Advertisement

Numerical Analysis and Applications

, Volume 1, Issue 4, pp 314–331 | Cite as

Estimation of derivatives with respect to parameters of a functional of a diffusion process moving in a domain with absorbing boundary

  • S. A. Gusev
Article

Abstract

A statistical method is proposed for estimating derivatives with respect to parameters of a functional of a diffusion process moving in a domain with absorbing boundary. The functional considered defines the probability representation of the solution of a corresponding parabolic first boundary-value problem. The problem posed is tackled by numerically solving stochastic differential equations (SDE) using the Euler method. An error of the proposed method is evaluated, and estimates of the variance of the resultant parametric derivatives are given. Some numerical results are presented.

Key words

diffusion process stochastic differential equations absorbing boundary derivatives with respect to parameters Euler method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Gobet, E. and Munos, R., Sensitivity Analysis Using Ito-Malliavin Calculus and Martingales. Application to Stochastic Optimal Control, SIAM J. Control Optim., 2005, vol. 43, no. 5, pp. 1676–1713.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Gobet, E. and Munos, R., Sensitivity Analysis Using Ito-Malliavin Calculus and Martingales. Numerical Implementation; http://www.cmap.polytechnique.fr/preprint/repository/520.pdf.
  3. 3.
    Gobet, E., Costantini, C., and Karoui, N.E., Boundary Sensitivities for Diffusion Processes in Time Dependent Domains, Appl. Math. Optim., 2006, vol. 54, no. 2, pp. 159–187.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fournie, E. et al., Applications of Malliavin Calculus to Monte Carlo Methods in Finance, Finance Stochast., 1999, no. 3, pp. 391–412.Google Scholar
  5. 5.
    Montero, M. and Kohatsu-Higa, A., Malliavin Calculus Applied to Finance, City: Physica A 320, 2003, pp. 548–570.Google Scholar
  6. 6.
    Gusev, S.A., Monte Carlo Estimates of Derivatives with Respect to Parameters of the Solution of the Parabolic Equation Based on the Numerical Solution of Stochastic Differential Equations, Sib. Zh. Vych. Mat., 2005, vol. 8, no. 4, pp. 297–306.MATHGoogle Scholar
  7. 7.
    Ladyzhenskaya, O.A., Solonnikov, V.A., and Ural’tseva, N.N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa (Linear and Quasilinear Parabolic-Type Equations), Moscow: Nauka, 1967.Google Scholar
  8. 8.
    Gikhman, I.I. and Skorokhod, A.V., Stokhasticheskie differentsial’nye uravneniya (Stochastic Differential Equations), Kiev: Naukova Dumka, 1968.Google Scholar
  9. 9.
    Gobet, E. and Menozzi, S., Stopped Diffusion Process: Overshoots and Boundary Correction, http://hal.archives-ouvertes.fr/hal-00157975/fr/.
  10. 10.
    Dokuchaev, N., Estimates for Distances between First Exit Times via Parabolic Equations in Unbounded Cylinders, Probab. Th. Relat. Fields, 2004, vol. 129, pp. 290–314.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2008

Authors and Affiliations

  1. 1.Institute of Computational Mathematics and Mathematical Geophysics, Siberian BranchRussian Academy of SciencesNovosibirskRussia

Personalised recommendations