Mario Annunziato is researcher in Mathematics, in the field of Numerical Analysis,
at the "Università degli Studi di Salerno" since the year 2004. He has been also member of the "Gruppo
Nazionale di Calcolo Scientifico, Istituto Nazionale di Alta Matematica" since 2005 till 2017.
His interests focuse to the numerical solution of Partial Differential Equations (PDE)
and Integral Equations, related to stochastic processes and stochastic optimal control, and to modeling and applications of stochastic processes.
The goals of the research are:
i) to find the probability density function (PDF) of a stochastic processes by numerically solving
PDEs of parabolic and hyperbolic type, or Volterra Integral equations,
by ensuring that the discrete PDF be positive (or monotone) and conservative.
ii) to develop numerical schemes for optimization problems with PDE constraints, related
to the optimal control of stochastic processes.
iii) to provide equation modeling for random phenomena.
From 2004 to 2013 he has taught to courses and exercises of Numerical Analysis at the Science Faculty
of the University.
Address: Dr. Mario Annunziato, Dipartimento di Matematica, Università degli
Studi di Salerno, Via Giovanni Paolo II, 132  84084 Fisciano (SA), ITALY
Office n. 8, email: mannunzi[at]unisa.it, Tel: +39 089 96 3372, Fax: +39 089 96 3303.

The "Università degli Studi di Salerno" provides "Fondi di Ateneo per la
ricerca di base (FARB ex 60%)" for the financial support of fundamental research.
FARB financial support:
 Metodi numerici innovativi per equazioni dinamiche o evolutive, associate a
processi stocastici, anche per problemi di controllo ottimale, years 2011  2016.

The "Gruppo Nazionale per il Calcolo Scientifico of the Istituto Nazionale di Alta Matematica
(IndamGNCS)" provides some financial support since the year 2007.

The European Science Foundation provided the exchange grants OPTPDE
n. 3875 and n. 4099 for the research project Optimal control with FokkerPlanck equation for stochastic
systems for the years 201213.

Partecipation to the
MultiITN STRIKE  Novel Methods in Computational Finance
(Marie Curie International Training Network) in the research unit of
Prof. Alfio Borzì
at the Würzburg University.

Governative financial support from "Fondo per il finanziamento delle attività base di ricerca"
year 2017 (L. 232/2016 Art. 1, comma 295).
"Piecewise Deterministic Processes" (PDP) are a general model for stochastic
point processes where a noise affects the motion of a state function, only at
some random point epochs. They have applications to queue systems, reliability
analysis and stochastic hybrid systems.
A stochastic processes can be almost completely described by its probability density (transition)
function (PDF). The ChapmanKolmogorov equation is an abstract equation for the PDF.
In the case of PDP the CK equation can take the form of system of first order hyperbolic PDEs
[
5,
7] or
Volterra integral equations [
9,
11,
12,
27].
These equations sometime are named Liouville Master Equation or generalized FokkerPlanck equations.
The system of hyperbolic PDEs has initial Cauchy conditions and the evolutory PDF solution must
be nonnegative and total probability conservative in time.
Further, depending on the type of point process, the PDEs can have special
nonlocal
boundary conditions [
5,
7,
12],
where the boundary conditions are not assigned functions, like classical Dirichelet or Neumann,
but depend on the unknown PDF by integrals over the interior of the domain.
PDEs with this problem formulation are very little investigated.
The development of stable, positive and mass preserving numerical methods is the aim of
this research task.
Refs.
27,
12,
11,
9,
7,
6,
5.
The possibility to control a stochastic process is a very interesting and stimulating research subject,
and have potentially many applications in science, engineering and finance.
In the optimal control theory the state of a system is controlled
by minimizing (maxmizing) an objective function of the state. Mathematically it is a constrained
minimization problem.
For stochastic models, in the current scientific leterature, the problem is formulated with
an average of the cost functional of the stochastic state. The solution of this optimal control problem
can be found by solving an HamiltonJacobiBellman equation.
In this research we propose to use the PDF as representative of the state of the system,
define the objective as a functional of the PDF, and use the FokkerPlanck equation
as constraint of the optimization problem. This is a new and unexplored framework in the field
of optimization.
The solution can be found by formulating the minimization problem as an optimality system of PDEs,
in order to find the reduced gradient of the objective and its vanishing point.
The aim of this research is to develop numerical methods to solve the optimality system and
special minimization techniques for this nonlinear optimization problem.
Refs.
30,
29,
28,
24,
23,
22,
21,
20,
19,
18,
17,
16,
15,
14,
13,
10,
8.
Publications

