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Estimating the parameters of stochastic differential equations

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  • Hurn, A.S.
  • Lindsay, K.A.

Abstract

Two maximum likelihood methods for estimating the parameters of stochastic differential equations (SDEs) from time-series data are proposed. The first is that of simulated maximum likelihood in which a nonparametric kernel is used to construct the transitional density of an SDE from a series of simulated trials. The second approach uses a spectral technique to solve the Kolmogorov equation satisfied by the transitional probability density. The exact likelihood function for a geometric random walk is used as a benchmark against which the performance of each method is measured. Both methods perform well with the spectral method returning results which are practically identical to those derived from the exact likelihood. The technique is illustrated by modelling interest rates in the UK gilts market using a fundamental one-factor term-structure equation for the instantaneous rate of interest.

Suggested Citation

  • Hurn, A.S. & Lindsay, K.A., 1999. "Estimating the parameters of stochastic differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 48(4), pages 373-384.
    • Handle: RePEc:eee:matcom:v:48:y:1999:i:4:p:373-384
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    References listed on IDEAS

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    1. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    2. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    3. Merton, Robert C., 1980. "On estimating the expected return on the market : An exploratory investigation," Journal of Financial Economics, Elsevier, vol. 8(4), pages 323-361, December.
    4. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(4), pages 627-627, November.
    5. Brennan, Michael J. & Schwartz, Eduardo S., 1979. "A continuous time approach to the pricing of bonds," Journal of Banking & Finance, Elsevier, vol. 3(2), pages 133-155, July.
    6. Michael J. Brennan and Eduardo S. Schwartz., 1979. "A Continuous-Time Approach to the Pricing of Bonds," Research Program in Finance Working Papers 85, University of California at Berkeley.
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    Cited by:

    1. Isambi Mbalawata & Simo Särkkä & Heikki Haario, 2013. "Parameter estimation in stochastic differential equations with Markov chain Monte Carlo and non-linear Kalman filtering," Computational Statistics, Springer, vol. 28(3), pages 1195-1223, June.
    2. A. Hurn & J. Jeisman & K. Lindsay, 2007. "Teaching an Old Dog New Tricks: Improved Estimation of the Parameters of Stochastic Differential Equations by Numerical Solution of the Fokker-Planck Equation," NCER Working Paper Series 9, National Centre for Econometric Research.
    3. Kathleen Goffey & Andrew Worthington, 2002. "Motor Vehicle Usage Patterns in Australia: A Comparative Analysis of Driver, Vehicle & Purpose Characteristics for Household & Freight Travel," School of Economics and Finance Discussion Papers and Working Papers Series 117, School of Economics and Finance, Queensland University of Technology.
    4. Stan Hurn & J.Jeisman & K.A. Lindsay, 2006. "Teaching an old dog new tricks: Improved estimation of the parameters of SDEs by numerical solution of the Fokker-Planck equation," Stan Hurn Discussion Papers 2006-01, School of Economics and Finance, Queensland University of Technology.
    5. Brandt, Michael W. & Santa-Clara, Pedro, 2002. "Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets," Journal of Financial Economics, Elsevier, vol. 63(2), pages 161-210, February.
    6. A. S. Hurn & J. I. Jeisman & K. A. Lindsay, 0. "Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations," Journal of Financial Econometrics, Oxford University Press, vol. 5(3), pages 390-455.
    7. Stan Hurn & J.Jeisman & K.A. Lindsay, 2006. "Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations. Working paper #2," NCER Working Paper Series 2, National Centre for Econometric Research.
    8. Tang, Sanyi & Heron, Elizabeth A., 2008. "Bayesian inference for a stochastic logistic model with switching points," Ecological Modelling, Elsevier, vol. 219(1), pages 153-169.

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