Stochastic Analysis

 

V4F1 - Summer semester 2016 

 

Wednesday 12.15-14.00 and Thursday 12.15-14.00, Kleiner Hörsaal, Wegelerstr. 10 

Some lectures will also be held on Tuesday 16.00-18.00,  Kleiner Hörsaal, Wegelerstr. 10. 

Remark (16/6/16): Next lectures are 29/6, 30/6, 5/7, 6/7, 12/7, 13/7, 14/7, 20/7, 21/7.  

Tutorial classes: Philipp Boos / Monday 16-18, SemR 0.008

Exam: Oral. 1-3 August 2016 / 21-23 September 2016.

NEW: Exam Schedule: (pdf) [updated 14/7/2016] Sample exam questions (pdf)

 

Topics

• SDEs: properties, weak solutions, transformations, stochastic flows, long time behaviour, large deviations, link with PDEs, variation of parameters, numerical schemes

Prerequisites 

Ito calculus for Brownian motion, see  e.g. Prof. Eberle's lecture notes on "Introduction to Stochastic Analysis" (pdf).

Lecture Notes

The first part of the course will be based on Prof. Eberle's lecture notes for Stochastic Analysis SS16 (pdf), in particular Chapters 2,3 but excluding processes with jumps. Some notes for material not covered by Prof. Eberle's lecture notes will be posted here:

• Note 1 : Stochastic differential equations : existence, uniqueness and martingale problems. (pdf) [version 1.1, posted 24/5/2016]
• Note 2 : Girsanov transform, Doob's h-transform. (pdf) [version 1.1, posted 24/5/2016]
• Note 3 : Brownian martingale representation theorem, Entropy and Girsanov transform, Boué-Dupuis formula, Large deviations. (pdf) [version 1.3, posted 16/6/2016]
• Note 4 : Kolmogorov theorem, Stochastic flows, Backward Ito formula. (pdf) [version 1.1, posted 29/6/2016]

Further References

• Rogers/Williams: Diffusions, Markov processes and martingales, Vol.2
• Bass: Stochastic processes
• Protter: Stochastic integration and differential equations

Problem sheets

• Sheet 1 (due on thursday 21/4, collected during the lecture)
• Sheet 2 (revised version of 25/4/16, due on thursday 28/4, collected during the lecture)
• Sheet 3 (due Mon May 9th, collected at the beginning of the tutorial)
• Sheet 4 (due Mon May 23rd, collected at the beginning of the tutorial)
• Sheet 5 (due Mon May 30th, collected at the beginning of the tutorial)
• Sheet 6 (due Mon Jun 6th, collected at the beginning of the tutorial)
• Sheet 7 (due Mon Jun 13th, collected at the beginning of the tutorial)
• Sheet 8 (due Mon Jun 20th, collected at the beginning of the tutorial)
• Sheet 9 (due Mon Jul 11th, collected at the beginning of the tutorial)
• Sheet 10 (due Mon Jul 18th, collected at the beginning of the tutorial)

Course Journal

• Lecture 13/4 : Overview of the course. Weak and strong solutions to SDEs, uniqueness, Yamada-Watanabe theorem. Levy's characterisation of Brownian motion.
• Lecture 14/4 : Finish proof of Levy's characterisation of Brownian motion. Orthogonal infinitesimal transformations of Brownian motion, Bessel processes, Tanaka's example of an SDE with weak solution but not strong solution not pathwise uniqueness. Ito-Doeblin formula, applications to PDEs. Martingale solutions to SDEs, equivalence between martingale and weak solutions (to be finished).
• Lecture 20/4 : End of the proof of the equivalence between martingale and weak solutions (with invertible diffusion matrix). Time change of continuous local martingales. Dubins-Schwartz theorem with proof.
• Lecture 21/4 : First example of change of time for SDEs. Example of non-uniqueness of weak solutions. Yamada-Watanabe uniqueness theorem. General result about change of time for SDEs. One dimensional diffusions : scale function and speed function.
• Lecture 27/4 : Knight's theorem, transformation of complex Brownian motion. Girsanov transform. Example in finite dimension. Change of probability on a filtered space. 
• Lecture 28/4 : Exponential martingale. Girsanov theorem for martingales. Novikov condition. The case of the Brownian motion. Adapted change of measure for Brownian motion.
• Lecture 11/5 : Doob's h-transform. Diffusion bridges. 
• Lecture 12/5 : Conditioning a diffusion not to leave a given domain. Brownian motion conditioned to stay positive. Exponential tilting. Weak solution to SDEs via Girsanov theorem.
• Lecture 2/6 : The martingale representation theorem.
• Lecture 8/6 : Entropy and Girsanov transform. Föllmer's drift. The Boué-Dupuis formula. 
• Lecture 9/6 : proof of the Boué-Dupuis formula, exponential integrability of Lipshitz functionals over the Wiener space.
• Lecture 14/6 : large deviations over the Wiener space, Laplace principle, small noise diffusions.
• Lecture 15/6 : Proof of the Laplace principle via Boué-Dupuis formula for a general class of Wiener functionals.
• Lecture 16/6 : Regularity of processes. Garcia-Rodemich-Rumsey lemma and Kolmogorov theorem.
• Lecture 29/6 : Stochastic flows. Regularity wrt parameters.
• Lecture 30/6 : Stochastic flows (continued).  Substitution in stochastic integrals.
• Lecture 5/7 : Stochastic flows (continued), cocycle property and homeomorphism property. 
• Lecture 6/7 : Differentiability of the stochastic flow.
• Lecture 12/7 : Stratonovich's integral and relation with Itô integral.
• Lecture 13/7 : Brownian motion on hypersurfaces and Doss-Sussmann transformation.
• Lecture 14/7 : Weak and strong error. Stochastic Taylor expansion.
• Lecture 20/7 : Numerical schemes for SDEs: Euler's and Milstein's.
• Lecture 21/7 :