Stochastic differential equations, modelling, parameter estimation, R programming

This UC aims to provide knowledge and skills in the theory and applications of stochastic differential equations (SDEs) for the analysis and modeling of phenomena that occur in random environments. Upon completion of this UC, the student should be able to

(i)                understand the fundamental techniques of computer simulation

(ii)                understand the main SDE models and investigate the existence, uniqueness, and properties of their solutions

(iii)               apply the previous models to practical cases in the areas of biology, ecology, economics, management, and finance

(iv)               solve SDEs analytically or numerically

(v)                estimate parameters of SDEs

1.       Simulation

a.       Monte Carlo simulation

b.       Generation of pseudorandom numbers

c.       Simulation of random variables

d.       Markov chain simulation

e.       Monte Carlo integration

f.        Advanced simulation methods

2.       Stochastic differential queations

a.       Existence and uniqueness of solutions

b.       Taxonomy of SDEs, their solutions and properties

c.       Properties of solutions and solutions to SDEs as Markov processes

d.       Stability

3.       Models of SDEs

a.       Applications to population growth

b.       Applications to economy and to management

c.       Applications to finance

4.       Approximation and estimation of solutions to SDEs

a.       Iterative schemes of Euler-Maruyama e Milstein

b.       Lamperti transformation

c.       Linearization: Ozaki and Shoji-Ozaki

d.       Stochastic Taylor expansions

5.       Estimation of parameters of SDEs

a.       Known and unknown transition density

b.       Maximum likelihood method

c.       Change of variable

1.       W.J. Braun, D.J. Murdoch, “A First Course in Statistical Programming with R”, Cambridge University Press, Cambridge, 2016. ISBN 978-1-107-57646-9.

2.       R.F. Bass, “Stochastic Processes”, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 33, Cambridge University Press, Cambridge, 2011. ISBN 978-1-107-00800-7.

3.       B. Øksendal, “Stochastic Differential Equations – An Introduction with Applications”, Sixth edition, Springer-Verlag, New York, 2003. ISBN 978-3-540-04758-2.

4.       M. J. Panik, “Stochastic Differential Equations,” John Wiley & Sons, Inc, New Jersey, 2017. ISBN 978-1-119-37738-2.

Teaching and assessment will be generally framed by the Virtual Pedagogical Model of Universidade Aberta, which advocates student-centered teaching and asynchronous approaches that encourage the development of collaborative work among students and strong student interaction with each other and with the teacher. Each of the five content items presented in the syllabus will have a working period in the virtual class of between 1 and 3 weeks, during which sets of problems will be proposed for students to work on, organized into groups. Discussions in forums will be evaluated and encouraged, as well as interaction with the professor in the forum to clarify doubts or provide any explanations of concepts, results, or techniques. In addition to the evaluation of the exercises worked in groups and interventions in the forums, there will be a final individual assignment (distinct from student to student) that will be part of the evaluation. 

Exercises and project

Enrolment in this curricular unit requires that the student has successfully completed the following subjects: probabilities, statistical inference, differential equations, stochastic processes, measure theory, and programming in a high-level language.