Nonparametric Estimation for Stochastic Differential Equations Driven by Mixed Fractional Brownian Motion with Random Ef
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Nonparametric Estimation for Stochastic Differential Equations Driven by Mixed Fractional Brownian Motion with Random Effects B. L. S. Prakasa Rao CR RAO Advanced Institute of Mathematics, Statistics and Computer Science, Hyderabad, India
Abstract We discuss nonparametric estimation of the density of random effects in models governed by a stochastic differential equation driven by a mixed fractional Brownian motion. AMS (2000) subject classification. Primary 62G20; Secondary 60G22. Keywords and phrases. Stochastic differential equation, random effects, nonparametric estimation, Kernel method, mixed fractional Brownian motion.
1 Introduction Calyampudi Radhakrishna Rao (aka) C.R. Rao has entered his centenary year on September 10, 2019 and it is a great pleasure for me to be one of the contributors to the special issue of Sankhya dedicated to honour Professor C.R. Rao. Professor C.R. Rao has made breakthroughs by his research contributions in several branches of theory and applications of statistics, in particular, to statistical inference. My interest has been in the area of statistical inference for stochastic processes over the last forty years and, as a student of Professor C. R. Rao, I learned the branch of statistics called the “small sample theory”, now known as the “linear models” directly from him during my master’s program (M.Stat 1960-62) at the Indian Statistical Institute, Kolkata. I was his student and later his colleague at the Indian Statistical Institute, Delhi Centre during the years 1976-79 and was inspired by his research works in the theory of estimation in statistical inference and his works in the area of characterizations of probability distributions. Dedicated to my teacher Professor C.R. Rao during his centenary year
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B. L. S. P. Rao
Stochastic modeling by processes driven by a fractional Brownian motion (fBm) has been used for studying phenomena with long range dependence. Statistical inference for stochastic processes satisfying stochastic differential equations driven by a fractional Brownian motion has been studied earlier and a comprehensive survey of various methods is given in Mishura (2008) and in Prakasa Rao (2010). It was noted that fBm cannot be used as an adequate model in all areas of applications and more complex fractional processes are needed to model real phenomenon. The fBm characterized by a single parameter, the Hurst index, may not be useful as a good model when there are several levels of fractionality. A mixed model can be used as linear combination of different fractional Gaussian processes The simplest case is a mixed model based on the standard Brownian motion and the fractional Brownian motion. In a recent monograph “Stochastic Analysis of Mixed Fractional Gaussian Proceseses”, Y. Mishura and Mishura and Zili (2018), ISTE Press and Elsevier, London give a comprehensive survey of properties of such processes. It was observed that modeling of the financial markets by processes driven by fractional Brownian motion lead to arbitrage opportunities which is contrary to the
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