Introduction to Stochastic Differential Equations: Theory and Numerics

1). Stochastic processes
a). Random sequences
b). Markov processes and Markov chains
c). Wiener processes and white noise

2). Stochastic Differential Equations
a). Stochastic integration
b). Ito and Stratonovich stochastic calculus
c). Stochastic differential equations
d). Stochastic Taylor expansion

3). Applications of stochastic differential equations

4). Fokker-Planck equation

5). Stochastic numerical methods
a). Pathwise Approximation and Strong Convergence
b). Approximation of Moments and Weak Convergence
c). Strong Taylor Schemes
d). Explicit Strong Schemes
e). Weak Taylor Schemes
f). Explicit Weak Schemes
g). Stochastic Runge-Kutta methods
h). Variance Reducing Approximations

6). Applications of stochastic numerical methods

After the successful completion of the module, students are able to understand and apply the basic notions, concepts, and methods of stochastic differential equations (SDEs). In particular, they understand the rigorous theoretical foundations of SDEs, can explicitly solve some simple SDEs, and can apply numerical schemes to find approximate solutions of general SDEs. Understanding the basic formalism of SDEs will allow the students to read more advanced textbooks in Mathematics, Numerical Analysis, Theoretical Physics, and Finance.

linear algebra; calculus; basic ordinary differential equations; basic numerical methods for ordinary differential equations; basic probability theory;

The module is offered as lectures. In the lectures, the concepts will be introduced with a mathematical rigor (definition, theorem, proof), but some theorems will be stated without a proof, and the emphasis will be put on explaining practical applications and presenting illustrative examples instead.

Blackboard

D. Higham, P. Kloeden, "An Introduction to the Numerical Simulation of Stochastic Differential Equations", SIAM, US, 2020

P. Kloeden, E. Platen, H. Schurz, "Numerical Solution of SDE Through Computer Experiments", Universitext, Springer-Verlag Berlin Heidelberg 1994

P. Kloeden, E. Platen, "Numerical Solution of Stochastic Differential Equations", Applications of Mathematics, Springer-Verlag Berlin Heidelberg 1992

The course will end with a written exam (~60 mins). By answering a few questions, the students will demonstrate that they have gained understanding of the basic concepts of stochastic differential equations, can state the definitions of basic stochastic processes, stochastic integrals, numerical schemes, etc., state theorems regarding their properties and explain under what assumptions these theorems hold. The students may also be asked to provide short proofs of simple theorems or perform simple practical calculations.