Resumen de Stochastic calculus and analytic characteristic functions: applications to finance
In this dissertation we have mainly studied probabilistic properties of stochastic processes through its characteristic function.
The first part proves that the log--spot in the Heston model has a infinitely smooth density and gives an expression of this density as an infinite convolution of Bessel type densities. Such properties are deduced from a factorization of the characteristic function, mainly obtained through an analysis of the complex moment generating function.
The second part proves that certain quotients of entire functions are characteristic functions. Under some conditions, the probability law corresponding to a characteristic function of that type has a density which can be expressed as a generalized Dirichlet series, whose coefficients are given in terms of the involved entire functions. As particular cases, conditions for the convergence of an infinite convolution of exponential or Laplace laws are deduced. This studies provide non trivial examples of meromorphic Lévy processes.
Finally, multiple integrals with respect to continuous centered Gaussian processes are considered to generalize the Lévy area. We showed that under certain condition on the finite variation of their covariance functions, the Lévy area can be defined as a double Wiener--Ito integral with respect to an isonormal Gaussian process. Moreover, some properties of the characteristic function of that generalized Lévy area are studied.
