We first study the drift parameter estimation of the fractional Ornstein-Uhlenbeck process (fOU) with periodic mean for every 1 2 < H < 1. More precisely, we extend the consistency proved in [6] for 1 2 < H < 3 4 to the strong consistency for any 1 2 < H < 1 on the one hand, and on the other, we also discuss the asymptotic normality given in [6]. In the second main part of the paper, we study the strong consistency and the asymptotic normality of the fOU of the second kind with periodic mean for any 1 2 < H < 1.

In this article we study the asymptotic behaviour of the realized quadratic variation of a process t 0 u s dY (1) s , where u is a β-Hölder continuous process with β > 1−H and Y (1) t = t 0 e −s dB H as , where a t = He t H and B H is a fractional Brownian motion, is connected to the fractional Ornstein-Uhlenbeck process of the second kind. We prove almost sure convergence uniformly in time, and a stable weak convergence for the realized quadratic variation. As an application, we construct strongly consistent estimator for the integrated volatility parameter in a model driven by Y (1) . t 0 u s dB H s with respect to a fractional Brownian motion was studied in [8]. In [8] the authors proved 1 that if u = {u t , t 0} has finite q-variation for some q < 1/(1 − H), then n −1+pH V n p (Z) t −→ c 1,p t 0

With the rapid urbanization occurring across African cities, the emergence of a middle class is exerting its influence on the urban form and structure. Matching their social status and drawing on global influences, the housing characteristics of this class are distinctive in spatial organization, material choice, and location, among others. Understanding these emerging typologies is critical for urban housing policy to be responsive to the needs and preferences of this class. The present paper aims first to develop a new approach for defining the middle class in the African context and second, to analyze its housing typology through a multidimensional analysis based on housing attributes and socio-economic characteristics in Casablanca City. A data-driven approach based on principal component analysis (PCA) has been used to define multidimensionally the middle class and its housing typology. Through the construction of a multidimensional composite index to measure middle class, the study highlighted that middle-class housing is characterized by an adequate condition, a suitable size (two to three rooms), and affordability between 8000 and 11,000 Moroccan dirhams (MAD) per square meter. In addition, although the Moroccan modern houses (two-floor single-family housing) and apartments (four floors or greater) are the most occupied by the middle class, the study showed that as income and social mobility increase, the apartment building and the villa are increasingly preferred to the Moroccan modern house.

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