M. Annunziato, A. Borzì,
A Fokker–Planck control framework for stochastic systems.
EMS Surveys in Mathematical Sciences 2018. DOI:10.4171/EMSS/27.

M. Annunziato, H. Gottschalk,
Calibration of Lévy processes using optimal control of Kolmogorov equations
with periodic boundary conditions,
Math. Modelling and Analysis,
Vol. 23 (2018) 390413. DOI:10.3846/mma.2018.024.

S. Roy, M. Annunziato, A. Borzì, C. Klingenberg,
A FokkerPlanck approach to control collective motion,
Comp. Optim. Appl. (2017) 137

M. Annunziato, E. Messina,
A positive and monotone numerical scheme for VolterraRenewal equations with space fluxes, Journal of Comp. Math. 37 (2019) 3347

B. Gaviraghi, M. Annunziato and A. Borzì,
A FokkerPlanck based approach to control jump processes.
Chapter 26 in: M. Ehrhardt, M. Günther and
J. ter Maten (eds) “Novel methods in Computational finance”,
Vol. 25 Mathematics in Industry 2017.

B. Gaviraghi, M. Annunziato and A. Borzì,
Splitting methods for FokkerPlanck equations
related to jumpdiffusion processes.
Chapter 25 in: M. Ehrhardt, M. Günther and
J. ter Maten (eds) “Novel methods in Computational finance”,
Vol 25 Mathematics in Industry 2017.

T. H. Breitenbach, M. Annunziato, A. Borzì,
On the optimal control of random walks with jumps and barriers.
Methodol. Comput. Appl. Probab. Vol. 20 (2018) pp. 435462

B. Gaviraghi, A. Schindele, M. Annunziato, A. Borzì,
On Optimal SparseControl Problems Governed by JumpDiffusion Processes.
Applied Mathematics, Vol. 7 (2016)
pp. 19782004 (DOI: 10.4236/am.2016.716162)

B. Gaviraghi, M. Annunziato, A. Borzì,
Analysis of splitting methods for solving a partial integrodifferential
FokkerPlanck equation.
Applied Mathematics and
Computation, Vol. 294 (2017) pp. 117

T. H. Breitenbach, M. Annunziato, A. Borzì,
On the optimal control of random walks.
IFAC papers online 49 (2016),
pp. 248253

S. Roy, M. Annunziato, A. Borzì,
A FokkerPlanck Feedback ControlConstrained Approach for Modeling Crowd Motion,
J. Comp. and Theor. Transport 45 (2016) pp. 442458

V. Thalhofer, M. Annunziato, A. Borzì,
Stochastic modelling and control of antibiotic subtilin production,
J. Math. Biol., Vol. 73 (2016), pp. 727749

M. Annunziato, A. Borzì, M. Magdziarz, A. Weron,
A fractional FokkerPlanck control framework for subdiffusion processes,
Optimal Contr. Appl. and Meth. Vol. 37 (2016), pp. 290304

M. Annunziato, A. Borzì, F. Nobile, R. F. Tempone,
On the connection between the HamiltonJacobiBellman and the FokkerPlanck control frameworks,
Applied Mathematics Vol. 5 (2014), pp. 24762484

M. Annunziato, A. Borzì,
Optimal control of a class of piecewise deterministic processes,
European Journal of Applied Mathematics
Vol. 25 (2014), pp. 125

M. Annunziato, A. Borzì,
FokkerPlanckbased control of a two level open quantum system,
Math. Models and Meth. in Appl. Sci. Vol. 23 (2013) No. 11,
pp. 20392064

