ON INFERENCE FOR FRACTIONAL DIFFERENTIAL EQUATIONS ALEXANDRA CHRONOPOULOU AND SAMY TINDEL Abstract. Based on Malliavin calculus tools and approximation results, we show how to compute a maximum likelihood type estimator for a rather general differential equa- tion 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. 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 pro- cesses is now a fairly well understood problem. In particular, during the last two decades, several advances have allowed to tackle the problem of inference based on discretely ob- served diffusions [10, 36, 40], which is of special practical interest. More specifically, consider a family of stochastic differential equations of the form Yt = a+ ∫ t 0 µ(Ys; ?) ds+ d∑ l=1 ∫ t 0 ?l(Ys; ?) dB l s, t ? [0, T ], (1) where a ? Rm, µ(·; ?) : Rm ? Rm and ?(·; ?) : Rm ? Rm,d
- invariant measure
- equations like
- situation can
- gaussian bounds
- mle methods used
- diffusion
- diffusion processes
- still hard
- mention just