Weak Dirichlet processes with a stochastic control perspective |
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Authors: | Fausto Gozzi Francesco Russo |
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Institution: | 1. Dipartimento di Scienze Economiche e Aziendali, Facolta’ di Economia, LUISS - Guido Carli, Viale Pola 12, I-00198 Roma, Italy;2. Université Paris 13, Institut Galilée, Mathématiques, 99, av. JB Clément, F-99430 Villetaneuse, France |
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Abstract: | The motivation of this paper is to prove verification theorems for stochastic optimal control of finite dimensional diffusion processes without control in the diffusion term, in the case where the value function is assumed to be continuous in time and once differentiable in the space variable (C0,1) instead of once differentiable in time and twice in space (C1,2), like in the classical results. For this purpose, the replacement tool of the Itô formula will be the Fukushima–Dirichlet decomposition for weak Dirichlet processes. Given a fixed filtration, a weak Dirichlet process is the sum of a local martingale M plus an adapted process A which is orthogonal, in the sense of covariation, to any continuous local martingale. The decomposition mentioned states that a C0,1 function of a weak Dirichlet process with finite quadratic variation is again a weak Dirichlet process. That result is established in this paper and it is applied to the strong solution of a Cauchy problem with final condition. |
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Keywords: | 60G05 60G44 60G48 60H05 60H10 60J60 35K15 35K55 35J15 93E20 |
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