Stochastic Modelling

Cod: 23058

Department: DCET

Department: DCET

ECTS: 10

Scientific area: Mathematics

Scientific area: Mathematics

Total working hours: 260

Total contact time: 20

Total contact time: 20

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.

Exercises and project