Non-Parametric Estimation of Stochastic Differential Equations



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Parametric estimation techniques are commonly used, in academia and industry, to estimate the drift and diffusion of Stochastic Differential Equations (SDE). Their major limitation is that they require a functional form for the drift and diffusion terms to be implemented. Such assumptions increase the risk of misspecification. Non-parametric methods on the other hand allow the estimation of both components without a priori assumptions. In this study, we first use a non-parametric technique based on eigenvalues and eigenfunctions to study the reconstruction of the drift and diffusion. In a second part with another non-parametric method, based on conditional expectation, we study the estimation errors generated by the space-time discretization necessary for the estimation. The work on these two non-parametric estimations constitute the two parts of this thesis. In the first part, we motivate the use of non parametric techniques to model time series data in real applications, explain the spectral reconstruction, and propose a methodology to extend its use to processes commonly used in finance. A real world application on intraday data is presented. In the latter, the components of a SDE driving the crude oil price are reconstructed for the following two periods: 2010-2013 and 2015-2016. A mean reverting process is identified for the first period whereas a random walk hypothesis failed to be rejected for the second period. The reconstruction is sensitive to the sample size and the discretization. Estimating drift and diffusion from discretely sampled data is fraught with the potential for errors from space-time discretization. Therefore, in the second part of this thesis we study in the L2 sense the impact of the space-time discretization on the estimation errors of conditional expectation based estimators. We concentrate this analysis on the case of Ornstein -Uhlenbeck process. We found that for time series data observed at fixed interval, the choice of an optimal space discretization influences the rate at which the errors decay. When the sample size is known, an upper bound for the errors is obtained analytically and verified using numerical simulations. We also propose upper bounds for the case of unknown sample size.



Stochastic differential equations (SDE), Estimation