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401-4657-00L 6 Credits BSC , DR , MSC D-MATH
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Numerical Solution of Stochastic Ordinary Differential Equations

Lecturers & Examiners: Prof. Dr. Christoph Schwab
Alternative course titles: "Numerical Analysis of Stochastic Ordinary Differential Equations" / "Computational Methods for Quantitative Finance: Monte Carlo and Sampling Methods"
VVZ CR n/a

Last Updated: 2026-06-01 11:30:59

Abstract

This course is on the numerical approximations of stochastic ordinary differential equations (SDEs) driven by Brownian motions and Lévy processes. SDEs have several applications, for example in financial engineering.The contents cover stochastic processes, stochastic calculus, well-posedness results for SDEs, strong and weak approximations of SDEs, and simulation via Monte Carlo methods.

Objective

The aim of this course is to enable the students to carry out simulations and their mathematical convergence analysis for stochastic models originating from applications such as mathematical finance. For this the course teaches a decent knowledge of the different numerical methods, their underlying ideas, convergence properties and implementation issues.

Content

Mathematical Formulation of random number generation Brownian motion and Lévy processes: definitions and basic properties Stochastic integration and stochastic Ito-calculus Stochastic ordinary differential equations (SDEs): existence, uniqueness, continuous dependence, integrability. Numerical approximations of SDEs: strong and weak convergence, Euler-Maruyama scheme, Single- and Multilevel Monte-Carlo, Milshtein-scheme, Stochastic Runge-Kutta, Talay-Tubaro Extrapolation. Stochastic simulation and Monte Carlo methods Euler-Maruyama for Lévy-driven SDEs: strong and weak convergence, Multi-level Monte Carlo, and efficient increment simulation.

Resources

Lecture Notes

There will be English, typed lecture notes for registered participants in the course.

Literature

P. E. Kloeden and E. Platen: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin, 1992. Bertoin, Jean: Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0 Cont, Rama; Tankov, Peter: Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004. xvi+535 pp. ISBN: 1-5848-8413-4

General Information

Language
English
Levels
BSC , DR , MSC
Frequency
Yearly recurring

Examination

Type
end-of-semester examination
Mode
written 120 minutes
Aids
None
Digital
The exam takes place on devices provided by ETH Zurich.
Learning tasks: Meaningful solutions to 70% of the weekly homework assignments can count as bonus of up to +0.25 of final grade.End-of-Semester examination will be *closed book*, 2hr in class, and will involve theoretical as well as MATLAB/Python programming problems.Examination will take place on ETH-workstations running MATLAB/Python.Own computer will NOT be allowed for the examination.

Registration & Places

Priority: Registration for the course unit is only possible for the primary target group

Course Components

Type Title Time & Place Hours
lecture Numerical Solution of Stochastic ODEs (Comp. Meth. Quant. Fin.: Monte Carlo and Sampling Methods)
  • Wed 14:15-16:00 (HG D 5.2)
  • Fri 14:15-15:00 (HG D 3.2)
3 h weekly
exercise Numerical Solution of Stochastic ODEs (Comp. Meth. Quant. Fin.: Monte Carlo and Sampling Methods)
Groups are selected in myStudies.
  • Fri 15:15-16:00 (CAB G 59)
  • Fri 15:15-16:00 (HG D 3.2)
1 h weekly

Offered In