Geodesic
Systems of Bellman Equations to Stochastic Differential Games with Discount Control
Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 663-681

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We consider two dimensional diagonal elliptic systems $\Delta u + au = H(x, u, \nabla u)$ which arise from stochastic differential games with discount control. The Hamiltonians $H$ have quadratic growth in $\nabla u$ and a special structure which has notyet been covered by regularity theory. Without smallness condition on $H$, the existence of a regular solution is established. 
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     author = {Bensoussan, Alain and Frehse, Jens},
     title = {Systems of {Bellman} {Equations} to {Stochastic} {Differential} {Games} with {Discount} {Control}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {663--681},
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     volume = {Ser. 9, 1},
     number = {3},
     year = {2008},
     zbl = {1190.49045},
     mrnumber = {2455338},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a8/}
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Bensoussan, Alain; Frehse, Jens. Systems of Bellman Equations to Stochastic Differential Games with Discount Control. Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 663-681. http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a8/