Stochastic Differential Equations

Basic information

Workload: 

45 hours

Prerequisite: 

Measure, Integration and Probability, Stochastic Calculation

Syllabus: 

Theorems of existence and uniqueness of solutions for Stochastic Differential Equations (EDEs). Dependence on solutions in relation to initial conditions and parameters. Strong and weak solutions. Analytical properties and moments of the solution. Solution like Markov and diffusion processes. Transition probabilities. General solution for linear EDE systems. The Ornstein-Uhlenbeck process. Qualitative theory. Stochastic stability. Moment stability. Connection between SDEs and PDEs. Hamiltonian systems. Stochastic oscillators. Modeling with EDEs and Applications. Approximation and simulation of trajectories.

Bibliography

Mandatory: 

· Øksendal, B. K. (2003) Stochastic Differential Equations. An Introduction with Applications. Universitext. Springer.
· Arnold, L. (1974) Stochastic Differential Equations. Wiley, New York.
· Gard, T. C. (1988) Introduction to Stochastic Differential Equations. Marcel Dekker, New York.

Complementary: 

·       Allen E. (2007) Modeling with Itô Stochastic Differential Equations. Springer.
·       Khasminskii, R (2012) Stochastic stability of Differential Equations. Springer.
·       Gikhman, I ;  Skorokhod A.V (1972). Stochastic Differential Equations. Springer.
·       Mao, X.  (2008) Stochastic Differential Equations and Applications. Elsevier.
·       Evans L. (2010) An Introduction to Stochastic Differential Equations. AMS.