1st Edition
Poisson Process and its Fractional Extensions with Applications
Part 1: Poisson processes
1. Poisson processes
1.1 The homogeneous Poisson process
1.2 The weighted Poisson process of order i
1.3 Skellam process: the difference between two Poisson processes
1.4 The time non-homogeneous Poisson process
1.5 Compound Poisson processes (CPP)
1.6 Poisson random fields
1.7 The integral of an homogeneous Poisson process
1.8 Iterated Poisson: composition of two independent Poisson processes
1.9 Shot noise and Hawkes processes
Part 2: Birth-and-Death processes
2. Birth processes
2.1 The nonlinear case
2.2 The linear case (Yule-Furry)
2.3 Estimation of the pure-birth process
3. Death processes
3.1 The nonlinear case
3.2 The linear case
3.3 The sublinear case
3.4 Estimation of the pure-death process
4. The birth-death process
4.1 The nonlinear case
4.2 The linear case
4.3 The extinction probability and the Riccati equation
4.4 Galton-Watson branching processes
4.5 Estimation of the birth-death process
Part 3: Fractional Poisson processes
5. Fractional extensions of the Poisson and birth and death processes
5.1 The space-fractional Poisson process: standard case
5.2 Generalized space-fractional Poisson processes
5.3 Another fractional generalization of the Poisson process
5.4 The time-fractional Poisson process
5.5 The space-time fractional Poisson process
5.6 Fractional birth-and-death processes
Biography
Enzo Orsingher is Emeritus Full Professor of Probability at the Sapienza University of Rome. He has written many seminal papers on Random motions, Random fields, Pseudo-processes governed by heat-type equations, Fractional calculus and fractional differential equations. Recent interests are in Diffusions with branching and Motions in non-euclidean spaces.
Riccardo Cesari is Full Professor of Mathematical Methods of Economics, Finance and Actuarial Sciences at the University of Bologna. He has a D.Phil from Oxford University. Between 2013 and 2025 he has been a member of the board of IVASS, the Italian Supervisory Authority for Insurance Companies. His academic interest and publications concern the term structure of interest rates, the valuation of structured securities, the analysis of unit trusts and pension funds, forecasting models of financial markets and strategic and tactical optimal financial allocation, implicit guarantees in insurance contracts and solvency problems in life and non-life insurance.
Vieri Mosco is an actuarial researcher at IVASS, the Italian Supervisory Authority for Insurance Companies. He has a Ph.D. from the Sapienza University of Rome, where he discussed a thesis on optimal management of insurance-segregated funds. His interest is in life and non-life insurance, quantitative actuarial models and empirical applications.
