IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v116y2006i11p1563-1583.html
   My bibliography  Save this article

Weak Dirichlet processes with a stochastic control perspective

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(06)00051-2
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. 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).
    4. Leão, Dorival & Ohashi, Alberto, 2010. "Weak Approximations for Wiener Functionals," Insper Working Papers wpe_215, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.
    5. 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.
    6. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    7. Issoglio, Elena & Jing, Shuai, 2020. "Forward–backward SDEs with distributional coefficients," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 47-78.
    8. 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.
    9. Bandini, Elena & Russo, Francesco, 2017. "Weak Dirichlet processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4139-4189.
    10. 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.
    11. Giorgio Fabbri & Francesco Russo, 2016. "Infinite Dimensional Weak Dirichlet Processes and Convolution Type Processes," Working Papers halshs-01309384, HAL.
    12. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Bandini, Elena & Russo, Francesco, 2017. "Weak Dirichlet processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4139-4189.
    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. 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.
    5. Bernt {O}ksendal & Elin R{o}se, 2015. "A white noise approach to insider trading," Papers 1508.06376, arXiv.org.
    6. Russo, Francesco & Tudor, Ciprian A., 2006. "On bifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 830-856, May.
    7. Almada Monter, Sergio Angel, 2015. "Quadratic covariation estimates in non-smooth stochastic calculus," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 343-361.
    8. Blandine, Bérard Bergery & Pierre, Vallois, 2008. "Approximation via regularization of the local time of semimartingales and Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 118(11), pages 2058-2070, November.
    9. 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.
    10. 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).
    11. Russo, Francesco & Vallois, Pierre, 1998. "Product of two multiple stochastic integrals with respect to a normal martingale," Stochastic Processes and their Applications, Elsevier, vol. 73(1), pages 47-68, January.
    12. Giorgio Fabbri & Francesco Russo, 2016. "Infinite Dimensional Weak Dirichlet Processes and Convolution Type Processes," Working Papers halshs-01309384, HAL.
    13. Čoupek, Petr & Duncan, Tyrone E. & Pasik-Duncan, Bozenna, 2022. "A stochastic calculus for Rosenblatt processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 853-885.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:116:y:2006:i:11:p:1563-1583. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.