In the deterministic context a series of well established results
allow to reformulate delay differential equations (DDEs) as evolution
equations in infinite dimensional spaces. Several models in the
theoretical economic literature have been studied using this
reformulation. On the other hand, in the stochastic case only few
results of this kind are available and only for specific problems.
The contribution of the present letter is to present a way to
reformulate in infinite dimension a prototype controlled stochastic
DDE, where the control variable appears delayed in the diffusion
term. As application, we present a model for quadratic risk
minimization hedging of European options with execution delay and
a time-to-build model with shock.
Some comments concerning the possible employment of the dynamic
programming after the reformulation in infinite dimension conclude the