M. Annunziato, A. Borzì,
A FokkerPlanck control framework for multidimensional stochastic processes,
Journal of Comp. and Appl. Math., Vol. 237 (2013) No. 1, pp. 487507

M. Annunziato, A. Borzì,
On a FokkerPlanck approach to control open quantum systems,
IEEE Xplore: Nonlinear Dynamics of Electronic Systems, Proceedings of NDES 2012

M. Annunziato,
On the Action of a SemiMarkov Process on a System of Differential Equations,
Math. Mod. Analysis Vol. 17 (2012) No. 5, pp. 650672

M. Annunziato, H. Brunner, E. Messina,
Asymptotic stability of solutions to Volterrarenewal Equation with Space Maps,
Journal of Mathematical Analysis and Applications, Vol. 395 (2012) No. 2, pp. 766775

M. Annunziato, A. Borzì,
Optimal control of probability density functions of stochastic processes,
Mathematical Modelling and Analysis, Vol. 15 (2010) No. 4, pp. 393407

M. Annunziato, E. Messina,
Numerical treatment of a Volterra Integral Equation with Space Maps,
APNUM 60 (2010) pp. 809815

M. Annunziato, A. Borzì,
Fast solvers of Fredholm optimal control problems,
Numer. Math. Theor. Meth. Appl. 3 (2010), pp. 431448.

M. Annunziato,
A finite difference method for piecewise deterministic processes with memory II,
Mathematical Modelling and Analysis, 14 (2009) pp. 139158

M. Annunziato, Analysis of upwind method for piecewise deterministic Markov processes,
Comp. Meth. Appl. Math. Vol. 8 (2008) No. 1, pp. 320

M. Annunziato, A finite difference method for piecewise deterministic processes with memory,
Mathematical Modelling and Analysis, 12 (2007) 157178

M. Annunziato, NonGaussian equilibrium distributions arising from the Langevin equation,
Physical Review E 65 (2002) 21113

M. Annunziato, P. Grigolini, B.J. West,
Canonical and NonCanonical Equilibrium Distribution,
Physical Review E 64 (2001) 11107.

M. Annunziato, P. Grigolini,
Stochastic versus dynamic approach to Lévy statistics in the presence of an external
perturbation,
Physics Letters A 269 (2000) 31.

