Parameter estimation for multiscale diffusions: an overview

There are many applications where it is desirable to fit reduced stochastic descriptions (e.g. SDEs) to data. These include molecular dynamics (Schlick (2000), Frenkel and Smit (2002)), atmosphere/ocean science (Majda and Kramer (1999)), cellular biology (Alberts et al. (2002)) and econometrics (Dacorogna, Gençay, Miiller, Olsen, and Pictet (2001)). The data arising in these problems often has a multiscale character and may not be compatible with the desired diffusion at small scales (see Givon, Kupferman, and Stuart (2004), Majda, Timofeyev, and Vanden-Eijnden (1999), Kepler and Elston (2001), Zhang, Mykland, and Aft-Sahalia (2005) and Olhede, Sykulski, and Pavliotis (2009)). The question then arises as to how to optimally employ such data to find a useful diffusion approximation.