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On the infinite-dimensional representation of stochastic controlled systems with delayed control in the diffusion term. (English) Zbl 1327.39009

Summary: 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.

MSC:

39A50 Stochastic difference equations
34K30 Functional-differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
60H99 Stochastic analysis
49L20 Dynamic programming in optimal control and differential games
91G20 Derivative securities (option pricing, hedging, etc.)
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