When Are HJB-Equations in Stochastic Control of Delay Systems Finite Dimensional?

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Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations (2011, background): “There are some papers dealing with this subject in the stochastic Brownian case: we refer the reader to [5], [9], [10], [11], and [12], The paper [3] represents an extension of [11] to the case when the equation is driven by a Levy noise.”
Stochastic optimal control of delay equations arising in advertising models (2004, result): “For example, by the results of Larssen and Risebro [25], (19) reduces to a finite dimensional PDE if and only if a0 = −a1.”