14.08.2018

Stochastic Analysis and SDEs

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We apply analytical methods to treat problems in probability, and - on the other hand - study analytical problems with probabilistic means. For example, we analyse properties of backwards stochastic differential equations, stochastic integrals, and stochastic processes with values in Banach spaces. We treat PDEs by stochastic methods, are interested in connections between harmonic analysis and probability, and use the theory of function spaces as well as interpolation theory.

Senior members

PhD Students

Research interests

Malliavin calculus and Besov spaces

(Stefan Geiss, Christel Geiss, Eija Laukkarinen, Thuan Nguyen)

Motivated by problems related to BSDEs and stochastic integrals we investigate differential properties of random variables, Besov spaces, properties of Itô's chaos expansion, and multiple integrals w.r.t. independent random measures and their moments. This also involves functional analysis, as interpolation theory and local theory of Banach spaces are used.

Backward stochastic differential equations

(Stefan Geiss, Christel Geiss Antti Luoto, Thuan Nguyen)

Backward stochastic differential equations (BSDEs) are a type of stochastic differential equations which is determined by a terminal condition. They appear in control problems, in connection to non-linear PDEs, and in many other applications. There are various types of BSDEs. We investigate BSDEs driven by the Brownian motion and by Lévy processes of Lipschitz and quadratic type, and are interested in Malliavin differentiability, existence, uniqueness, comparison theorems, variational properties of their solutions, and chaos expansions of their solutions. Tools from stochastic analysis and decoupling techniques are used.

Probability in Banach spaces, Harmonic Analysis, and Interpolation

(Stefan Geiss, Henri Ylinen)

Probability theory and analysis go hand in hand. We use methods from harmonic analysis to study backward SDEs and stochastic integrals and interpolation spaces on the Wiener-Itô space. Moreover, extrapolation techniques for moment inequalities regarding Banach space valued stochastic processes are investigated that are used in the construction of infinite dimensional stochastic integrals. These techniques are relevant for solving SPDEs.

Nonlinear PDEs and SDEs

(Stefan Geiss, Christel Geiss, Mikko Parviainen)

Backward SDEs are connected with viscosity solutions of systems of semi-linear second order PDEs of parabolic and elliptic type. Another type of non-linear PDEs, we consider, is based on the p-Laplace operator and on related operators. Operators of this type are associated to stochastic differential games.

Approximation and SDEs

(Christel Geiss, Stefan Geiss, Eija Laukkarinen, Antti Luoto)

Our research comprises approximation of stochastic integrals driven by Lévy processes in various function spaces, approximations schemes for backward SDEs with jumps based on chaos expansions, and random walk approximation of backward SDEs.

Publications

Publications of the group can be found on the arXiv preprint server and the TUTKA database. The most up-to-date information is available on the members' personal web pages.

Events

Links

Funding

Stochastic Analysis and Nonlinear Partial Differential Equations, Interactions and Applications (PI Stefan Geiss)