%0 Book Section %1 nunno2013chaos %A Di Nunno, Giulia %A Sjursen, Steffen %B Seminar on Stochastic Analysis, Random Fields and Applications VII %D 2013 %E Dalang, Robert C. %E Dozzi, Marco %E Russo, Francesco %I Springer Basel %K Poisson_process Poisson_random_field chaos_expansion orthogonal_polynomials %P 23-54 %R 10.1007/978-3-0348-0545-2_2 %T On Chaos Representation and Orthogonal Polynomials for the Doubly Stochastic Poisson Process %U http://dx.doi.org/10.1007/978-3-0348-0545-2_2 %V 67 %X In an L 2-framework, we study various aspects of stochastic calculus with respect to the centered doubly stochastic Poisson process. We introduce an orthogonal basis via multilinear forms of the value of the random measure and we analyze the chaos representation property. We review the structure of non-anticipating integration for martingale random fields and in this framework we study non-anticipating differentiation. We present integral representation theorems where the integrand is explicitly given by the non-anticipating derivative. Stochastic derivatives of anticipative nature are also considered: The Malliavin type derivative is put in relationship with another anticipative derivative operator here introduced. This gives a new structural representation of the Malliavin derivative based on simple functions. Finally we exploit these results to provide a Clark–Ocone type formula for the computation of the nonanticipating derivative. %@ 978-3-0348-0544-5

@incollection{nunno2013chaos, abstract = { In an L 2-framework, we study various aspects of stochastic calculus with respect to the centered doubly stochastic Poisson process. We introduce an orthogonal basis via multilinear forms of the value of the random measure and we analyze the chaos representation property. We review the structure of non-anticipating integration for martingale random fields and in this framework we study non-anticipating differentiation. We present integral representation theorems where the integrand is explicitly given by the non-anticipating derivative. Stochastic derivatives of anticipative nature are also considered: The Malliavin type derivative is put in relationship with another anticipative derivative operator here introduced. This gives a new structural representation of the Malliavin derivative based on simple functions. Finally we exploit these results to provide a Clark–Ocone type formula for the computation of the nonanticipating derivative. }, added-at = {2015-04-10T02:02:01.000+0200}, author = {Di Nunno, Giulia and Sjursen, Steffen}, biburl = {https://www.bibsonomy.org/bibtex/23a454617f0721f21fcd7d5e39ed0944b/peter.ralph}, booktitle = {Seminar on Stochastic Analysis, Random Fields and Applications VII}, doi = {10.1007/978-3-0348-0545-2_2}, editor = {Dalang, Robert C. and Dozzi, Marco and Russo, Francesco}, interhash = {7253b6e146b5049fe88591e0826e790b}, intrahash = {3a454617f0721f21fcd7d5e39ed0944b}, isbn = {978-3-0348-0544-5}, keywords = {Poisson_process Poisson_random_field chaos_expansion orthogonal_polynomials}, language = {English}, pages = {23-54}, publisher = {Springer Basel}, series = {Progress in Probability}, timestamp = {2015-04-10T02:02:01.000+0200}, title = {On Chaos Representation and Orthogonal Polynomials for the Doubly Stochastic {Poisson} Process}, url = {http://dx.doi.org/10.1007/978-3-0348-0545-2_2}, volume = 67, year = 2013 }