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Weak Dirichlet processes with a stochastic control perspective


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  • Gozzi, Fausto
  • Russo, Francesco


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. Applications to the proof of verification theorems will be addressed in a companion paper.

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Bibliographic Info

Article provided by Elsevier in its journal Stochastic Processes and their Applications.

Volume (Year): 116 (2006)
Issue (Month): 11 (November)
Pages: 1563-1583

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Handle: RePEc:eee:spapps:v:116:y:2006:i:11:p:1563-1583

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Keywords: Stochastic calculus via regularization Weak Dirichlet processes Stochastic optimal control Cauchy problem for parabolic partial differential equations;


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  1. Russo, Francesco & Vallois, Pierre, 1995. "The generalized covariation process and Ito formula," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 81-104, September.
  2. Errami, Mohammed & Russo, Francesco, 2003. "n-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes," Stochastic Processes and their Applications, Elsevier, vol. 104(2), pages 259-299, April.
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Cited by:
  1. Cristina Di Girolami & Giorgio Fabbri & Francesco Russo, 2013. "The covariation for Banach space valued processes and applications," Documents de recherche 13-01, Centre d'Études des Politiques Économiques (EPEE), Université d'Evry Val d'Essonne.
  2. Leão, Dorival & Ohashi, Alberto, 2010. "Weak Approximations for Wiener Functionals," Insper Working Papers wpe_215, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.


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