Nonparametric estimation of forward-backward stochastic differential equations with random terminal time

Abstract

This paper investigates the nonparametric estimation of the functional coefficients of the forward-backward stochastic differential equations with random terminal time, focusing on both local constant and local linear estimators. We establish the asymptotic properties of these estimators under both long observation time spans and short sampling intervals, providing precise expressions for the bias and variance terms. Moreover, we propose an empirical likelihood method to construct data-driven confidence intervals for these functional coefficients. We conduct numerical simulations to examine the finite-sample properties of the estimators and to compare the performance of the empirical likelihood method with the conventional approach for constructing confidence intervals based on asymptotic normality.

Keywords

Backward stochastic differential equations / Nonparametric estimation / Asymptotic normality / Empirical likelihood

Acknowledgements

This work was supported by the NSF of China (Grant No.12071071) and the Jiangsu Provincial Scientific Research Center of Applied Mathematics (Grant No. BK20233002).

The author is very grateful to Professor Ying Hu for valuable discussions. Besides, the author thanks two reviewers since their suggestions and comments allowed her to improve the results and the presentation of this paper.

References

[1] Aït-Sahalia, Y. and Park, J. Y., Bandwidth selection and asymptotic properties of local nonparametric estimators in possibly nonstationary continuous-time models, Journal of Econometrics, 2016, 192(1): 119-138.

[2] Bandi, F. M. and Phillips, P. C., Fully nonparametric estimation of scalar diffusion models, Econometrica, 2003, 71(1): 241-283.

[3] Bandi, F. M. and Moloche, G., On the functional estimation of multivariate diffusion processes, Econometric Theory, 2018, 34(4): 896-946.

[4] Bandi, F. M. and Renò R., Nonparametric stochastic volatility, Econometric Theory, 2018, 34(6): 1207-1255.

[5] Borkovec, M. and Klüppelberg, C., Extremal behavior of diffusion models in finance, Extremes, 1998, 1(1): 47-80.

[6] Chen, X. and Lin, L., Nonparametric estimation for FBSDEs models with applications in finance, Communications in Statistics-Theory and Methods, 2010, 39(14): 2492-2514.

[7] Fan, J. and Zhang, C., A reexamination of diffusion estimators with applications to financial model validation, Journal of the American Statistical Association, 2003, 98(461): 118-134.

[8] Florens-Zmirou, D., On estimating the diffusion coefficient from discrete observations, Journal of Applied Probability, 1993, 30(4): 790-804.

[9] Jacod, J., Non-parametric kernel estimation of the coefficient of a diffusion, Scandinavian Journal of Statistics, 2000, 27(1): 83-96.

[10] Kutoyants, Y. A., Approximation of the solution of the backward stochastic differential equation: Small noise, large sample and high frequency cases, Proceedings of the Steklov Institute of Mathematics, 2014, 287(1): 133-154.

[11] Kutoyants, Y. A., On approximation of BSDE and multi-step MLE-processes, Probability, Uncertainty and Quantitative Risk, 2016, 1: 4.

[12] Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 1990, 14(1): 55-61.

[13] Pardoux, E., Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, In: DecreusefondL., ØksendalB., GjerdeJ. and ÜstünelA. S.(eds.), Stochastic Analysis and Related Topics VI. Progress in Probability, Vol. 42, Birkhäuser, Boston, MA, 1998.

[14] Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastics Reports, 1991, 37(1-2): 61-74.

[15] Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion, Springer, Berlin, Heidelberg, 1999.

[16] Stanton, R., A nonparametric model of term structure dynamics and the market price of interest rate risk, The Journal of Finance, 1997, 52(5): 1973-2002.

[17] Song, Y. and Wang, H., Central limit theorems of local polynomial threshold estimator for diffusion processes with jumps, Scandinavian Journal of Statistics, 2018, 45(3): 644-681.

[18] Su, Y. and Lin, L., Semi-parametric estimation for forward-backward stochastic differential equations, Communications in Statistics-Theory and Methods, 2009, 38(11): 1759-1775.

[19] Xu, K. L., Empirical likelihood-based inference for nonparametric recurrent diffusions, Journal of Econometrics, 2009, 153(1): 65-82.

[20] Xu, K. L., Reweighted functional estimation of diffusion models, Econometric Theory, 2010, 26(2): 541-563.

[21] Zhang, D., Local linear estimators for reflected diffusions, Journal of Nonparametric Statistics, 2024, https://doi.org/10.1080/10485252.2024.2426099.