Stochastic optimal control with delays: $C^{1,\alpha}$-regularity of the value function and optimal synthesis

de Feo Filippo, Politecnico di Milano, Università Luiss Guido Carli

Stochastic optimal control theory focuses on governing a dynamical system, called the state equation, using a control process to optimize a specific performance measure. While classical optimal control problems involving Markovian stochastic differential equations have been extensively studied, many real-world applications necessitate the consideration of path-dependent non-Markovian dynamics. However, when dealing with path-dependent models, significant mathematical difficulties arise and the theory is not yet well developed. In this talk, we consider an optimal control problem of stochastic delay differential equations with delays only in the state. To regain Markovianity and use the dynamic programming approach, we lift the problem on a suitable Hilbert space. Then we characterize the value function of the problem as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation, which is a fully non-linear second-order partial differential equation on a Hilbert space with an unbounded operator. Since no regularity results are available for viscosity solutions of fully non-linear second-order HJB equations on Hilbert spaces, using a finite-dimensional reduction procedure and the regularity theory for finite-dimensional PDEs, we prove a partial $C^{1,\alpha}$-regularity result of the value function. When the diffusion is independent of the control, this regularity result allows to define a candidate optimal feedback control. However, due to the lack of $C^2$-regularity of the value function, we cannot prove a verification theorem using standard techniques based on Ito’s formula. Then, using a technical double approximation procedure, we construct regular approximants of the value function, which are supersolutions of perturbed HJB equations and regular enough so that they satisfy a non-smooth Ito’s formula. This allows us to prove a verification theorem and construct optimal feedback controls. The talk is based on the following manuscripts: \cite{defeo_federico_swiech} (in collaboration with S. Federico and A. Święch) and \cite{deFeoSwiech} (in collaboration with A. Święch).

Area: CS6 - Stochastic optimal control, BSDEs, and applications (Fulvia Confortola and Alessandro Calvia)

Keywords: Stochastic optimal control

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