# Stochastic functional differential equations and sensitivity to their initial path

## TL;DR

This paper studies stochastic functional differential equations with memory, focusing on how initial past history influences evaluations like financial prices, and introduces a new way to compute sensitivity (Delta) using a functional directional derivative and Malliavin calculus.

## Contribution

It proposes a novel definition of Delta as a functional directional derivative and develops a representation formula for its computation without differentiability assumptions.

## Key findings

- Derived a representation formula for the functional Delta.
- Linked Malliavin derivative to the functional directional derivative.
- Introduced the technique of randomisation of initial conditions.

## 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}.