 113-129, 2017.
  • Olver, F.W.J., Lozier, D.W., Boisvert, R.F. and Clark, C.W. Nist handbook of mathematical functions, Model Assisted Statistics and Applications, National Institute of Standards and Technology, United States, 2010.
  • Patie, P. and Yor, M. On some first passage time problems motivated by financial applications, Doctoral thesis ETH, No. 15834, Zurich, 2004.
  • Vasicek, O. An equilibrium characterization of the term structure, Journal of Financial Economics 5, 177-188, 1977.
  • Veestraeten, D. On the inverse transform of Laplace transforms that contain (products of) the parabolic cylinder function, Integral Transforms and Special Functions 26(11), 859-871, 2015.
  • Veestraeten, D. Some integral representations and limits for (products of) the parabolic cylinder function, Integral Transforms and Special Functions 27(1), 64-77, 2016.
  • Zani, M. Large deviations for squared radial Ornstein-Uhlenbeck processes, Stochastic Pro- cesses and their Applications 102, 25-42, 2002.
Year 2017, Volume: 46 Issue: 3, 409 - 417, 01.06.2017

Abstract

References

  • Abramowitz, M. and Stegun, I. A. Handbook of mathematical functions with formulas, graphs, and mathematical tables, 9th printing, Dover, New York, 1972.
  • Alili, L., Patie, P. and Pedersen, J. L. Representations of the first hitting time density of an Ornstein-Uhlenbeck process, Stochastic Models 21(4), 967-980, 2005.
  • Ansari, A. Remarks on the green function of space-fractional biharmonic heat equation using Ramanujan's master theorem, Kuwait Journal of Science Accepted, 2016.
  • Ansari, A. Fractional exponential operators and nonlinear partial fractional differential equations in the Weyl fractional derivatives, Applied Mathematics and Computation 220, 149-154, 2013.
  • Ansari, A., Refahi Sheikhani, A. and Saberi Naja, H. Solution to system of partial fractional differential equation using the fractional exponential operators, Mathematical Methods in the Applied Sciences 35, 119-123, 2012.
  • Ansari, A., Ahamadi Darani, M. and Moradi, M. On fractional Mittag-Leer operators, Reports on Mathematical Physics 70(1), 119-131, 2012.
  • Ansari, A. Fractional exponential operators and time-fractional telegraph equation, Boundary Value Problems 125, 2012.
  • Ansari, A., Refahi Sheikhani, A. and Kordrostami, S. On the generating function $e^{xt+y\phi(t)}$ and its fractional calculus, Central European Journal of Physics 11(10), 1457-1462, 2013.
  • Bennati, E., Rosa-Clot, M. and Taddei, S. A path integral approach to derivative security pricing: 1. Formalism and analytical results, International Journal of Theoretical and Applied Finance 2(4), 381-407, 1999.
  • Chandrasekhar, S. Dynamic friction. I. General considerations: The cofficient of dynamical friction, Astrophysical Journal 97, 255-262, 1943.
  • Dattoli, G., Srivastava, H.M. and Zhukovsky, K. Operational methods and differential equations with applications to initial value problems, Applied Mathematics and Computation 184, 979-1001, 2007.
  • Dattoli, G., Ottaviani, P.L. and Vazquez, L. Evolution operators equations: integration with algebric and nite difference methods. Applications to physical problems in classical and quantum mechanics and quantum eld theory, Rivista del Nuovo Cimento 20, 1-133, 1997.
  • Dattoli, G., Ricci, P.E. and Khomasuridze, I. Operational methods, special polynomial and functions and solution of partial differential equations, Integral Transforms and Special Functions 15, 309-321, 2004.
  • Dattoli, G., Ricci, P.E. and Sacchetti, D. Generalized shift operators and pseudo-polynomials of fractional order, Applied Mathematics and Computation 141, 215-224, 2003.
  • Gao, F. and Jiang, H. Moderate deviations for squared radial Ornstein-Uhlenbeck process, Statistics and Probability Letters 79, 1378-1386, 2009.
  • Gluss, B. A model for neuron ring with exponential decay of potential resulting in diffusion equations for probability density, Bulletin of Mathematical Biophysics 29, 233-243, 1967.
  • Going-Jaeschke, A. and Yor, M. A survey and some generalizations of Bessel processes, Bernoulli 9(2), 313-349, 2003.
  • Graczyk, P. and Jakubowski, T. Exit time and poisson kernels of the Ornstein-Uhlenbeck diffusion, Stochastic Models 24, 314-337, 2008.
  • Jun, Y. Quasi-Stationary distributions for the radial Ornstein-Uhlenbeck processes, Acta Mathematica Scientia 28B(3), 513-522, 2008.
  • Lamberton, D. and Lapeyre, B. Introduction au calcul stochastique appliqué à la nance, Model Assisted Statistics and Applications, Ellipses:, Paris, 1997.
  • Leonenkoa, N.N., Meerschaert, M.M. and Sikorskii, A. Fractional Pearson diffusions, Jour- nal of Mathematical Analysis and Applications 403, 532-546, 2013.
  • Melnikov, Y.A. and Melnikov, M.Y. Green's functions construction and applications, Model Assisted Statistics and Applications, De Gruyter, Berlin, 2012.
  • Moslehi, L. and Ansari, A. On M-Wright transforms and time-fractional diffusion equations, Integral Transforms and Special Functions 28(2) 113-129, 2017.
  • Olver, F.W.J., Lozier, D.W., Boisvert, R.F. and Clark, C.W. Nist handbook of mathematical functions, Model Assisted Statistics and Applications, National Institute of Standards and Technology, United States, 2010.
  • Patie, P. and Yor, M. On some first passage time problems motivated by financial applications, Doctoral thesis ETH, No. 15834, Zurich, 2004.
  • Vasicek, O. An equilibrium characterization of the term structure, Journal of Financial Economics 5, 177-188, 1977.
  • Veestraeten, D. On the inverse transform of Laplace transforms that contain (products of) the parabolic cylinder function, Integral Transforms and Special Functions 26(11), 859-871, 2015.
  • Veestraeten, D. Some integral representations and limits for (products of) the parabolic cylinder function, Integral Transforms and Special Functions 27(1), 64-77, 2016.
  • Zani, M. Large deviations for squared radial Ornstein-Uhlenbeck processes, Stochastic Pro- cesses and their Applications 102, 25-42, 2002.
There are 29 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Leila Moslehi This is me

