The Annals of Applied Probability

Pathwise stability of likelihood estimators for diffusions via rough paths

Joscha Diehl, Peter Friz, and Hilmar Mai

Full-text: Open access


We consider the classical estimation problem of an unknown drift parameter within classes of nondegenerate diffusion processes. Using rough path theory (in the sense of T. Lyons), we analyze the Maximum Likelihood Estimator (MLE) with regard to its pathwise stability properties as well as robustness toward misspecification in volatility and even the very nature of the noise. Two numerical examples demonstrate the practical relevance of our results.

Article information

Ann. Appl. Probab., Volume 26, Number 4 (2016), 2169-2192.

Received: March 2015
First available in Project Euclid: 1 September 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M05: Markov processes: estimation 62F99: None of the above, but in this section
Secondary: 60H99: None of the above, but in this section

Maximum likelihood estimation robust estimation rough paths analysis


Diehl, Joscha; Friz, Peter; Mai, Hilmar. Pathwise stability of likelihood estimators for diffusions via rough paths. Ann. Appl. Probab. 26 (2016), no. 4, 2169--2192. doi:10.1214/15-AAP1143.

Export citation


  • [1] Avellaneda, M., Levy, A. and Parás, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2 73–88.
  • [2] Bailleul, I. and Diehl, J. (2014). Recovering a signal. Preprint. Available at arXiv:1407.2768.
  • [3] Caramellino, L. (1998). Strassen’s law of the iterated logarithm for diffusion processes for small time. Stochastic Process. Appl. 74 1–19.
  • [4] Comte, F., Coutin, L. and Renault, E. (2012). Affine fractional stochastic volatility models. Ann. Finance 8 337–378.
  • [5] Decreusefond, L. and Üstünel, A. S. (1999). Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 177–214.
  • [6] Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett Publishers, Boston, MA.
  • [7] Friz, P., Gassiat, P. and Lyons, T. (2015). Physical Brownian motion in a magnetic field as a rough path. Trans. Amer. Math. Soc. 367 7939–7955.
  • [8] Friz, P. K. and Hairer, M. (2014). A Course on Rough Paths: With an Introduction to Regularity Structures. Universitext. Springer, Cham.
  • [9] Friz, P. K. and Victoir, N. B. (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge Studies in Advanced Mathematics 120. Cambridge Univ. Press, Cambridge.
  • [10] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. Elsevier, Amsterdam.
  • [11] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [12] Kutoyants, Y. A. (2004). Statistical Inference for Ergodic Diffusion Processes. Springer Series in Statistics. Springer, London.
  • [13] Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes, Vol. I, II. Springer, Berlin.
  • [14] Lyons, T. (1991). On the nonexistence of path integrals. Proc. Roy. Soc. London Ser. A 432 281–290.
  • [15] Lyons, T. (1995). Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2 117–133.
  • [16] Lyons, T. and Qian, Z. (2002). System Control and Rough Paths. Oxford Mathematical Monographs. Oxford Univ. Press, Oxford.
  • [17] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 215–310.
  • [18] Lyons, T. J., Caruana, M. and Lévy, T. (2007). Differential Equations Driven by Rough Paths. Lecture Notes in Math. 1908. Springer, Berlin.
  • [19] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422–437.
  • [20] Papavasiliou, A. and Ladroue, C. (2011). Parameter estimation for rough differential equations. Ann. Statist. 39 2047–2073.
  • [21] Pavliotis, G. A. and Stuart, A. M. (2008). Multiscale Methods: Averaging and Homogenization. Texts in Applied Mathematics 53. Springer, New York.
  • [22] Schwartz, E. S. (1997). The stochastic behavior of commodity prices: Implications for valuation and hedging. J. Finance 52 923–973.
  • [23] Shao, Y. (1995). The fractional Ornstein–Uhlenbeck process as a representation of homogeneous Eulerian velocity turbulence. Phys. D 83 461–477.
  • [24] Vasicek, O. (1977). An equilibrium characterisation of the term structure. Journal of Financial Economics 5 177–188.