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Weak approximation of martingale representations

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  • Cont, Rama
  • Lu, Yi

Abstract

We present a systematic method for computing explicit approximations to martingale representations for a large class of Brownian functionals. The approximations are obtained by computing a directional derivative of the weak Euler scheme and yield a consistent estimator for the integrand in the martingale representation formula for any square-integrable functional of the solution of an SDE with path-dependent coefficients. Explicit convergence rates are derived for functionals which are Lipschitz-continuous in the supremum norm. Our results require neither the Markov property, nor any differentiability conditions on the functional or the coefficients of the stochastic differential equations involved.

Suggested Citation

  • Cont, Rama & Lu, Yi, 2016. "Weak approximation of martingale representations," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 857-882.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:3:p:857-882
    DOI: 10.1016/j.spa.2015.10.002
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    References listed on IDEAS

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    1. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
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    3. Jakša Cvitanić & Jin Ma & Jianfeng Zhang, 2003. "Efficient Computation of Hedging Portfolios for Options with Discontinuous Payoffs," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 135-151, January.
    4. Chen, Nan & Glasserman, Paul, 2007. "Malliavin Greeks without Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1689-1723, November.
    5. Davis, Mark H.A. & Johansson, Martin P., 2006. "Malliavin Monte Carlo Greeks for jump diffusions," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 101-129, January.
    6. Elliott, Robert J. & Kohlmann, Michael, 1988. "A short proof of a martingale representation result," Statistics & Probability Letters, Elsevier, vol. 6(5), pages 327-329, April.
    7. Arturo Kohatsu & Montero Miquel, 2003. "Malliavin calculus in finance," Economics Working Papers 672, Department of Economics and Business, Universitat Pompeu Fabra.
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    Cited by:

    1. Dmitry Kramkov & Sergio Pulido, 2016. "Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model," Post-Print hal-01181147, HAL.
    2. Cont, Rama & Kalinin, Alexander, 2020. "On the support of solutions to stochastic differential equations with path-dependent coefficients," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2639-2674.
    3. Privault, N. & Yam, S.C.P. & Zhang, Z., 2019. "Poisson discretizations of Wiener functionals and Malliavin operators with Wasserstein estimates," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3376-3405.
    4. Marc Sabate-Vidales & David v{S}iv{s}ka & Lukasz Szpruch, 2020. "Solving path dependent PDEs with LSTM networks and path signatures," Papers 2011.10630, arXiv.org.

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