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Weak Dirichlet processes with a stochastic control perspective

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  • Gozzi, Fausto
  • Russo, Francesco

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

Suggested Citation

  • Gozzi, Fausto & Russo, Francesco, 2006. "Weak Dirichlet processes with a stochastic control perspective," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1563-1583, November.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:11:p:1563-1583
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    References listed on IDEAS

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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. Elena Issoglio & Francesco Russo, 2024. "Stochastic Differential Equations with Singular Coefficients: The Martingale Problem View and the Stochastic Dynamics View," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2352-2393, September.
    2. Bruno Bouchard & Gr'egoire Loeper & Xiaolu Tan, 2021. "A $C^{0,1}$-functional It\^o's formula and its applications in mathematical finance," Papers 2101.03759, arXiv.org.
    3. Fabbri, Giorgio & Russo, Francesco, 2017. "Infinite dimensional weak Dirichlet processes and convolution type processes," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 325-357.
    4. Fabbri, G. & Russo, F., 2017. "HJB equations in infinite dimension and optimal control of stochastic evolution equations via generalized Fukushima decomposition," Working Papers 2017-07, Grenoble Applied Economics Laboratory (GAEL).
    5. Leão, Dorival & Ohashi, Alberto, 2010. "Weak Approximations for Wiener Functionals," Insper Working Papers wpe_215, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.
    6. Bruno Bouchard & Grégoire Loeper & Xiaolu Tan, 2021. "A C^{0,1}-functional Itô's formula and its applications in mathematical finance," Working Papers hal-03105342, HAL.
    7. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    8. Issoglio, Elena & Jing, Shuai, 2020. "Forward–backward SDEs with distributional coefficients," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 47-78.
    9. Cristina Girolami & Giorgio Fabbri & Francesco Russo, 2014. "The covariation for Banach space valued processes and applications," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(1), pages 51-104, January.
    10. Bandini, Elena & Russo, Francesco, 2017. "Weak Dirichlet processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4139-4189.
    11. Bouchard, Bruno & Loeper, Grégoire & Tan, Xiaolu, 2022. "A ℂ0,1-functional Itô’s formula and its applications in mathematical finance," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 299-323.
    12. Giorgio Fabbri & Francesco Russo, 2016. "Infinite Dimensional Weak Dirichlet Processes and Convolution Type Processes," Working Papers halshs-01309384, HAL.
    13. Bruno Bouchard & Grégoire Loeper & Xiaolu Tan, 2022. "A C^{0,1}-functional Itô's formula and its applications in mathematical finance," Post-Print hal-03105342, HAL.

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