Fractional calculus and fractional processes with applications to financial economics. Theory and applications (Q2825434)

The book has two parts: Part I (including Chapters 1--5) and Part II (Chapters 6--8). Chapter 1 (Fractional calculus and fractional processes: an overview) discuss how fractional calculus and fractional processes are used in economics and finance: proceeds on defining fractional calculus and fractional processes, link between long memory, fractals, self-similarity, Brownian motion, fractional Brownian motion, continuous time random walk. Chapter 2 (Fractional calculus) brings the different definitions of fractional derivatives: Riemann-Liouville, Caputo, Grunwald-Letnikov and fractional derivatives based on the Fourier transform. It also discusses how to implement fractional derivatives in MatLab. Chapter 3 (Fractional Brownian motion) defines fractional Brownian motion and discusses its application in finance and, in particular, how it is not suitable for asset prices. Discusses long range dependency, self-similarity and arbitrage considerations. Chapter 4 (Fractional diffusion and heavy-tail distributions: stable distributions) establishes the connection between stable distributions and fractional calculus. The author obtains some analytical-numerical approximations for the density function of univariate and multivariate stable distributions, namely: homotopy perturbation method, Adomian decomposition method and variational iteration method. Chapter 5 (Fractional diffusion and heavy-tail distributions: geo-stable distributions) introduces geometric-stable (geo-stable) distributions as suitable alternatives for the normal distribution. These distributions also do not have closed forms for the density and distribution functions. It is shown how fractional calculus provide suitable tools for obtaining analytical-numerical approximations for these functions. Connections between multivariate geo-stable distributions and fractional PDEs are also presented. Chapter 6 (Fractional PDE and option pricing) makes a bridge between a SDE and ODE and PDE through forward and backward Kolmogorov equations and the Feynman-Kac formula. The chapter also discusses the details of option pricing with the Black-Scholes framework and the proposition of the classical tempered stable (or CGMY) process and its advantage in comparison with the geometric BM. Finally, the authors show an application of fractional calculus in option pricing by constructing a fractional PDE using the CGMY process. Chapter 7 (Continuous-time random walk and fractional calculus) discusses the continuous-time random walk and then moves on to its application in financial economics, namely modeling the behavior of the dynamics of high frequency asset prices and the rate of growth of a firm. Chapter 8 (Applications of fractional processes) discusses financial applications other than option pricing where fractional processes are used. In particular, the authors consider long memory processes, such as volatility, interest rates and order arrivals. To do this, fractionally integrated time series are introduced.