Research team

Expertise

Reviews of mathematical papers and books.

Models in financial mathemactics with inertia and jumps. 01/01/2008 - 31/12/2008

Abstract

Starting with the works by Nadine Bellamy (1999), Damien Lamberton (1997), Steve E. Shreve (2004) as well as that of Erhan Bayraktar and others (2003), respectïvely about stochastic processes with jumps in finance and market models with inertia, we intend to define basic differential equations (with jumps) which reflect at the same time the effect of inertia and jumps in view of evaluation and hedging of options in a financial market.

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Infinite-dimensional analysis and stochastics. 01/10/2006 - 31/12/2008

Abstract

An important part of the program will consist of a study of Markov semi-groups and propagators, by employing techniques taken from functional analysis and from the theory of stochastic processes. For locally compact spaces there is a one-to-one correspondence between strongly continuous Feller-Dynkin semi-groups and strong Markov processes (with certain continuity properties. Feller-Dynkin semigroups leave the space of bounded continuous functions vanishing at infinity invariant. This correspondence gives rise to an interaction between stochastic analysis and classical operator semi-group theory. However, many interesting spaces are not locally compact. Nevertheless these more general spaces are interesting and important from the viewpoint of stochastic analysis and possibilities for applications. Examples of such spaces are: Wiener space, loop spaces, Fock space. Many of these spaces are Polish. The idea is to develop an analysis which includes these Polish spaces. In this context we also want to investigate the martingale problem. In the commutative case this leads to problems as described in the paper: J.A. van Casteren, Some problems in stochastic analysis and semigroup theory (in Proceedings of the First International Conference of Semigroups of Operators: Theory and Applications, December 1998, Newport Beach, California The main organizer/chairman is A.V. Balakrishnan). Series: Progress in Nonlinear Differential Equations and Their Applications, Vol. 42; Publisher: Birkhauser Verlag, Basel, Switzerland, 2000; pp. 43--60. These problems were for a part repeated and further elaborated in J.A. Van Casteren, Markov processes and Feller semigroups "Conferenze del Seminario di Matematica dell'Universita di Bari" Estratti Dalle Conferenze del Seminario di Matematica, Proceedings of the Summer School Operator methods for Evoluton Equations and Approximation Problems (OMEEAP) 2002, Roma 2003, pages 19-93. In the non-commutative situation some other but related problems were described in the same publication (here ``positive'' is replaced with ``completely positive'', and Feller generator is substituted by Lindblad generator). Shortly the book Jan A. Van Casteren, Markov processes, Feller semigroups and evolution equations, 650 pages will be resubmitted for publication. This book contains many of the results described above.

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Innfinite-dimensional stochastic analysis. 01/01/2005 - 31/12/2007

Abstract

The idea is to generalize and improve theorems, which exist for Markov processes with locally compact state space, to a more general topological setting. In fact the state space should be replaced with a not necessarily locally compact polish (or more general) topological space. Some new techniques and methods will be developed. These methods will have a functional analytic as well as a stochastic aspect. One of the problems will be to give an adequate definition of a generator of such a process. The corresponding martingale problem also requires a new approach. We hope to apply our results to concrete infinite-dimensional problems in mathematical analysis.

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Feynman-Kac propagators and other topics in stochastic analysis and semigroup theory. 01/01/2004 - 31/12/2006

Abstract

A central issue in this project is the use of stochastic methods and techniques (like Markov Processes, Brownian motion, martingale techniques) to investigate forward and backward (stochastic) differential equations. A crucial role are played by the linear (and non-linear) Feynman-Kac formula, as well as by related formulae like the Girsanov (or drift) transformation (Cameron-Martin formula). Operator semi-groups are extensively employed throughout the whole project.

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Some problems in stochastic analysis and semi group theory. 01/01/2002 - 31/12/2004

Abstract

The following topics wil be discussed. (1) Establishing and better understanding the intimate relationship between Euclidean Quantum Mechanics, martingale measures and generators of diffusions. (2) Extend the existing results on generators of strong Markov processes (with locally compact state spaces) to Polish and other topological spaces. The strict topology plays a central role here. Do something similar in the non-commutative context: again a notion of strict topology is available. (3) Try to improve the knowledge about stochastic partial differential equations and of backward stochastic differential equations. In particular, indicate the existing relationship between Neumann semigroups, reflected Markov processes, backward stochastic differential equations, and singular limits of quadratic forms.

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Backward Stochastic Differential Equations and their connections with Partial Differential Equations: Approximation results. 01/01/2002 - 31/10/2002

Abstract

The idea is to approximate reflected backward stochastic differential equations by backward stochastic differential equations without reflection. It is feasible that the problem could also be solved by approaching a parabolic problem of Neumann type by a parabolic problem without Neumann boundary conditions. However, this is not sure. That is why an attempt is done to employ backward stochastic differential equations.

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    Infinite-dimensional stochastic and classical analysis. 01/09/2001 - 31/08/2003

    Abstract

    The main aim of this project is to investigate a number of issues, which were subject to research in a infinite-dimensional situation (like Feyman-Kac semigroups, the relationship between Markov processes and Feller semigroups, Neumann semigroups; scattering theory, generators of completely positive semigroups) in an infinite-dimensional setting. Another topic of research is to replace perturbations of order zero by perturbations of higher order (in particular of order 1).

