Transition density and likelihood approximations for diffusion processes: a gentle introduction using MATLAB
U. Picchini
back to Umberto Picchini's home
This page contains some material I prepared for the course "Statistical methods for diffusion models" (spring 2008, blok 4, prof. S. Ditlevsen),
Department of Mathematical Sciences, University of Copenhagen.
These lessons treat the problem of approximating the transition density of a diffusion process from discrete data when the anaytical expression
for the exact density is not available. There are several ways to do that: I consider Monte-Carlo methods based
on the numerical approximation of the SDE solution and the Y. Ait-Sahalia's method that approximates the density in closed-form, using the Hermitian expansion. Of course,
when an approximation to the transition density is obtained, it is possible to make inference on the parameters of the process using
maximum likelihood. In these lessons I consider both merits and drawbacks of the different methods, using
an applied-user-friendly approach based on MATLAB numerical experiments. No previous knowledge of MATLAB is required, since
the codes (see below) are heavily commented and html documentation is provided for each file (hint: if you want to modify the codes, you can easily produce the corresponding html documentation
using the MATLAB publishing capabilities). Finally, recent applications of the closed-form approximation method in the SDMEM framework
will be proposed (SDMEM = stochastic differential mixed-effects models, i.e. stochastic differential equations containing random parameters).
Slides and Codes
Lesson 1: transition densities and likelihood approximation via Monte-Carlo simulations and maximum likelihood estimation of stochastic differential equations from
discrete data; an application to the modelization of glucose-insulin dynamics is presented.
[Slides]; [Codes and html docs]
Lesson 2: closed-form approximation of transition densities.
[Slides]; [Codes and html docs]
Lesson 3: Mixed-models defined by stochastic differential equations.
Hints and References
During the lessons I often make reference to the articles below: take a look at them if you are interested in
theoretical details, and especially if you find that my exposition of the problems is not good enough! The links
point to the pdf preprints, when freely available.
WARNING: if you are puzzled by the analytical expressions that I used for the coefficients of the transition density expansion, as provided in the MATLAB codes,
that's because these coefficients are not the same coefficients given in Ait-Sahalia (1999,2002), which parametrize the expansion using a different notation;
in the codes I always follow (and adapt to the one-dimensional case) the more compact notation given in Ait-Sahalia (2008).
- Y. Ait-Sahalia (1999). Transition densities for interest rate and other nonlinear diffusions, Journal of Finance, 54, 1361-1395.
- Y. Ait-Sahalia (2002). Maximum-likelihood estimation of discretely-sampled diffusions: a closed-form approximation approach, Econometrica, 70, 223-262.
- Y. Ait-Sahalia (2008). Closed-form likelihood expansions for multivariate diffusions, The Annals of Statistics, 36, 906-937.
- M.W. Brandt and P. Santa-Clara (2002). Simulated likelihood estimation of diffusions with an application to exchange rate dynamics
in incomplete markets. Journal of Financial Economics 63, 161-210.
- A.R. Pedersen (1995). A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations.
Scand. J. Stat., 22, 55-71.
- A.R. Pedersen (2001). Likelihood inference by Monte Carlo methods for incompletely discretely observed diffusion processes. Department of Biostatistics,
University of Aarhus, Denmark.
- U. Picchini, S. Ditlevsen and A. De Gaetano (2006). Modeling the euglycemic hyperinsulinemic clamp by stochastic differential equations. Journal of Mathematical Biology, 53(5), 771-796.
The following is a primitive version of the methodology presented during the lessons: though, it should be sufficient for a first introduction to the modelization via SDMEMs.
- U. Picchini, A. De Gaetano and S. Ditlevsen (2006). Parameter estimation in stochastic differential mixed-effects
models. Research Report 06/12, Department of Biostatistics, University of Copenhagen.
back to Umberto Picchini's home