M. Annunziato, P. Grigolini, J. Riccardi,
A FluctuationDissipation Process without Time Scale,
Physical Review E 61 (2000) 4801.
 ControlPV2018  Pavia, Sept 19th 2018,
Optimal Control and Mean Field Games.
Optimal control of multi particle stochastic system with mean field approximation.
 FGI 2017  Paderborn, Sept 25th 2017,
18th French  German  Italian Conference on Optimization.
A numerical solver for the FokkerPlanck optimal control of stochastic jumpdiffusion processes.
Abstract.
 CPDE 2016  Bertinoro, June 14th 2016,
2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations.
On the optimal control of random walks.
Abstract.
 WIAS  Berlin, Nov 12th 2015,
Direct and Inverse Problems for PDEs with Random Coefficients,
An application of the FokkerPlanckKolmogorov optimal control framework to the calibration of
Lévy processes (invited lecture).
Abstract.
 INRIA  Sophia Antipolis, Jul 1st 2015,
27 th IFIP TC7 Conference 2015 on System Modelling and Optimization,
Optimal control of subdiffusion processes with the fractional FokkerPlanck control framework.
Abstract.
 CWI  Amsterdam, Feb. 2nd 2015, FokkerPlanck optimal control of anomalous diffusion
processes (invited lecture).
 Universität Würzburg, Oct. 2nd 2014,
MultiITN STRIKE MiniWorkshop in Stochastic Computing and Optimization,
A new trend in optimal control of stochastic processes (invited lecture)
Abstract
 Würzburg July 2014.
Internationale Tage der Nachhaltigkeit (International sustainability Days) "EuroIbsa".
Best Control of Uncertainty (invited lecture).
 NetCO2014 Tours. New Trends in Optimal Control 2014,
Optimal control of stochastic processes via probability density distribution function control.
 ItalienGerman training for stochastic modeling of financial
crisis at the University of Wuppertal (Invited lecture, Dec 9th  16th 2013).
 InterDyn2013 Paris.
Workshop on Modeling and Control of Large Interacting Dynamical Systems 1012 Sept. 2013,
Dauphine University. An innovative framework for the optimal
control of stochastic processes (invited lecture).
 International School of Mathematics "G. Stampacchia"
59th Workshop Nonlinear Optimization: a Bridge from Theory to Applications 1162013. An optimal control framework for piecewise deterministic
processes.
 Oberwolfach 3112013. Probability density function optimal control of piecewise determistic processes.
 MMA2012 Tallinn. On the action of semiMarkov processes on differential equations.
Abstract
 7th European Conference on Elliptic and Parabolic Problems, Gaeta 2012.
Fokker Planckbased stochastic optimal control.
 Universität Würzburg, June 30th 2011, Seminar: Numerical solution to the Liouville Master
Equation for piecewise deterministic processes.
Abstract
 MMA 2011 Sigulda, An Optimal Control of Probability Density Function of
OneDimensional Stochastic Processes
Conference Abstracts
 SIAM OP 2011 Darmstadt, An Optimal Control of Probability Density Functions of Stochastic Processes with
the FokkerPlanck Formulation
Conference Abstracts
 EMG 2010 Ischia, Fast numerical schemes for optimal control problems with Fredholm constraints
 SIMAI 2010 Cagliari, Robust and fast method for Fredholm optimal control problem
 MMA 2010 Druskininkai, Numerical schemes for the solution to differential equations driven by semiMarkov processes
Conference Abstract
 Convegno Parma 2009  Equazioni Integrali:
recenti sviluppi numerici e nuove applicazioni, Fast solvers
of Fredholm optimal control problems (with A. Borzì).
 Lecture at Trier Universität 2009, Numerical solution of the Liouville Master Equation for Piecewise Deterministic Processes.
 ENUMATH 2009 Uppsala, High order numerical method for piecewise deterministic processes
with ENO scheme.
 GNCS 2009 Montecatini Terme, Trattamento numerico di un'equazione integrale di Volterra con
mappa sullo spazio (with E. Messina).
 ICCAM 2008 Ghent, Numerical treatment of a Volterra Integral Equation
with Space Maps (with E. Messina).
Conference Abstract
 MMAAMOE 2008 Tartu, A finite difference method for piecewise deterministic processes
with memory: monotonicity and conservativity.
 XVIII Congresso UMI Bari, Alcuni risultati sul trattamento numerico della
``Liouville Master Equation'' per Processi Deterministici a Tratti.
 SciCADE'07 Saint Malo, Multidimensional piecewise deterministic Markov
processes: a first order numerical treatment.
Conference abstract
 MMA2007 Trakai, A numerical treatment of the LiouvilleMaster Equation for piecewise
deterministic processes with memory: convergence and monotonicity
(invited lecture).Conference abstract
 NTSEE 2006 Bielefeld, On a finite difference scheme for
piecewisedeterministic processes with memory and its parallel
algorithm implementation.
 Innovative Methods for Solving Evolutionary Problems with Memory
CAPRI, A finite difference scheme for
piecewise deterministic processes with memory.
Preprints
 M. Annunziato, H. Gottschalk, Calibration of Lévy Processes
using Optimal Control of Kolmogorov Equations with Periodic Boundary Conditions,
eprint:arXiv:1506.08439
 M. Annunziato, E. Messina, Numerical treatment of a Volterra Integral Equation
with Space Maps, Preprint n. 202008 DMI  Università di Salerno (outdated).
 M. Annunziato, A Legendre polynomials series for the distribution functions of
a piecewise deterministic relaxation Markov process,
Preprint n. 142008 DMI  Università di Salerno.
 M. Annunziato, On a finite difference method for
piecewise deterministic processes with memory and its parallel algorithm
implementation,
Preprint n. 122006 DMI  Università di Salerno (outdated).
 M. Annunziato, A finite difference method for
piecewise deterministic Markov processes,
Preprint n.102006 DMI  Università di Salerno,
eprint: arxiv.org/abs/math.NA/0606588 (outdated).
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