Estimation of the local Hurst function of multifractional Brownian motion A second difference increment ratio estimator Simon Edvinsson Simon Edvinsson Vt 2015 Bachelor thesis, mathematics 15 hp Abstract In this thesis, a specific type of stochastic processes displaying time-dependent regularity is studied. Specifically, multifractional Brownian motion processes are examined. Due to their properties, these processes have gained interest in various fields of research. An important aspect when modeling using such processes are accurate estimates of the time-varying pointwise regularity. This thesis proposes a moving window ratio estimator using the distributional properties of the second difference increments of a discretized multifractional Brownian motion. The estimator captures the behaviour of the regularity on average. In an attempt to increase the accuracy of single trajectory pointwise estimates, a smoothing approach using nonlinear regression is employed. The proposed estimator is compared to an estimator based on the Increment Ratio Statistic. Sammanfattning I denna uppsats studeras en specifik typ av stokastiska processer, vilka uppvisar tidsberoende regel- bundenhet. Specifikt behandlas multifraktionella Brownianska r¨orelserd˚aderas egenskaper f¨oranlett ett ¨okat forskningsintresse inom flera f¨alt.Vid modellering med s˚adanaprocesser ¨arnoggranna estimat av den punktvisa, tidsberoende regelbundenheten viktig. Genom att anv¨andade distributionella egen- skaperna av andra ordningens inkrement i ett r¨orligtf¨onster,¨ar det m¨ojligtatt skatta den punktvisa regelbundenheten av en s˚adanprocess. Den f¨oreslagnaestimatorn uppn˚ari genomsnitt precisa resultat. Dock observeras h¨ogvarians i de punktvisa estimaten av enskilda trajektorier. Ickelinj¨arregression ap- pliceras i ett f¨ors¨okatt minska variansen i dessa estimat. Vidare presenteras ytterligare en estimator i utv¨arderingssyfte. Acknowledgements I would first of all like to thank my supervisor at Ume˚auniversity, Oleg Seleznjev for his unyielding support, expertise, and generous guidance. A special thanks to Yuliya Mishura at Kiev university for her indespensable advice, fruitful discussions, and kind encouragement. I would also like to thank the examinator Konrad Abramowicz, whose remarks and comments have significantly improved this thesis. Finally, I would like to thank Olof Lundgren for his time spent reading this thesis and supplying constructive criticism. Without his support, this thesis would never have been written. Contents 1 Introduction 5 1.1 Background . .5 1.2 Problem definition . .5 2 Fractional Brownian processes and local pointwise regularity 6 2.1 Stochastic processes . .6 2.2 Fractional Brownian motion . .6 2.3 Multifractional Brownian motion . .7 2.4 MBm process regularity estimation . .9 2.4.1 Approximate local behaviour . .9 2.4.2 Derivation of the Second Difference Ratio estimator . 11 2.4.3 Increment Ratio Statistic estimator . 12 2.4.4 Nonlinear regression . 13 3 Main results 14 3.1 Fractional Brownian motion evaluation . 14 3.2 Multifractional Brownian motion evaluation . 15 3.3 Smoothing using nonlinear regression . 18 4 Conclusion 21 5 Appendix 24 1 Introduction Multifractional Brownian motion (mBm) processes have become an area of interest in various fields of research, such as finance, internet traffic, image processing, and terrain modeling (see, Bianchi (2005); Ayache et al. (2000); Echelard et al. (2010)). The mBm is a special case of a process with variable smoothness as independently introduced by Peltier and V´ehel (1995) and Benassi et al. (1997). Described by Mandelbrot (1983), fractal geometry can be used to describe the irregularities observed in nature. This notion motivates the application of stochastic fractal processes to model various phenomena as observed in the real world. This thesis proposes a pointwise Second Difference Ratio (SDR) estimator for the estimation of the time-varying irregularity function of mBm processes. 1.1 Background A classic example of a stochastic fractal process is the fractional Brownian motion (fBm) first studied by Kolmogorov (1940) and later defined by Mandelbrot and Van Ness (1968). The fBm is characterized by the Hurst exponent, which describes the regularity of the process. This property makes the fBm suitable to model various processes in a parsimonious way, in the sense that their properties can be adjusted by varying a single parameter. In image processing, fractal processes have been used to generate images, classify textures, and calculate the length of coastlines (see, e.g., Fournier et al. (1982); Peleg et al. (1984); Mandelbrot (1975)). Furthermore, the property of long term dependence makes the fBm a more realistic model for modeling financial assets compared to classical models (see Mandelbrot and Van Ness (1968)). However, a limitation of the fBm is that the regularity of the process remains constant at all time points. Generalizing the fBm to incorporate time-varying pointwise regularity enables a more natural modeling environment. Multifractional Brownian motion (mBm) is a generalization of the fBm, introducing a time- varying Hurst function enabling the fractional process to dynamically change its pointwise regularity. The mBm process was independently introduced by Peltier and V´ehel(1995) and Ayache and Vehel (2000) and has since then attracted increased interest. The research regarding mBm processes is an area that has gained traction as an increasing number of fields of research are considering the possibility of time-varying regularity in processes. The mBm process, in the same sense as the fBm, provides a parsimonious modeling environment using a time-varying Hurst function, which enables a wide range of possible applications which makes it a interesting area of research. 