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n-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes

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  • Errami, Mohammed
  • Russo, Francesco

Abstract

In this paper, we introduce first a natural generalization of the concept of Dirichlet process, providing significant examples. The second important tool concept is the n-covariation and the related n-variation. The n-variation of a continuous process and the n-covariation of a vector of continuous processes, are defined through a regularization procedure. We calculate explicitly the n-variation process, when it exists, of a martingale convolution. For processes having finite cubic variation, a basic stochastic calculus is developed. We prove an Itô formula and we study existence and uniqueness of the solution of a stochastic differential equation, in a symmetric-Stratonovich sense, with respect to those processes.

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  • 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.
    • Handle: RePEc:eee:spapps:v:104:y:2003:i:2:p:259-299
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    References listed on IDEAS

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    1. Bardina, Xavier & Jolis, Maria, 2000. "Weak convergence to the multiple Stratonovich integral," Stochastic Processes and their Applications, Elsevier, vol. 90(2), pages 277-300, December.
    2. 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.
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    1. 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.
    2. 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).
    3. 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.
    4. Russo, Francesco & Tudor, Ciprian A., 2006. "On bifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 830-856, May.
    5. 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.
    6. Bandini, Elena & Russo, Francesco, 2017. "Weak Dirichlet processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4139-4189.
    7. 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.
    8. Giorgio Fabbri & Francesco Russo, 2016. "Infinite Dimensional Weak Dirichlet Processes and Convolution Type Processes," Working Papers halshs-01309384, HAL.

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