Abstract:
We present a review of some recently obtained results on estimation of the solution of a backward stochastic differential equation (BSDE) in the Markovian case. We suppose that the forward equation depends on some finite-dimensional unknown parameter. We consider the problem of estimating this parameter and then use the proposed estimator to estimate the solution of the BSDE. This last estimator is constructed with the help of the solution of the corresponding partial differential equation. We are interested in three observation models admitting a consistent estimation of the unknown parameter: small noise, large samples and unknown volatility. In the first two cases we have a continuous time observation, and the unknown parameter is in the drift coefficient. In the third case the volatility of the forward equation depends on the unknown parameter, and we have discrete time observations. The presented estimators of the solution of the BSDE in the three casesmentioned are asymptotically efficient.
Citation:
Yury A. Kutoyants, “Approximation of the solution of the backward stochastic differential equation. Small noise, large sample and high frequency cases”, Stochastic calculus, martingales, and their applications, Collected papers. Dedicated to Academician Albert Nikolaevich Shiryaev on the occasion of his 80th birthday, Trudy Mat. Inst. Steklova, 287, MAIK Nauka/Interperiodica, Moscow, 2014, 140–161; Proc. Steklov Inst. Math., 287:1 (2014), 133–154
\Bibitem{Kut14}
\by Yury~A.~Kutoyants
\paper Approximation of the solution of the backward stochastic differential equation. Small noise, large sample and high frequency cases
\inbook Stochastic calculus, martingales, and their applications
\bookinfo Collected papers. Dedicated to Academician Albert Nikolaevich Shiryaev on the occasion of his 80th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2014
\vol 287
\pages 140--161
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm3586}
\crossref{https://doi.org/10.1134/S0371968514040098}
\elib{https://elibrary.ru/item.asp?id=22681992}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2014
\vol 287
\issue 1
\pages 133--154
\crossref{https://doi.org/10.1134/S0081543814080094}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000348379600009}
\elib{https://elibrary.ru/item.asp?id=24030860}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84921925224}
Linking options:
https://www.mathnet.ru/eng/tm3586
https://doi.org/10.1134/S0371968514040098
https://www.mathnet.ru/eng/tm/v287/p140
This publication is cited in the following 9 articles:
Yury A. Kutoyants, “Volatility estimation of hidden Markov processes and adaptive filtration”, Stochastic Processes and their Applications, 173 (2024), 104381
Yoshida N., “Quasi-Likelihood Analysis and Its Applications”, Stat. Infer. Stoch. Proc., 25:1 (2022), 43–60
Kutoyants Yu.A., “On Multi-Step Estimation of Delay For Sde”, Bernoulli, 27:3 (2021), 2069–2090
R. Z. Khasminskii, Yu. A. Kutoyants, “On parameter estimation of hidden telegraph process”, Bernoulli, 24:3 (2018), 2064–2090
A. S. Dabye, A. A. Gounoung, Yu. A. Kutoyants, “Method of moments estimators and multi-step MLE for Poisson processes”, J. Contemp. Math. Anal.-Armen. Aca., 53:4 (2018), 237–246
Yusuke Kaino, Masayuki Uchida, “Hybrid estimators for stochastic differential equations from reduced data”, Stat Inference Stoch Process, 21:2 (2018), 435
Yu. A. Kutoyants, “On the multi-step MLE-process for ergodic diffusion”, Stoch. Process. Their Appl., 127:7 (2017), 2243–2261
Yu. A. Kutoyants, A. Motrunich, “On multi-step MLE-process for Markov sequences”, Metrika, 79:6 (2016), 705–724
Yu A. Kutoyants, “On approximation of BSDE and multi-step MLE-processes”, Probab Uncertain Quant Risk, 1:1 (2016)
Citation in format AMSBIB
Contact us: