last update...09/07/2005

by John Nikas

Dimitsana Summer School

Stochastic Differential Geometry

and applications in finance

10-17 July 2005, Thermo, Aitoloakarnania, Greece

Sponsored by the National Bank of Greece

Under the auspices of University of Patras

Speaker: Marc Yor

Title: Basic Facts about Brownian Motion, Stochastic Integration and Stochastic Differential Equations

Speaker: Michel Emery

Title: Second order stochastic differential geometry in probability theory

Abstract:

Second order stochastic differential geometry and its use in probability theory were introduced 25 years ago by L. Schwartz,  to provide an intrinsic view of stochastic processes, for instance solutions of stochastic differential equations, taking their values in a manifold that has no linear structure, for instance a curved surface.

I have chosen this topic for two reasons. First, these 2nd-order objects provide a very elegant language, which leads to a much deeper insight into the behaviour of stochastic processes than the more restricted, 1st-order viewpoint using Stratonovich integration. Second, it is not only a  language, but also a theory, necessary for further advances in the study of manifold-valued martingales.
 

Syllabus:

1. Second-order tangent and cotangent vectors. Acceleration of curves, second differential of a function, product of two first-order forms. Continuous semimartingales with values in a differentiable manifold.

2. Itô's formula; Schwartz' principle. Intrinsic integral of 2nd-order
covectors along semimartingales. Itô and Stratonovich intrinsic stochastic integrals of 1st-order forms.

3. Stochastic differential equations between manifolds: their algebraic structure (Schwartz morphisms), existence, uniqueness up to explosion time.

4. Examples of stochastic differential equations; the special case of
 parallel transport.

5. Itô and Stratonovich transfer principles. Discrete-time approximations of Itô and Stratonovich stochastic differential equations.

6. Martingales, their behaviour. Harmonic mappings, their smoothness.
 

References:

• Laurent Schwartz, Semimartingales and their stochastic calculus on manifolds. Presses de l'Universite de Montreal (1984).

• Laurent Schwartz, Geometrie differentielle du 2eme ordre, semi-martingales et equations diffé rentielles stochastiques sur une variete differentielle. In: Seminaire de probabilites XVI (Supplementary volume), Lect. Notes Math. 921, 1-150 (1982).

• P.A. Meyer, Geometrie stochastique sans larmes. In: Seminaire de probabilites XV, Lect. Notes Math. 850, 44-102(1981).

• M. Emery, Stochastic calculus in manifolds. With an appendix by P.A. Meyer. Universitext, Springer (1989).

Speaker: Xue-Mei Li

Title: Techniques and Topics in Stochastic Differential Equations on  Geometric Spaces.

Abstract:

In this course we discuss aspects of stochastic differential  equations which either have connections with analysis and geometry or find  applications somewhere. The state space will be non-compact smooth  manifolds. We shall view the basic questions in stochastic differential  equations from the point of view of the derivative flows. Elliptic  differential Equations will be discussed together with their associated  semi-groups on functions and on differential forms and the underlying  Riemannian structure. We shall also touch  various Bismut type formulae  and the equivalent pillar structure on path spaces: the integration by  part formulae and beyond.
 

Syllabus.

1. Linearised SDE, estimates on the derivative flow, relate the derivative  flow to  the following: non-explosion, smooth dependence of the solution  on initial data,  stability, and to answer the question "when does the  solution of SDE sends smooth curves to smooth curves" and when d P_t  equals P_t d.

2. Analytic Aspects of SDEs. Elliptic differential operators, Markov property of the solutions, differentiation of the associated semi-groups on functions and on differential forms. Application to geometric and topological problems. Various Bismut type formulae. A touch of hypoellipticity by basic instinct.

3. Derivative of solution flows to SDEs in the sample variable. Integration by parts formulae related to Bismut type formulae. Related analysis on path spaces.
 

References:

• Riemannian Geometry, By M. Do Carmo.

• Stochastic Differential Equations on Noncompact Manifolds.
   Xue-Mei Li. 1992 on <www.xuemei.org/bib.html>.

• P.A. Meyer, Geometrie stochastique sans larmes. In: Seminaire de probabilites XV, Lect. Notes Math. 850, 44-102(1981).

• K. D. Elworthy. Geometric aspects of diffusions on manifolds. LNIM1362,   pages 276–425.  1988.

 

Speaker: Damir  Filipovic

Title: The Geometry of  Interest Rate Models.

Abstract:

We consider interest rate models of the Heath-Jarrow-Morton (HJM) type, where the forward rates are driven by a finite dimensional Wiener process. Such models can be realized as stochastic equations in a Hilbert space H of forward curves. Within this framework we give necessary and sufficient conditions for the existence of finite dimensional realizations (FDRs). FDRs arise in connection with the estimation of the forward curve by parametrized curve families.
From a geometric point of view, a FDR corresponds to a finite dimensional invariant submanifold for the HJM equation in H. Thus we are led to the problem of characterization and existence of finite dimensional invariant submanifolds for stochastic equations in infinite dimension.

Syllabus.

1. Stochastic equations in Hilbert spaces: mild, weak and strong solutions

2. Estimation of forward curves, consistency problems

3. Finite dimensional invariant submanifolds for weak solutions to stochastic equations: characterization and regularity of the viable solutions

4. Existence of finite dimensional invariant submanifolds: Frobenius theorem in infinite dimension

5. Appliations to interest rate models: complete characterization of generically finite dimensional models
 

References:

• Bjork T.: On the Geometry of Interest Rate Models, Springer LNM 1847,  p.133-215, 2004

• Da Prato G.: Kolmogorov Equations for Stochastic PDEs, CRM Barcelon, Birkhauser, 2004

• Filipovic D.: Consistency Problems for Heath-Jarrow-Morton Interest Rate Models, Springer LNM 1760, 2001

• Teichmann J.: Stochastic Evolution Equations in Infinite Dimension with ppliations to Term Structure Problems, Unpublished Manuscript, TU Vienna, 2005