Harnack Inequalities for G-SDEs with Multiplicative Noise

Abstract

The Harnack inequality for stochastic differential equation driven by G-Brownian motion with multiplicative noise is derived by means of the coupling by change of measure, which extends the corresponding results derived in Wang (Probab. Theory Related Fields 109:417–424) under the linear expectation. Moreover, we generalize the gradient estimate under nonlinear expectation appeared in Song (Sci. China Math. 64:1093–1108).

Keywords

Harnack inequality / Gradient estimate / Multiplicative noise / G-Brownian motion / SDEs

References

[1.] Bao J, Wang F-Y, Yuan C. Derivative formula and Harnack inequality for degenerate functionals SDEs. Stoch. Dyn.. 2013, 13 1-22

[2.] Cohen,S., Ji,S., Peng,S.: Sublinear expectations and Martingales in discrete time. (2011). arXiv:1104.5390

[3.] Denis L, Hu M, Peng S. Function spaces and capacity related to a sublinear expectation: application to $G$-Brownian motion pathes. Potential Anal.. 2011, 34 139-161

[4.] Hu M, Ji S, Peng S, Song Y. Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by $G$-Brownian motion. Stoch. Process. Appl.. 2014, 124 1170-1195

[5.] Hu M, Wang F, Zheng G. Quasi-continuous random variables and processes under the $G$-expectation framework. Stoch. Process. Appl.. 2016, 126 2367-2387

[6.] Huang,X., Yang,F.-F.: Harnack inequality and gradient estimate for $G$-SDEs with degenerate noise, Sci. China Math. (2021). https://doi.org/10.1007/s11425-020-1784-0

[7.] Li X, Peng S. Stopping times and related Itô’s calculus with $G$-Brownian motion. Stoch. Process. Appl.. 2011, 121 1492-1508

[8.] Liu G. Exit times for semimartingales under nonlinear expectation. Stochastic Process. Appl.. 2020, 130 7338-7362

[9.] Osuka E. Girsanov’s formula for $G$-Brownian motion. Stoch. Process. Appl.. 2013, 123 1301-1318

[10.] Peng,S.: $G$-Brownian motion and dynamic risk measures under volatility uncertainty, (2007) arXiv:0711.2834v1

[11.] Peng, S.:$G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type, in: Stochastic Analysis and Applications, in: Abel Symp., vol. 2, Springer, Berlin, (2007), pp.541–567

[12.] Peng S. Nonlinear expectations and stochastic calculus under uncertainty with robust CLT and G-Brownian motion, Probability Theory and Stochastic Modelling. 2019 Berlin: Springer

[13.] Ren P, Yang F-F. Path independence of additive functionals for stochastic differential equations under $G$-framework. Front. Math. China. 2019, 14 135-148

[14.] Song Y. Gradient estimates for nonlinear diffusion semigroups by coupling methods. Sci. China Math.. 2021, 64 1093-1108

[15.] Song Y. Properties of hitting times for $G$-martingales and their applications. Stoch. Process. Appl.. 2011, 121 1770-1784

[16.] Song Y. Some properties on $G$-evaluation and it’s applications to $G$-martingale decomposition. Sci. China Math.. 2011, 54 287-300

[17.] Wang F-Y. Estimates for invariant probability measures of degenerate SPDEs with singular and path-dependent drifts. Probab. Theory Related Fields. 2018, 172 1181-1214

[18.] Wang,F.-Y.: Harnack inequalities for stochastic partial differential equations, Springer Briefs in Mathematics, Springer, New York, (2013), pp, ISBN: 978–1–4614–7933-8, 978–1–4614–7934–5

[19.] Wang F-Y. Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds, (English summary). Ann. Probab.. 2011, 39 1449-1467

[20.] Wang F-Y. Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields. 1997, 109 417-424

[21.] Xu J, Shang H, Zhang B. A Girsanov type theorem under $G$-framework. Stoch. Anal. Appl.. 2011, 29 386-406

[22.] Yang F-F. Harnack inequality and applications for SDEs driven by $G$-Brownian motion. Acta Math. Appl. Sin. Engl. Ser.. 2020, 36 1-9

Funding

Young Scientists Fund(11801406)