Stochastic analysis and stochastic finance

The mathematical models of this section capture real world phenomena arising from, for instance, high frequency financial data or climate data; these occur on a predominantly mesoscopic level. We study both their dynamics and calibrate them to reality through statistical inference. Randomness enters through central or local (functional) limits producing the important paradigms of Lévy processes, in particular

• light-tailed Brownian motion,
• more heavy-tailed processes with jumps,
• or heavy-tailed Gaussian processes such as fractional Brownian motion.

As a unifying feature, the characteristic function of their marginal distributions may be of an exponential affine form, characterizing the class of affine processes, with particular relevance in stochastic finance. The description of the models’ dynamics on this level originates from the microscopic view – the focus of the second main topic of our research program – through central, local or diffusion limit theorems, and leads to the main objects of stochastic analysis: (backward) stochastic (partial) differential equations, driven by Gaussian or Lévy, or more generally semi-martingale dynamics, or in a still more general context by rough paths. Beyond Markovian models, systems with time delayed feedback involve in particular stochastic differential equations with delay.