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In our last lesson on Thursday, 18. May, we discussed the reconstruction theorem of M. Hairer, which can be regarded as a generalization of the Sewing Lemma. As an application of this theorem, we recovered the theorems of T. Lyons and M. Gubinelli on rough path integration as a particular case. See Ch. 13 in Friz, Hairer.
The lecture notes (part 5) on the construction of solutions to the KPZ equation via renormalization will be available on this website, soon.
Muntlig eksamen p? fredag, 9. juni, rom 637, NHA fra kl. 09:00 !
Eksamensform: Eksamenen som tar 45 minutter best?r av to deler:
1. Foredrag av fritt valg om et emne som er knyttet til r?ffstiteori. Foredragsemnet som foresl?s skal meddeles meg senest 2 dager f?r eksamenen (via e-mail). Dessuten m? emnet bli godkjent av meg. Foredraget skal vare i omtrent 20 minutter og fremf?ringsformen er opp til kandidatene (ved tavle med manus eller ikke, beamer,...).
2. Generelle sp?rsm?l knyttet til r?ffstiteori.
Kap. 1, 2, 3 , 4.1, 4.2, 4.3 og Th. 4.6 i manuset mitt er kurspensum (eller se p? de tilsvarende emnene i boken til Friz, Hairer).
Manuset mitt er lagt ut p? fagsiden.
Et sett av eksamensrelevante emner kan nedlastes her: emner
&nbs...
In the last weeks we discussed the existence and uniqueness of rough differential equations, a priori estimates of solutions and the continuity of the Ito-Lyons map (see Ch. 8 in the Friz, Hairer). As an application of those results, we studied the link between solutions of Ito-SDE?s and RDE?s. Further we obtained the Wong-Zakai theorem on the approximation of SDE-solutions by means of solutions to ODE?s (see Ch. 9). In our next lesson (4. May) we aim at proving the support theorem of Stroock-Varadhan, which provides a characterization of the support of the distribution of Ito-SDE-solutions (see Ch. 9). Another application of the above mentioned results, we want to look at in the next lesson, pertains to the construction of solutions to a certain class of rough path and stochastic partial differential equations (see Ch. 12). Our plan is to start with Ch. 13 on an introduction to regularity structures as...
There will be also no lecture on Thursday, 23. March because of illness !
We will continue with our course on Thursday, 30. March and discuss the construction of solutions of rough path differential equations (Chapter 8 in the book of Friz, Hairer).
We also aim at making up for the lost time with respect to the two cancelled lectures.
There will be no lecture on Thursday, 16. March !
After a general introduction to rough path theory (26. Jan., 2. Feb.), we discussed Chen?s relation and defined the space of (geometric) rough paths. Further, we studied the link beween rough path theory and geometry and gave a characterization of geometric rough paths by means of Lie groups (9., 16. Feb.). See Chapter 2 in the book of Friz, Hairer. In our next lesson (23. Feb.) we aim at giving the construction of a rough path integral of 1-forms by using the so-called Sewing Lemma. See Chapter 4 in Friz, Hairer.
NB ! Kurset STK4290 avsluttes med muntlig eksamen.
The exam in STK4290 at end of the course will be oral.
NB ! Vi skal starte opp med kurset torsdag, 26. jan., kl. 12:15-15:00, Niels Henrik Abels hus, rom 801 !
Our course will start on Thursday, 26. Jan., 12:15-15:00, Niels Henrik Abels hus, room 801 !
Rough path theory, whose foundation was laid by Terry Lions in the 1990?s, has proved to be a very useful and innovative tool in the analysis of stochastic differential equations, stochastic partial differential equations and in applications to statistics, financial data analysis or machine learning.
Rough path theory, which aims at "removing" probability from stochastic systems to the degree possible, enables e.g. the analysis of solutions to stochastic (partial) differential equations in a deterministic way, that is path by path.
This theory, which has links to other branches of mathematics (e.g. Malliavin calculus or Dirichlet forms), provides non-probabilistic techniques with simplified proofs of crucial results in stochastic analysis and their generalizations (e.g. Wong-Zakai theorem or limit theorems of stochastic flows).
The course gives a basic introduction to rough path analysis with applicat...