Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
STOCHASTIC INTEGRALS AND ABELIAN PROCESSES CLAUDIO ALBANESE
 

Summary: STOCHASTIC INTEGRALS AND ABELIAN PROCESSES
CLAUDIO ALBANESE
Abstract. We study triangulation schemes for the joint kernel of a diffusion process with
uniformly continuous coefficients and an adapted, non-resonant Abelian process. The proto-
typical example of Abelian process to which our methods apply is given by stochastic integrals
with uniformly continuous coefficients. The range of applicability includes also a broader class
of processes of practical relevance, such as the sup process and certain discrete time summa-
tions we discuss.
We discretize the space coordinate in uniform steps and assume that time is either con-
tinuous or finely discretized as in a fully explicit Euler method and the Courant condition is
satisfied. We show that the Fourier transform of the joint kernel of a diffusion and a stochastic
integral converges in a uniform graph norm associated to the Markov generator. Convergence
also implies smoothness properties for the Fourier transform of the joint kernel. Stochastic
integrals are straightforward to define for finite triangulations and the convergence result gives
a new and entirely constructive way of defining stochastic integrals in the continuum. The
method relies on a reinterpretation and extension of the classic theorems by Feynman-Kac,
Girsanov, Ito and Cameron-Martin, which are also reobtained.
We make use of a path-wise analysis without relying on a probabilistic interpretation. The
Fourier representation is needed to regularize the hypo-elliptic character of the joint process
of a diffusion and an adapted stochastic integral. The argument extends as long as the Fourier

  

Source: Albanese, Claudio - Department of Mathematics, King's College London

 

Collections: Mathematics