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We study stochastic delay differential equations (SDDE) where thecoefficients depend on the moving averages of the state process. As a firstcontribution, we provide sufficient conditions under which a linear pathfunctional of the solution of a SDDE admits a finite-dimensional Markovianrepresentation. As a second contribution, we show how approximatefinite-dimensional Markovian representations may be constructed when theseconditions are not satisfied, and provide an estimate of the errorcorresponding to these approximations. These results are applied to optimalcontrol and optimal stopping problems for stochastic systems with delay.

Finite-dimensional representations for controlled diffusions with delay