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      Multiparametric Markov (Ornstein-Uhlenbeck) processes and families of operators. 01/10/1999 - 30/06/2000

      Abstract

      Relationships between the following concepts should be established: multiparametric Markov processes, operator families (multiparametric semigroups), (higher order) stochastic differential equations, higher order capacities, coninuity properties of such processes.

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        Research related to Stochastic analysis. 01/11/1998 - 31/10/2000

        Abstract

        In this project the following topics will be discussed: regular and singular first order perturbations of quantum mechanical Hamiltonians, quantum mechanical scattering theory for such systems, Kullback information, and its relationship with scattering theory and non-commutative probability theory. Auxiliaries are among other things: Markov processes, martingales, and operator theory (semigroups).

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          Fundamental Mathematics. 01/01/1997 - 31/12/1997

          Abstract

          The library is, in all its aspects, the research lakoratory in mathematica. This is certainly the case for fundamental mathematica. With the promised money books will be purchased. As such the research in algebra end analysis/stochastics will be enhanced.

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            Local times and Neumann semigroups. 01/10/1996 - 30/04/1997

            Abstract

            The concept of local time for Markov processes is to be defined using the Doob-Meyer decomposition theorem (semi-martingale theory). This theory can be applied in conjunction with the natural distance function coming from the carre du champ operator (squared radient operator). The link with singular first order problems, in particular, with Neumann semigroups should be made.

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              Stochastic analysis and mathematical modelling of physical processes. 01/01/1996 - 31/12/1996

              Abstract

              The underlying idea behind this project is to use techniques and develop methods from stochastic and from mathematical analysis to model physical processes. In particular the following topics will be considered : completeness of scattering systems, the use of loop spaces, properties of stationary states, non-equilibrium states and long-range correlations in many particle systems. Among others the theory of Markov processes, Malliavin calculus and random walks will be used. The construction of Gibbs states and potential functions will play an important role.

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                Discretization and approximation of ill-posed problems. 30/09/1995 - 31/03/1996

                Abstract

                Let the operator A be the generator of an analytic semi group. The Cauchy problem u'(t) = - Au(t) (1), u(0) = u', t > 0, is called ill-posed: solutions need not be strongly continuous. To (1) one can associate a stochastic differential equation of a certain form (2). It is known that (2) yields a regularization of (1). We intend to apply semi-discretization and full discretization to approximate solutions of (1) by solutions of (2) as ,u tends to 0. In particular we like to obtain the optimal order of convergence.

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                  Computer intensive and probabilistic methods. 01/01/1995 - 31/12/1996

                  Abstract

                  The idea behind this project is to provide the participants with the technical and other means that are necessary to do fundamental and/or applied mathematical research. The subjects to be studied include : reliability regions for the location parameter, median computations, efficiency and robustness, algebraic and probabilistic techniques. To carry out this kind of research the participants need to have at their disposal the right software (SPLUS and MATHEMATICA) and hardware (X-terminal). In order to perform research of high qualitaty, it is necessary that the participants have contacts to external scientists.

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                    Stochastic analysis and mathematical modelling of physical processes. 01/01/1994 - 31/12/1994

                    Abstract

                    The underlying idea behind this project is to use techniques and develop methods from stochastic and from mathematical analysis to model physical processes. In particular the following topics will be considered : completeness of scattering systems, the use of loop spaces, properties of stationary states, non-equilibrium states and long-range correlations in many particle systems. Among others the theory of Markov processes, Malliavin calculus and random walks will be used. The construction of Gibbs states and potential functions will play an important role.

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                      Stochastic spectral analysis. 01/01/1994 - 31/01/1994

                      Abstract

                      Prof. Demuth will give some lectures on stochastic spectral analysis. The main object will be pertrubation theory. An unperturbed Hamiltonian will be perturbed by a potential function. The resulting operator can be used to describe the quantum mechanical behaviour of a particle under the influence of a potential.

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                        Topological sensitivity of Markov processes 30/09/1991 - 29/09/1994

                        Abstract

                        We investigate Markov processes carried by a differtiable manifold, usually Riemannian. More specifically we verify to what extent the finite-dimensional distributions restrict the manifold's topology (such as homotopy and homology groups)

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                          Operator semigroups, Markov processes, population dynamics and Malliavin Calculus 30/09/1991 - 29/09/1993

                          Abstract

                          The purpose of this project is to treat problems that occur in population dynamics and in the description of flows on subvarieties of the Wiener space and also on finite dimensional manifolds. This will be done via analytics methods (semigroups) and by stochastic methods (Markov processes and stochastic calculus).

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                            Operator semigroups, Markov processes, population dynamics and Malliavin Calculus 30/09/1991 - 29/09/1993

                            Abstract

                            The purpose of this project is to treat problems that occur in population dynamics and in the description of flows on subvarieties of the Wiener space and also on finite dimensional manifolds. This will be done via analytics methods (semigroups) and by stochastic methods (Markov processes and stochastic calculus).

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                              01/01/1991 - 31/12/1991

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                                30/09/1989 - 29/09/1991

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                                  30/09/1989 - 29/09/1991

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