Transition density estimation for stochastic differential equations via forward-reverse representations

The general reverse diffusion equations are derived and applied to the problem of transition density estimation of diffusion processes between two fixed states. For this problem we propose density estimation based on forward–reverse representations and show that this method allows essentially better results to be achieved than the usual kernel or projection estimation based on forward representations only.

[1]  D. Talay,et al.  The law of the Euler scheme for stochastic differential equations , 1996 .

[2]  B. Silverman.  Density estimation for statistics and data analysis , 1986 .

[3]  Mireille Bossy,et al.  A stochastic particle method for the McKean-Vlasov and the Burgers equation , 1997, Math. Comput..

[4]  L. Devroye,et al.  Nonparametric Density Estimation: The L 1 View. , 1985 .

[5]  C. D. Kemp,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[6]  Ü. Rannik,et al.  Evaluation of mean concentration and fluxes in turbulent flows by Lagrangian stochastic models , 2001 .

[7]  Denis Talay,et al.  The law of the Euler scheme for stochastic differential equations , 1996, Monte Carlo Methods Appl..

[8]  D. Thomson.  Criteria for the selection of stochastic models of particle trajectories in turbulent flows , 1987, Journal of Fluid Mechanics.

[9]  Arturo Kohatsu-Higa,et al.  High order Itô-Taylor approximations to heat kernels , 1997 .

[10]  J. Bretagnolle,et al.  Estimation des densités: risque minimax , 1978 .

[11]  R. Seydel.  Numerical Integration of Stochastic Differential Equations , 2004 .

[12]  Leslie Greengard,et al.  The Fast Gauss Transform , 1991, SIAM J. Sci. Comput..

[13]  David W. Scott,et al.  Multivariate Density Estimation: Theory, Practice, and Visualization , 1992, Wiley Series in Probability and Statistics.

[14]  G. N. Milstein,et al.  Monte Carlo construction of hedging strategies against multi-asset European claims , 2002 .

[15]  Nigel J. Newton.  Variance Reduction for Simulated Diffusions , 1994, SIAM J. Appl. Math..

[16]  E. Helfand.  Numerical integration of stochastic differential equations , 1979, The Bell System Technical Journal.

[17]  D. Talay,et al.  Expansion of the global error for numerical schemes solving stochastic differential equations , 1990 .

[18]  Karl Sabelfeld,et al.  Direct and Adjoint Monte Carlo Algorithms for the Footprint Problem , 1999, Monte Carlo Methods Appl..

[19]  W. Wagner.  Monte carlo evaluation of functionals of solutions of stochastic differential equations. variance reduction and numerical examples , 1988 .