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Quadratic covariation estimates in non-smooth stochastic calculus

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  • 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
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    References listed on IDEAS

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    1. 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.
    2. Day, Martin V., 1995. "On the exit law from saddle points," Stochastic Processes and their Applications, Elsevier, vol. 60(2), pages 287-311, December.
    3. 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.
    4. 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.
    5. 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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