On inference for fractional differential equations
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On inference for fractional differential equations Alexandra Chronopoulou · Samy Tindel
Received: 13 December 2011 / Accepted: 4 October 2012 / Published online: 2 March 2013 © Springer Science+Business Media Dordrecht 2013
Abstract Based on Malliavin calculus tools and approximation results, we show how to compute a maximum likelihood type estimator for a rather general differential equation driven by a fractional Brownian motion with Hurst parameter H > 1/2. Rates of convergence for the approximation task are provided, and numerical experiments show that our procedure leads to good results in terms of estimation. Keywords Fractional brownian motion · Stochastic differential equations · Malliavin calculus · Inference for stochastic processes Mathematics Subject Classification (2000) 62M09
60H35 · MSC 60H07 · 60H10 · 65C30 ·
1 Introduction In this introduction, we first try to motivate our problem and outline our results. We also argue that only a part of the question can be dealt with in a single paper. We briefly sketch a possible program for the remaining tasks in a second part of the introduction. 1.1 Motivations and outline of the results The inference problem for diffusion processes is now a fairly well understood problem. In particular, during the last two decades, several advances have allowed to tackle the problem A. Chronopoulou (B) · S. Tindel Institut Élie Cartan Nancy, B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France e-mail: [email protected] Present address: A. Chronopoulou Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106-3110, USA S. Tindel e-mail: [email protected]
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Stat Inference Stoch Process (2013) 16:29–61
of inference based on discretely observed diffusions (Durham and Gallant 2002; Pedersen 1995; Sorensen 2009), which is of special practical interest. More specifically, consider a family of stochastic differential equations of the form t d t Yt = a + μ(Ys ; θ ) ds + σ l (Ys ; θ ) d Bsl , t ∈ [0, T ], (1) 0
l=1
0
where a ∈ Rm , μ(·; θ ) : Rm → Rm and σ (·; θ ) : Rm → Rm,d are smooth enough functions, B is a d-dimensional Brownian motion with Hurst parameter H > 1/2 (the stochastic integral in (1) being understood in the Young sense) and θ is a parameter varying in a subset ⊂ Rq . If one wishes to identify θ from a set of discrete observations of Y , most of the methods which can be found in the literature are based on (or are closely linked to) the maximum likelihood principle. Indeed, if B is a Brownian motion and Y is observed at some equally distant instants ti = iτ for i = 0, . . . , n, then the log-likelihood of a sample (Yt1 , . . . , Ytn ) can be expressed as n (θ ) =
n
ln p τ, Yti−1 , Yti ; θ ,
(2)
i=1
where p stands for the transition semi-group of the diffusion Y . If Y enjoys some ergodic properties, with invariant measure νθ0 under Pθ0 , then we get 1 n (θ ) = Eθ0 [ p (τ, Z 1 , Z 2 ; θ )] Jθ0 (θ ), (3) n where Z 1 ∼ νθ0 and L(Z 2 | Z 1 ) = p(τ, Z 1 , · ; θ ). Furthermore, it
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