Title: Stochastic functional differential equations and sensitivity to their initial path,
Abstract: We consider systems with memory represented by stochastic functional
differential equations. Substantially, these are stochastic differential
equations with coefficients depending on the past history of the process
itself. Such coefficients are hence defined on a functional space. Models with
memory appear in many applications ranging from biology to finance. Here we
consider the results of some evaluations based on these models (e.g. the prices
of some financial products) and the risks connected to the choice of these
models. In particular we focus on the impact of the initial condition on the
evaluations. This problem is known as the analysis of sensitivity to the
initial condition and, in the terminology of finance, it is referred to as the
Delta. In this work the initial condition is represented by the relevant past
history of the stochastic functional differential equation. This naturally
leads to the redesign of the definition of Delta. We suggest to define it as a
functional directional derivative, this is a natural choice. For this we study
a representation formula which allows for its computation without requiring
that the evaluation functional is differentiable. This feature is particularly
relevant for applications. Our formula is achieved by studying an appropriate
relationship between Malliavin derivative and functional directional
derivative. For this we introduce the technique of {\it randomisation of the
initial condition}.