Browsing by Author "Frehse, J."
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Item Parabolic Bellman Equations with Risk Control(Society for Industrial and Applied Mathematics Publications) Bensoussan, Alain; Breit, D.; Frehse, J.; 0000000113231180 (Bensoussan, A); 0000-0003-0743-498X (Bensoussan, A); Bensoussan, AlainWe consider stochastic optimal control problems with an additional term representing the variance of the control functions. The latter one may serve as a risk control. We present and treat the problem in a purely analytical way via a Vlasov-McKean functional and Bellman equations with mean field dependence. We obtain global existence and, essentially, optimal global regularity for the solutions of the Bellman equation and the minimizing control. Surprisingly, the risk term simplifies the analysis to a certain extend.Item Stochastic Differential Games with a Varying Number of Players(American Institute of Mathematical Sciences) Bensoussan, Alain; Frehse, J.; Grün, C.; 0000 0001 1323 1180 (Bensoussan, A); 2002119562 (Bensoussan, A)We consider a non zero sum stochastic differential game with a maximum n players, where the players control a diffusion in order to minimisena certain cost functional. During the game it is possible that present players may die or new players may appear. The death, respectively the birth time of a player is exponentially distributed with intensities that depend on the diffusion and the controls of the players who are alive. We show how the game is related to a system of partial differential equations with a special coupling in the zero order terms. We provide an existence result for solutions in appropriate spaces that allow to construct Nash optimal feedback controls. The paper is related to a previous result in a similar setting for two players leading to a parabolic system of Bellman equations [4]. Here, we study the elliptic case (infinite horizon) and present the generalisation to more than two players.