<abstract xmlns="http://www.w3.org/1999/xhtml">The main objective of this paper is to investigate the problem of estimating the trend function St = S(xt) for process satisfying stochastic differential equations of the type
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2020-0008_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mrow><mtext>dX</mtext></mrow></mrow><mtext>t</mtext></msub><mo>=</mo><mtext>S</mtext><mrow><mo>(</mo><mrow><msub><mrow><mtext>X</mtext></mrow><mtext>t</mtext></msub></mrow><mo>)</mo></mrow><mtext>dt</mtext><mo>+</mo><mi>ε</mi><msubsup><mrow><mrow><mtext>dB</mtext></mrow></mrow><mtext>t</mtext><mrow><mtext>H</mtext><mo>,</mo><mtext>K</mtext></mrow></msubsup><mo>,</mo><mi> </mi><msub><mrow><mtext>X</mtext></mrow><mtext>0</mtext></msub><mo>=</mo><msub><mrow><mtext>x</mtext></mrow><mtext>0</mtext></msub><mo>,</mo><mi> </mi><mn>0</mn><mo>≤</mo><mtext>t</mtext><mo>≤</mo><mtext>T</mtext><mo>,</mo></mrow></math><tex-math>{\rm{d}}{{\rm{X}}_{\rm{t}}} = {\rm{S}}\left( {{{\rm{X}}_{\rm{t}}}} \right){\rm{dt + }}\varepsilon {\rm{dB}}_{\rm{t}}^{{\rm{H,K}}},\,{{\rm{X}}_{\rm{0}}} = {{\rm{x}}_{\rm{0}}},\,0 \le {\rm{t}} \le {\rm{T,}}</tex-math></alternatives></disp-formula>where {<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2020-0008_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mtext>B</mtext></mrow><mtext>t</mtext><mrow><mtext>H</mtext><mo>,</mo><mtext>K</mtext></mrow></msubsup><mo>,</mo><mtext>t</mtext><mo>≥</mo><mtext>0</mtext></mrow></math><tex-math>{\rm{B}}_{\rm{t}}^{{\rm{H,K}}},{\rm{t}} \ge {\rm{0}}</tex-math></alternatives></inline-formula>} is a bifractional Brownian motion with known parameters H <italic>∈</italic> (0, 1), K <italic>∈</italic> (0, 1] and HK <italic>∈</italic> (1/2, 1). We estimate the unknown function S(xt) by a kernel estimator ̂St and obtain the asymptotic properties as ε → 0. Finally, a numerical example is provided.</abstract>

The main objective of this paper is to investigate the problem of estimating the trend function St = S(xt) for process satisfying stochastic differential equations of the type
dXt=S(Xt)dt+εdBtH,K, X0=x0, 0≤t≤T,{\rm{d}}{{\rm{X}}_{\rm{t}}} = {\rm{S}}\left( {{{\rm{X}}_{\rm{t}}}} \right){\rm{dt + }}\varepsilon {\rm{dB}}_{\rm{t}}^{{\rm{H,K}}},\,{{\rm{X}}_{\rm{0}}} = {{\rm{x}}_{\rm{0}}},\,0 \le {\rm{t}} \le {\rm{T,}}where {BtH,K,t≥0{\rm{B}}_{\rm{t}}^{{\rm{H,K}}},{\rm{t}} \ge {\rm{0}}} is a bifractional Brownian motion with known parameters H ∈ (0, 1), K ∈ (0, 1] and HK ∈ (1/2, 1). We estimate the unknown function S(xt) by a kernel estimator ̂St and obtain the asymptotic properties as ε → 0. Finally, a numerical example is provided.

Nonparametric estimation of trend function for stochastic differential equations driven by a bifractional Brownian motion

Laboratory of Stochastic Models, Statistic and Applications

Djillali Liabes University of Sidi Bel Abbes

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

The main objective of this paper is to investigate the problem of estimating the trend function St = S(xt) for process satisfying stochastic differential...

Nonparametric estimation of trend function for stochastic differential equations driven by a bifractional Brownian motion

Published Online: Jul 16, 2020

Page range: 128 - 145

Received: Jun 15, 2019

DOI: https://doi.org/10.2478/ausm-2020-0008

Keywords
bifractional Brownian motion, trend function, kernel estimator, stochastic differential equations, nonparametric estimation

© 2020 Abdelmalik Keddi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Nonparametric estimation of trend function for stochastic differential equations driven by a bifractional Brownian motion

Published Online: Jul 16, 2020

Page range: 128 - 145

Received: Jun 15, 2019

DOI: https://doi.org/10.2478/ausm-2020-0008

Keywordsbifractional Brownian motion, trend function, kernel estimator, stochastic differential equations, nonparametric estimation

© 2020 Abdelmalik Keddi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
bifractional Brownian motion, trend function, kernel estimator, stochastic differential equations, nonparametric estimation