Parameter Inference for Stochastic Differential Equations with Density Tracking by Quadrature

  • Harish S. Bhat
  • R. W. M. A. Madushani
  • Shagun Rawat
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 231)


We derive and experimentally test an algorithm for maximum likelihood estimation of parameters in stochastic differential equations (SDEs). Our innovation is to efficiently compute the transition densities that form the log likelihood and its gradient, and to then couple these computations with quasi-Newton optimization methods to obtain maximum likelihood estimates. We compute transition densities by applying quadrature to the Chapman–Kolmogorov equation associated with a time discretization of the original SDE. To study the properties of our algorithm, we run a series of tests involving both linear and nonlinear SDE. We show that our algorithm is capable of accurate inference, and that its performance depends in a logical way on problem and algorithm parameters.


Stochastic differential equations Parameter inference Maximum likelihood estimation 



This work was partly supported by a grant from the Committee on Research at UC Merced.


  1. 1.
    Aït-Sahalia, Y.: Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica 70(1), 223–262 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bally, V., Talay, D.: The law of the Euler scheme for stochastic differential equations. II. Convergence rate of the density. Monte Carlo Methods Appl. 2(2), 93–128 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bhat, H.S., Madushani, R.W.M.A.: Density tracking by quadrature for stochastic differential equations (2016). Accessed 22 July 2017
  4. 4.
    Bhat, H.S., Madushani, R.W.M.A.: Nonparametric adjoint-based inference for stochastic differential equations. In: 2016 IEEE International Conference on Data Science and Advanced Analytics (DSAA), pp. 798–807 (2016)Google Scholar
  5. 5.
    Bhat, H.S., Madushani, R.W.M.A., Rawat, S.: Rdtq: density tracking by quadrature (2016). Accessed 22 July 2017 (R package version 0.1)
  6. 6.
    Bhat, H.S., Madushani, R.W.M.A., Rawat, S.: Scalable SDE filtering and inference with Apache Spark. Proc. Mach. Learn. Res. 53, 18–34 (2016)Google Scholar
  7. 7.
    Bhat, H.S., Madushani, R.W.M.A., Rawat, S.: Bayesian inference of stochastic pursuit models from basketball tracking data. In: Bayesian Statistics in Action: BAYSM 2016, Florence, Italy, June 19–21, pp. 127–137. Springer, Berlin (2017)Google Scholar
  8. 8.
    Bhattacharya, R.N., Waymire, E.C.: Stochastic Processes with Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2009)Google Scholar
  9. 9.
    Fuchs, C.: Inference for Diffusion Processes: With Applications in Life Sciences. Springer, Berlin (2013)CrossRefGoogle Scholar
  10. 10.
    Iacus, S.M.: Simulation and Inference for Stochastic Differential Equations: With R Examples. Springer Series in Statistics. Springer, New York (2009)Google Scholar
  11. 11.
    Johnson, S.G.: The NLopt nonlinear-optimization package (2016). Accessed 22 July 2017
  12. 12.
    Nocedal, J.: Updating quasi-Newton matrices with limited storage. Math. Comput. 35, 773–782 (1980)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Nocedal, J., Liu, D.C.: On the limited memory BFGS method for large scale optimization. Math. Program. 45(3), 503–528 (1989)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Pedersen, A.R.: A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Stat. 22(1), 55–71 (1995)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Santa-Clara, P.: Simulated likelihood estimation of diffusions with an application to the short term interest rate. Working Paper 12-97, UCLA Anderson School of Management (1997)Google Scholar
  16. 16.
    Sørensen, H.: Parametric inference for diffusion processes observed at discrete points in time: a survey. Int. Stat. Rev. 72(3), 337–354 (2004)CrossRefGoogle Scholar
  17. 17.
    Svanberg, K.: A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J. Optim. 12(2), 555–573 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ypma, J.: NLoptr: R interface to NLopt (2014). Accessed 22 July 2017

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of California MercedMercedUSA

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