On coupled systems of Kolmogorov equations with applications to stochastic differential games

Addona, Davide; Angiuli, Luciana; Lorenzi, Luca; Tessitore, Gianmario

doi:10.1051/cocv/2016019

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On coupled systems of Kolmogorov equations with applications to stochastic differential games

Addona, Davide ¹ ; Angiuli, Luciana ² ; Lorenzi, Luca ³ ; Tessitore, Gianmario ¹

¹ Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via Cozzi 55, 20125 Milano, Italy.
² Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, Via per Arnesano, 73100 Lecce, Italy.
³ Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 937-976

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Résumé

We prove that a family of linear bounded evolution operators ${(𝐆 (t, s))}_{t \geq s \in I}$ can be associated, in the space of vector-valued bounded and continuous functions, to a class of systems of elliptic operators $𝒜$ with unbounded coefficients defined in $I \times ℝ^{d}$ (where $I$ is a right-halfline or $I = ℝ$ ) all having the same principal part. We establish some continuity and representation properties of ${(𝐆 (t, s))}_{t \geq s \in I}$ and a sufficient condition for the evolution operator to be compact in $C_{b} (ℝ^{d}; ℝ^{m})$ . We prove also a uniform weighted gradient estimate and some of its more relevant consequence.

MR Zbl

DOI : 10.1051/cocv/2016019

Classification : 35K45, 35K58, 47B07, 60H10, 91A15
Keywords: Nonautonomous parabolic systems, unbounded coefficients, evolution operators, compactness, gradient estimates, semilinear systems, stochastic games

Affiliations des auteurs :

Addona, Davide ¹ ; Angiuli, Luciana ² ; Lorenzi, Luca ³ ; Tessitore, Gianmario ¹

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     title = {On coupled systems of {Kolmogorov} equations with applications to stochastic differential games},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {937--976},
     publisher = {EDP-Sciences},
     volume = {23},
     number = {3},
     year = {2017},
     doi = {10.1051/cocv/2016019},
     zbl = {1371.35144},
     mrnumber = {3660455},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2016019/}
}

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Addona, Davide; Angiuli, Luciana; Lorenzi, Luca; Tessitore, Gianmario. On coupled systems of Kolmogorov equations with applications to stochastic differential games. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 937-976. doi: 10.1051/cocv/2016019

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