General information

Course Description

First lecture: Thursday 24.08.
Last lecture: Wednesday 15.11.

• Every 2-3 weeks there will be homework exercises.
• Solutions will be posted on the webpage.
• See "Homework" in left menu.

• Espen Robstad Jakobsen
  
• Espen [dot] Jakobsen [at] ntnu [dot] no
  
• Office 1148, SBII
  
• Office hours: Wednesday 10:30-11:00, room 1148 SB2 (my office)

The course is about stochastic processes and differential equations with noisy/uncertain coefficients, so-called stochastic differential equations (SDEs): mathematical/probabilistic background, Ito integral, Ito calculus, SDEs, diffusions, and some applications. Of the multitude of applications in science, engineering and other disciplines, we mention diffusions in Physics (Einstein, Ornstein-Uhlenbeck, Langevin) and option pricing in finance (Black-Scholes).

• Interested students at Master or PhD level.
• The level should be suitable for good 4th year students in the industrial mathematics program.
• It can be taken as a regular course or a 'fordypningsemne' (TMA4505).

• Øksendal: Stochastic Differential Equations, Springer Verlag.
  
  Available on Springer Link: https://link.springer.com/book/10.1007/978-3-642-14394-6

• Measure And Probability
• Lebesgueintegralet
• Brownian Motion
• Martingales And Ito
• Mean Square Continuous Processes and Proof of Ito's formula
• Linear SDEs and Physical Brownian Motion
• Informal Comments on Ito Diffusions
• Kolmogorov and Fokker-Planck equations

• Strong solution of SDEs. Uniqueness and Existence

• Easy introduction:
  • T. Mikosch: Elementary Stochastic Calculus with Finance in View, World Scientific, 1998.
• Intermediate level:
  • L. C. Evans: An Introduction to Stochastic Differential Equations, 2013, AMS (a very readable lecture note)
  • J. Jacod and P. Protter: Probability Essentials, 2004, Spinger.
  • R. Schilling and L. Partzsch: Brownian Motion, 2012, De Gruyter.
  • H.-H. Kuo: Introduction to Stochastic Integration,2006, Springer.
• Advanced level:
  • D. Revuz and M. Yor: Continuous Martingales and Brownian Motion, 2005, Springer.
  • I. Karatzas and S. E. Shreve: Brownian Motion and Stochastic Calculus, 1991, Springer.
• Numerical solution of SDEs:
  • P. E. Kloden and E. Platen: Numerical Solution of Stochastic Differential Equations, 1992, Springer
  • G.N. Milstein, M.V. Tretyakov: Stochastic Numerics for Mathematical Physics, 2010, Springer.

• Probability and measure theory (background)
• Independence and conditional expectation (main theorems)
• Differential equations with stochastic loading
• Brownian motion
• Martingale theory
• The Itô integral
• Itô calculus
• Stochastic differential equations
• Optimal stopping
• Diffusions
• Limit theorems
• Stochastic modelling applications

From Øksendal:

• Chp. 2
• Chp. 3
• Sec. 4.1-4.2
• Sec. 4.3 ideas and results, no proofs (self-study)
• Sec. 5.1-5.2
• Sec. 5.3 ideas and results, no proofs (self-study)
• Sec. 7.1
• Sec. 7.2 (up till but not including Thm. 7.2.4 and its proof)
• Sec. 7.3
• Sec. 7.4
• Sec. 8.1
• Sec. 8.2 (not the proof)
• Sec. 8.3 (only Thm. 8.3.1)
• Appendix A
• Appendix B

Notes:

• Measure And Probability
• Brownian Motion
• Martingales And Ito
• Mean Square Continuous Processes and Proof of Ito's formula
• Strong solution of SDEs. Uniqueness and Existence
• Linear SDEs and Physical Brownian Motion
• Informal Comments on Ito Diffusions
• Kolmogorov and Fokker-Planck equations

All homework problem sets.

• More Brownian motion: See e.g. the book Revuz-Yor Continuous martingales and Brownian motions.
• Levy processes: The easiest generalisation of Brownian motion.
  (Classical book: Sato Levy processes and infinitely divisible distributions)
• The martingale problem: Weak solutions of SDEs, fine analysis of diffusion processes using PDE/potential theory/harmonic analaysis.
  (Classical book: Stroock-Varadhan Multidimensional diffusion processes)
• Stochastic control theory: Controling SDEs, Bellman principle, optimal controls, Bellman PDE.
  (Books: Young-Zhou Stochastic Controls, Fleming-Soner Controlled markov processes and viscosity solutions)
• General Markov processes: Many more processes than Brownian motion and Ito processes. Potential theory, semigroup theory, Dirichlet forms.
  (Classical book: Blumenthal-Getoor Markov processes and pontial theory, Fukushima et al. Dirichlet forms and symmetric markov processes)
• Pseudo differential operators and Markov processes: See book series by Niels Jacob.
• Numerical methods: See e.g. the book Kloden-Platen Numerical solution of stochastic differential equations.
• Backward SDEs: 1 equation, 2 unknowns! Additional unknown process needed to get adapted solutions (by the Martingale representation theorem). Applications in economy and PDEs. There are several books and lecture notes on the topic, e.g. by Carmona and Pardoux.
• Random fields: Generalisation - from processes in time to fields (multi-d) in space. Used e.g. in spatial statistics and PDEs with noise.
• Stochastic PDEs: Noisy or uncertain PDEs. Space-time white noise and Ito integrals. PDEs as SDEs in Hilbert spaces. Many and very different theories. See this page for some literature on linear PDEs.
• Other: Rough path theory, applications in economy, physics, engineering, sciences, …