Exponentials, diffusions, finance, entropy and information

This monograph presents some interconnected topics arising from the fields of continuous-time stochastic processes, finance, generalized entropy concepts, statistical decision and information theory, and potential theory. Designed to be readable for anyone with a solid stochastic background, the text is divided into three parts. The first part deals with widely-used models for describing the time-evolution of stochastic phenomena, namely Markov processes, and corresponding exponentials of “time-additive” functionals. Such quantities appear e. g. in the solution-construction of stochastic resp. ordinary resp. partial differential equations, in models of accumulations of rewards (such as interest rates) or environmental damage costs, in time-averages of the kinetic energy of diffusion particles, and many other situations. The second part investigates some construction methods of diffusion processes by means of special “time-additive” functionals. The last part is devoted to the study of properties and applications of these diffusion processes, for instance in connection with financial abitrage theory and option pricing, optimal statistical decisions concerning the risk of choosing a wrong description model, generalizations of the widely-used concept of relative entropy (cross-entropy measure, Kullback-Leibler information), and the quantification of “closeness” between two stochastic models.