1.2 Problem definition The starting point to mBm modeling is obtaining accurate estimates of the process time-varying regularity. There are several proposed estimators (see, e.g.,Benassi et al. (1998); Istas and Lang (1997)). A pointwise moving window approximation method using separate estimation of a scaling constant has been proposed by Bianchi et al. (2013). Another example of these estimators is the Increment Ratio Statistic (IRS) as developed by Surgailis et al. (2008) and applied to multifractional Brownian motion processes by Bardet and Surgailis (2013). The IRS estimator will be used for comparison purposes in this thesis. In this thesis the distributional properties of the second difference increments of mBm processes and the moving window approach of Bianchi et al. (2013) are combined to derive the SDR estimator, which belongs to the class of ratio estimators defined by Benassi et al. (1998). The reason for basing the estimator on the second difference increments is that it will be possible to estimate the Hurst function on its whole range. This would not be possible if the estimator was based on first difference increments (see Bardet and Surgailis (2011)). The idea of deriving the SDR estimator was originally proposed by professor Yuliya Mishura. When evaluating the SDR estimator, it can be concluded that it manages on average to estimate the Hurst function. The computed error measures for the SDR estimator and the IRS estimator are comparable. However, the single trajectory estimates are volatile, which motivates a smoothing approach in an attempt to increase the accuracy of the pointwise estimates. The application of nonlinear regression as a smoothing technique manages to reduce the error of the pointwise estimates when the Hurst function is specified as a power function. 5 This thesis is organized as follows. Firstly in Section 2, fractional Brownian motion and multifractional Brownian motion processes are defined. Secondly, local properties of a increment mBm process and distri- butional properties of the increments are presented. Lastly, the section is concluded with the definition of the SDR estimator as well as the introduction of the IRS estimator. In Section 3, the main results are pre- sented. Firstly, the results for estimation of the Hurst exponent are evaluated. Secondly, the estimators are evaluated for various Hurst functions. Lastly, the section is concluded with the results from the smoothing of the pointwise estimates using nonlinear regression. The thesis is concluded with a discussion of the results and future research. The appendix concludes with a complete collection of figures and tables that are not displayed in the main body of the thesis. 2 Fractional Brownian processes and local pointwise regularity In this section, the underlying theory needed for deriving the estimator is presented. With the properties of fractional Brownian motion (fBm) as an outset, a generalization to multifractional Brownian motion (mBm) with a time-varying Hurst function is made. In the following section, the distributional properties of the second difference increments of a discretized mBm process are shown. Finally, the SDR estimator is defined and presented as well as an estimator based on the Increment Ratio Statistic. 2.1 Stochastic processes With the definition of stochastic processes as an outset, some important properties are defined that will be of use in the proceeding sections of this thesis. Definition 1. A stochastic process X = fX(t; !); t 2 T;! 2 Ωg, is a set of random variables on a common probability space (Ω; F; P) indexed by a parameter t 2 T ⊂ R. Clearly, if a fixed t is considered fX(t; ·); t 2 T g will be a collection of random variables. Correspondingly, if a fixed ! is considered fX(·;!);! 2 Ωg is a realization of the stochastic process. For notational convenience we will use fX(t); t 2 T g = fX(t; !); t 2 T;! 2 Ωg when referring to a stochastic process. Definition 2. We say that a stochastic process fX(t); t 2 T g is a Gaussian process if for every n 2 N+ and every finite subset ft1; : : : ; tng of T , the random vector (X(t1);:::;X(tn)) is multivariate normally distributed. A property of Gaussian processes is that they are uniquely defined by their mean and covariance functions. Definition 3. We say that a stochastic process fX(t); t 2 T g is selfsimilar with index H > 0 if for any α > 0, fX(αt)g =d fαH X(t)g.