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The dynamic programming equation for a stochastic volatility optimal control problem. (English) Zbl 1429.93414

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
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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
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