### Miljenko Huzak, Snježana Lubura Strunjak and Andreja Vlahek Štrok

Department of Mathematics, Faculty of Science, University of Zagreb, 10 000 Zagreb, Croatia
e-mail:miljenko.huzak@math.hr

Department of Mathematics, Faculty of Science, University of Zagreb, 10 000 Zagreb, Croatia
e-mail:snjezana.lubura.strunjak@math.hr

Faculty of Chemical Engineering and Technology, University of Zagreb, 10 000 Zagreb, Croatia
e-mail:avlahek@fkit.hr

Abstract.   For a fixed $$T$$ and $$k \geq 2$$, a $$k$$-dimensional vector stochastic differential equation $$dX_t=\mu(X_t, \theta)\,dt+\nu(X_t)\,dW_t,$$ is studied over a time interval $$[0,T]$$. Vector of drift parameters $$\theta$$ is unknown. The dependence in $$\theta$$ is in general nonlinear. We prove that the difference between approximate maximum likelihood estimator of the drift parameter $$\overline{\theta}_n\equiv \overline{\theta}_{n,T}$$ obtained from discrete observations $$(X_{i\Delta_n}, 0 \leq i \leq n)$$ and maximum likelihood estimator $$\hat{\theta}\equiv \hat{\theta}_T$$ obtained from continuous observations $$(X_t, 0\leq t\leq T)$$, when $$\Delta_n=T/n$$ tends to zero, converges stably in law to the mixed normal random vector with covariance matrix that depends on $$\hat{\theta}$$ and on path $$(X_t, 0 \leq t\leq T)$$. The uniform ellipticity of diffusion matrix $$S(x)=\nu(x)\nu(x)^T$$ emerges as the main assumption on the diffusion coefficient function.

2020 Mathematics Subject Classification.   62M05, 62F12, 60J60

Key words and phrases.   Multidimensional diffusion processes, maximum likelihood estimation, uniform ellipticity, asymptotic mixed normality

Full text (PDF) (access from subscribing institutions only)

References:

1. Y. Aït-Sahalia, Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach, Econometrica 70 (2002), 223–262.
MathSciNet    CrossRef

2. Y. Aït-Sahalia, Closed-form likelihood expansions for multivariate diffusions, Ann. Statist. 36 (2008), 906–937.
MathSciNet    CrossRef

3. A. Alfonsi, B. Jourdain and A. Kohatsu-Higa, Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme, Electron. J. Probab. 20 (2015), no. 70, 31 pp.
MathSciNet    CrossRef

4. M. Barczy and G. Pap, Asymptotic properties of maximum-likelihood estimators for Heston models based on continuous time observations, Statistics 50 (2016), 389–417.
MathSciNet    CrossRef

5. I. V. Basawa and B. L. S. Prakasa Rao, Statistical inference for stochastic processes, Academic Press, London, 1980.
MathSciNet

6. P. Billingsley, Convergence of probability measures, John Wiley & Sons, New York, 1999.
MathSciNet    CrossRef

7. Y. G. Borisovich, N. M. Bliznyakov, T. N. Fomenko and Y. A. Izrailevich, Introduction to differential and algebraic topology, Kluwer Academic Publishers Group, Dordrecht, 1995.
MathSciNet    CrossRef

8. P. J. Brockwell and R. A. Davis, Time series: theory and methods, Springer, New York, 2006.
MathSciNet

9. J. Chang and S. X. Chen, On the approximate maximum likelihood estimation for diffusion processes, Ann. Statist. 39 (2011), 2820–2851.
MathSciNet    CrossRef

10. Q. Clairon and A. Samson, Optimal control for estimation in partially observed elliptic and hypoelliptic linear stochastic differential equations, Stat. Inference Stoch. Process. 23 (2020), 105–127.
MathSciNet    CrossRef

11. R. Durrett, Probability: theory and examples, Cambridge University Press, Cambridge, 2010.
MathSciNet    CrossRef

12. J. E. Gentle, Matrix algebra, Springer, New York, 2007.
MathSciNet    CrossRef

13. E. Gobet, Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach, Bernoulli 7 (2001), 899–912.
MathSciNet    CrossRef

14. E. Gobet, LAN property for ergodic diffusions with discrete observations, Ann. Inst. Henri Poincaré Probab. Stat. 38 (2002), 711–737.
MathSciNet    CrossRef

15. E. Gobet and R. Munos, Sensitivity analysis using Itô-Malliavin calculus and martingales, and application to stochastic optimal control, SIAM J. Control Optim. 43 (2005), 1676–1713.
MathSciNet    CrossRef

16. M. Huzak, A general theorem on approximate maximum likelihood estimation, Glas. Mat. Ser. III 36(56) (2001), 139–153.
MathSciNet

17. M. Huzak, Estimating a class of diffusions from discrete observations via approximate maximum likelihood method, Statistics  52 (2018), 239–272.
MathSciNet    CrossRef

18. J. Jacod, On continuous conditional Gaussian martingales and stable convergence in law, in Séminaire de Probabilitiés, Springer, Berlin, 1997, 232–246.
MathSciNet    CrossRef

19. J. Jacod and P. Protter, Discretization of processes, Springer, Heidelberg, 2012.
MathSciNet    CrossRef

20. R. Khasminskii, Stochastic stability of differential equations, Springer, Heidelberg, 2012.
MathSciNet    CrossRef

21. P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Springer-Verlag, Berlin, 1992.
MathSciNet    CrossRef

22. H. Lee and G. Trutnau, Existence and uniqueness of (infinitesimally) invariant measures for second order partial differential operators on Euclidean space, J. Math. Anal. Appl. 507 (2022), no. 125778, 31 pp.
MathSciNet    CrossRef

23. C. Li, Maximum-likelihood estimation for diffusion processes via closed-form density expansions, Ann. Statist. 41 (2013), 1350–1380.
MathSciNet    CrossRef

24. R. S. Liptser and A. N. Shiryaev, Statistics of random processes: General theory, Springer-Verlag, Berlin, 2001.
MathSciNet

25. S. Lubura Strunjak, Local asymptotic properties of approximate maximum likelihood estimator of drift parameters in diffusion model, Ph.D. thesis, University of Zagreb, 2015 (in Croatian).

26. S. Lubura Strunjak and M. Huzak, Local asymptotic mixed normality of approximate maximum likelihood estimator of drift parameters in diffusion model, Glas. Mat. Ser. III 52(72) (2017), 377–410.
MathSciNet    CrossRef

27. B. Øksendal, Stochastic differential equations, Springer-Verlag, Berlin, 2003.
MathSciNet    CrossRef

28. D. Revuz and M. Yor, Continuous martingales and Brownian motion, Springer-Verlag, Berlin, 1999.
MathSciNet    CrossRef

29. D. W. Stroock and S. R. S. Varadhan, Multidimensional diffusion processes, Springer-Verlag, Berlin, 2006.
MathSciNet

30. M. E. Taylor, Partial differential equations. I. Basic theory, Springer, New York, 2011.
MathSciNet    CrossRef

31. A. W. van der Vaart, Asymptotic statistics, Cambridge University Press, Cambridge, 1998.
MathSciNet    CrossRef

32. N. Yang, N. Chen and X. Wan, A new delta expansion for multivariate diffusions via the Itô-Taylor expansion, J. Econometrics, 209 (2019), 256–288.
MathSciNet    CrossRef