Barbu, Viorel The dynamic programming equation for a stochastic volatility optimal control problem. (English) Zbl 1429.93414 Automatica 107, 119-124 (2019). Summary: In this note, one constructs a distributional solution to the \(d\)-dimensional dynamic programming equation, \(d\geq 3\), for an optimal control problem governed by a stochastic volatility model. The approach is based on nonlinear semigroup theory in the space \(L^1(\mathbb{R}^d)\). MSC: 93E20 Optimal stochastic control 93B52 Feedback control 90C39 Dynamic programming 37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations Keywords:Brownian motions; accretive operator; optimal feedback controller PDFBibTeX XMLCite \textit{V. Barbu}, Automatica 107, 119--124 (2019; Zbl 1429.93414) Full Text: DOI References: [1] Barbu, V., Nonlinear differential equations of monotone types in banach spaces (2010), Springer Science & Business Media · Zbl 1197.35002 [2] Barbu, V., Generalized solutions to nonlinear Fokker-Planck eq1uations, Journal of Differential Equations, 261, 2446-2471 (2016) · Zbl 1342.35395 [3] Barbu, V.; Benazzoli, C.; Di Persio, L., Mild solutions to the dynamic programming equation for stochastic optimal control problems, Automatica, 93, 520-526 (2018) · Zbl 1400.93328 [4] Barbu, V.; Benazzoli, C.; Di Persio, L., Feedback optimal controllers for the heston model, Applied Mathematics and Optimization (2018) [5] Benilan, Ph.; Brezis, H.; Crandall, M. G., A semilinear equation in \(L^1(R^n)\), Annali Scuola Normale Sup. Pisa, 2, 4, 523-555 (1975) · Zbl 0314.35077 [6] Benilan, Ph.; Crandall, M. G., The continuous dependence on \(\varphi\) of solutions of \(\mu_t - \Delta \varphi(u) = 0\), Indiana University Mathematics Journal, 30, 161-177 (1981) · Zbl 0482.35012 [7] Brezis, H., Functional analysis Sobolev spaces and partial differential equations (2011), Springer Science + Bussiness Media · Zbl 1220.46002 [8] Crandall, M. G.; Ishii, H.; Lions, P. L., User guide to viscosity solutions of second order partial differential equations, Bulletin American Mathematical Society, 27, 1, 1-67 (1992) · Zbl 0755.35015 [9] Fleming, W. H.; Rishel, R. W., Deterministic and stochastic optimal control, Vol. I (2012), Springer Science & Business Media [10] Oksendal, B., Stochastic differential equations (2009), Springer Verlag, Heidelberg: Springer Verlag, Heidelberg New York [11] Shreve, S. E., Stochastic calculus for finance. Continuous time models (2004), Springer-Finance · Zbl 1068.91041 [12] Stein, J., Stochastic optimal control, international finance and debt crises, Vol. 1 (2006), Oxford University Press: Oxford University Press Oxford · Zbl 1181.91004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.