IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v125y2015i1p343-361.html
   My bibliography  Save this article

Quadratic covariation estimates in non-smooth stochastic calculus

Author

Listed:
  • Almada Monter, Sergio Angel

Abstract

Given a Brownian Motion W, in this paper we study the asymptotic behavior, as ε→0, of the quadratic covariation between f(εW) and W in the case in which f is not smooth. Among the main features discovered is that the speed of the decay in the case f∈Cα is at least polynomial in ε and not exponential as expected. We use a recent representation as a backward–forward Itô integral of [f(εW),W] to prove an ε-dependent approximation scheme which is of independent interest. We get the result by providing estimates to this approximation. The results are then adapted and applied to generalize the results of Almada Monter and Bakhtin (2011) and Bakhtin (2011) related to the small noise exit from a domain problem for the saddle case.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:1:p:343-361
    DOI: 10.1016/j.spa.2014.09.005
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414914002105
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2014.09.005?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    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. Day, Martin V., 1995. "On the exit law from saddle points," Stochastic Processes and their Applications, Elsevier, vol. 60(2), pages 287-311, December.
    2. Moret, S. & Nualart, D., 2001. "Generalization of Itô's formula for smooth nondegenerate martingales," Stochastic Processes and their Applications, Elsevier, vol. 91(1), pages 115-149, January.
    3. Bakhtin, Yuri, 2008. "Exit asymptotics for small diffusion about an unstable equilibrium," Stochastic Processes and their Applications, Elsevier, vol. 118(5), pages 839-851, May.
    4. 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.
    5. 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.
    Full references (including those not matched with items on IDEAS)

    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. Bernt {O}ksendal & Elin R{o}se, 2015. "A white noise approach to insider trading," Papers 1508.06376, arXiv.org.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. Bandini, Elena & Russo, Francesco, 2017. "Weak Dirichlet processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4139-4189.
    7. 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.
    8. Ohashi, Alberto & Simas, Alexandre B., 2015. "A note on the sharp Lp-convergence rate of upcrossings to the Brownian local time," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 137-141.
    9. 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:125:y:2015:i:1:p:343-361. 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.