# Infinite dimensional stochastic calculus via regularization.

april, 2010

Publication type:

Lecture note

Journal:

Arxiv

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Keywords :

Calculus via regularization, Infinite dimensional analysis,
Fractional Brownian motion, Tensor analysis, Clark-Ocone formula,
Dirichlet processes; Itô formula; Quadratic variation,
Hedging theory without semimartingales.

Abstract:

This small monograph develops some aspects of stochastic calculus via regularization
to Banach valued processes.
An original concept
of $\chi$-quadratic variation is introduced, where $\chi$ is a
subspace of the dual of a tensor product $B \otimes B$ where $B$ is the
values space of some process $X$ process. Particular interest
is devoted to the case when $B$ is
the space of real continuous
functions defined on $[-\tau,0]$, $\tau>0$.
It\^o formulae and stability of finite $\chi$-quadratic variation
processes are established. Attention is deserved to a
finite real quadratic
variation (for instance Dirichlet, weak Dirichlet) process $X$.
The $C([-\tau,0])$-valued process $X(\cdot)$ defined by
$X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$, is called {\it window} process.
Let $T >0$. If $X$ is a finite quadratic variation process
such that $[X]_t = t$ and $h = H(X_T(\cdot))$ where $H:C([-T,0])\longrightarrow \R$
is $L^{2}([-T,0])$-smooth or $H$ non smooth but finitely based
it is possible to represent
$h$ as a sum of a real $H_{0}$ plus a forward integral of type $\int_0^T \xi d^-X$ where $H_{0}$ and $\xi$ are explicitly given.
This representation result will be strictly linked with a
function $u:[0,T]\times C([-T,0])\longrightarrow \R$ which in general
solves an infinite dimensional partial differential equation
with the property
$H_{0}=u(0, X_{0}(\cdot))$, $\xi_{t}=D^{\delta_{0}}u(t, X_{t}(\cdot)):=Du(t, X_{t}(\cdot))(\{0\})$.
This decomposition generalizes the Clark-Ocone formula
which is true when $X$ is the standard Brownian motion $W$.
The financial perspective of this work is related to hedging theory of path dependent
options without semimartingales.

BibTeX:

@misc{DiG-Rus-2010, title={Infinite dimensional stochastic calculus via regularization. }, journal={Arxiv }, year={2010 }, month={4}, comment={{umatype:'cours'}}, }