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The Riccati System and a Diffusion-Type Equation

Author

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  • Erwin Suazo

    (School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287–1804, USA
    Department of Mathematical Sciences, University of Puerto Rico, Mayagüez, Call Box 9000,Puerto Rico 00681–9018, USA)

  • Sergei K. Suslov

    (School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287–1804, USA)

  • José M. Vega-Guzmán

    (Department of Mathematics, Howard University, 223 Academic Support Building B, Washington,DC 20059, USA)

Abstract

We discuss a method of constructing solutions of the initial value problem for diffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered. Examples include, but are not limited to the Fokker-Planck equation in physics, the Black-Scholes equation and the Hull-White model in finance.

Suggested Citation

  • Erwin Suazo & Sergei K. Suslov & José M. Vega-Guzmán, 2014. "The Riccati System and a Diffusion-Type Equation," Mathematics, MDPI, vol. 2(2), pages 1-23, May.
    • Handle: RePEc:gam:jmathe:v:2:y:2014:i:2:p:96-118:d:36135
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    References listed on IDEAS

    as
    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    2. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    3. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    4. Merton, Robert C, 1976. "The Impact on Option Pricing of Specification Error in the Underlying Stock Price Returns," Journal of Finance, American Finance Association, vol. 31(2), pages 333-350, May.
    5. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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