On the infinite-dimensional representation of stochastic controlled systems with delayed control in the diffusion term

Giorgio Fabbri (1) , Salvatore Federico (2)

1 EPEE - Centre d'Etudes des Politiques Economiques
2 Dipartimento di economia, management e metodi quantitativi

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 letter.

Keywords: Stochastic delay differential equations; Dynamic programming; Evolution equations in Hilbert space

Domains: Economics and Finance; Portfolio Management [q-fin.PM]

Giorgio Fabbri, Salvatore Federico. On the infinite-dimensional representation of stochastic controlled systems with delayed control in the diffusion term. Mathematical Economics Letters, 2014, 2 (3/4), pp.33-43. ⟨10.1515/mel-2014-0011⟩. ⟨hal-01038088⟩