Alireza Ansari

Publication Date June 1, 2017
Published in Issue Year 2017 Volume: 46 Issue: 3

Cite

APA Moslehi, L., & Ansari, A. (2017). Squared radial Ornstein-Uhlenbeck processes and inverse Laplace transforms of products of confluent hypergeometric functions. Hacettepe Journal of Mathematics and Statistics, 46(3), 409-417.
AMA Moslehi L, Ansari A. Squared radial Ornstein-Uhlenbeck processes and inverse Laplace transforms of products of confluent hypergeometric functions. Hacettepe Journal of Mathematics and Statistics. June 2017;46(3):409-417.
Chicago Moslehi, Leila, and Alireza Ansari. “Squared Radial Ornstein-Uhlenbeck Processes and Inverse Laplace Transforms of Products of confluent Hypergeometric Functions”. Hacettepe Journal of Mathematics and Statistics 46, no. 3 (June 2017): 409-17.
EndNote Moslehi L, Ansari A (June 1, 2017) Squared radial Ornstein-Uhlenbeck processes and inverse Laplace transforms of products of confluent hypergeometric functions. Hacettepe Journal of Mathematics and Statistics 46 3 409–417.
IEEE L. Moslehi and A. Ansari, “Squared radial Ornstein-Uhlenbeck processes and inverse Laplace transforms of products of confluent hypergeometric functions”, Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 3, pp. 409–417, 2017.
ISNAD Moslehi, Leila - Ansari, Alireza. “Squared Radial Ornstein-Uhlenbeck Processes and Inverse Laplace Transforms of Products of confluent Hypergeometric Functions”. Hacettepe Journal of Mathematics and Statistics 46/3 (June 2017), 409-417.
JAMA Moslehi L, Ansari A. Squared radial Ornstein-Uhlenbeck processes and inverse Laplace transforms of products of confluent hypergeometric functions. Hacettepe Journal of Mathematics and Statistics. 2017;46:409–417.
MLA Moslehi, Leila and Alireza Ansari. “Squared Radial Ornstein-Uhlenbeck Processes and Inverse Laplace Transforms of Products of confluent Hypergeometric Functions”. Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 3, 2017, pp. 409-17.
Vancouver Moslehi L, Ansari A. Squared radial Ornstein-Uhlenbeck processes and inverse Laplace transforms of products of confluent hypergeometric functions. Hacettepe Journal of Mathematics and Statistics. 2017;46(3):409-17.