2010/05/20 by Cristina Di Girolami, Di Girolami, Cristina, Francesco Russo +1
Economics, Econometrics and Finance · Mathematics · #Complex Systems and Time Series Analysis #FOS: Mathematics #Financial Risk and Volatility Modeling #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis #Probability (math.PR) #Stochastic processes and financial applications #math.PR
paper · pdf · doi:10.48550/arxiv.1005.3608
openalex publication_date 2010/05/20 · arxiv created 2010/10/26 · arxiv updated 2010/10/27 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We provide a suitable framework for the concept of finite quadratic variation for processes with values in a separable Banach space B using the language of stochastic calculus via regularizations, introduced in the case B= \R by the second author and P. Vallois. To a real continuous process X we associate the Banach valued process X(⋅), called \it window process, which describes the evolution of X taking into account a memory τ>0. The natural state space for X(⋅) is the Banach space of continuous functions on [-τ,0]. If X is a real finite quadratic variation process, an appropriated Itô formula is presented, from which we derive a generalized Clark-Ocone formula for non-semimartingales having the same quadratic variation as Brownian motion. The representation is based on solutions of an infinite dimensional PDE.