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Laplace approximation of transition densities posed as Brownian expectations

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  • Markussen, Bo

Abstract

We construct the Laplace approximation of the Lebesgue density for a discrete partial observation of a multi-dimensional stochastic differential equation. This approximation may be computed integrating systems of ordinary differential equations. The construction of the Laplace approximation begins with the definition of the point of minimum energy. We show how such a point can be defined in the Cameron-Martin space as a maximum a posteriori estimate of the underlying Brownian motion given the observation of a finite-dimensional functional. The definition of the MAP estimator is possible via a renormalization of the densities of piecewise linear approximations of the Brownian motion. Using the renormalized Brownian density the Laplace approximation of the integral over all Brownian paths can be defined. The developed theory provides a method for performing approximate maximum likelihood estimation.

Suggested Citation

  • Markussen, Bo, 2009. "Laplace approximation of transition densities posed as Brownian expectations," Stochastic Processes and their Applications, Elsevier, vol. 119(1), pages 208-231, January.
  • Handle: RePEc:eee:spapps:v:119:y:2009:i:1:p:208-231
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    1. de Falco, Diego & Khandekar, Dinkar C., 1988. "Applications of white noise calculus to the computation of Feynman integrals," Stochastic Processes and their Applications, Elsevier, vol. 29(2), pages 257-266, September.
    2. Streit, L. & Hida, T., 1984. "Generalized Brownian functionals and the Feynman integral," Stochastic Processes and their Applications, Elsevier, vol. 16(1), pages 55-69, January.
    3. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    4. Baudoin, Fabrice, 0. "Conditioned stochastic differential equations: theory, examples and application to finance," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 109-145, July.
    5. Alexandros Beskos & Omiros Papaspiliopoulos & Gareth O. Roberts & Paul Fearnhead, 2006. "Exact and computationally efficient likelihood‐based estimation for discretely observed diffusion processes (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 333-382, June.
    6. Yacine Ait-Sahalia, 2002. "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-form Approximation Approach," Econometrica, Econometric Society, vol. 70(1), pages 223-262, January.
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