Research

My current research interests are in pure and applied mathematics.


Research interests

Analysis and Applications asymptotic analysis, special functions, singular perturbations
Partial Differential Equations subelliptic operators, subRiemannian geometries, applications in several complex variables, degenerate and singular parabolic equations
Scientific Computing and Numerical Analysis numerical eigenvalue problems, fractional diffusion equations, imaging sciences
Quantitative and Computational Finance option pricing models, fast algorithms, big data and machine learning in finance

Publications

The research papers are mainly in four areas:

Analysis and Applications Partial Differential Equations Scientific Computing Quantitative Finance


Analysis and Applications

  • [A12] Wen-Gao Long, Yu-Tian Li*, and Qing-hai Wang, Connection problem of the first Painlevé transcendents transcendent between poles and negative infinity, SIAM Journal on Mathematical Analysis, 55 (2023), 6676-6706. doi: 10.1137/21M1465251
  • [A11] Wen-Gao Long and Yu-Tian Li*, Connection problem of the first Painlevé transcendents with large initial data, Journal of Physics A: Mathematical and General, 56 (2023), article id: 175201, 20pages. doi: 10.1088/1751-8121/acc620
  • [A10] Yu-Tian Li, Xiang-Sheng Wang*, and Roderick Wong. Asymptotics of the Wilson polynomials, Analysis and Applications, 18 (2020), 237-270. doi: 10.1142/S0219530519500076
  • [A9] Wen-Gao Long, Dan Dai, Yu-Tian Li*, and Xiang-Sheng Wang. Asymptotics of orthogonal polynomials with asymptotic Freud-like weights, Studies in Applied Mathematics, 144 (2020), 133-163. doi: 10.1111/sapm.12291
  • [A8] Li-Hua Cao, Yu-Tian Li*, and Yu Lin. Asymptotic approximations of the continuous Hahn polynomials, Journal of Approximation Theory, 247 (2019), 32-47. doi: 10.1016/j.jat.2019.07.001
  • [A7] A.D. Alhaidari* and Y. Li: Quantum systems associated with the Hahn and continuous Hahn polynomials, Reports on Mathematical Physics, 82 (2018), 285-301. doi: 10.1016/S0034-4877(19)30002-3
  • [A6] W.-G. Long, Y.-T. Li, S.-Y. Liu, and Y.-Q. Zhao*: Real solutions of the first Painlevé equation with large initial data, Studies in Applied Mathematics 139 (2017), 505-532. doi: 10.1111/sapm.12171
  • [A5] Y. Li, S. Liu, S. Xu, and Y. Zhao*: Asymptotics of Landau constants with optimal error bounds, Constructive Approximation 40 (2014), 281-305. doi: 10.1007/s00365-014-9259-x
  • [A4] L.-H. Cao and Y.-T. Li*: Linear difference equations with a transition point at the origin, Analysis and Applications 12 (2014), 75-106. doi: 10.1142/S0219530513500371
  • [A3] Y. T. Li* and R. Wong: Global asymptotics for Stieltjes–Wigert polynomials, Analysis and Applications 11 (2013), 1350028, 12 pages. doi: 10.1142/S0219530513500280
  • [A2] Y. Li, S. Liu, S. Xu, and Y. Zhao*: Full asymptotic expansions of the Landau constants via a difference equation approach, Applied Mathematics and Computation 219 (2012), 988-995. doi: 10.1016/j.amc.2012.07.003
  • [A1] Y. T. Li and R. Wong*: Integral and series representations of the Dirac delta function, Communications on Pure and Applied Analysis 7 (2008), 229-247. doi: 10.3934/cpaa.2008.7.229 This becomes a section in NIST Handbook of Mathematical Functions.

Partial Differential Equations

  • [P11] Yingshu Zhang and Yutian Li*: Dynamics of a Leslie-Gower predator-prey model with advection and free boundaries Discrete and Continuous Dynamical Systems, series B 29 (2024), 319-350. doi: 10.3934/dcdsb.2023097
  • [P10] Shanming Ji, Jingxue Yin*, and Yutian Li: Positive periodic solutions of the weighted $p$-Laplacian with nonlinear sources, Discrete and Continuous Dynamical Systems, series A 38 (2018), 2411-2439. doi: 10.3934/dcds.2018100
  • [P9] P. Greiner* and Y. Li: A fundamental solution of a nonelliptic operator, II, Analysis and Applications 16 (2018), 407-433. doi: 10.1142/S0219530516500196
  • [P8] S. Ji, Y. Li, R. Huang*, and J. Yin: Singular periodic solutions for the $p$-Laplacian in a punctured domain, Communications on Pure and Applied Analysis 16 (2017), 373-392. doi: 10.1934/cpaa.2017019
  • [P7] P. Greiner* and Y. Li: Heat kernels, old and new, Bulletin of the Institute of Mathematics Academia Sinica, New Series 12 (2017), 1-37. doi: 10.21915/BIMAS.2017101
  • [P6] D.-C. Chang and Y. Li*: Small time asymptotics of the heat kernels for the Heisenberg subLaplacian and step two Grushin operator, Proceedings of the Royal Society A 471 (2015), 20140943. 19 pages. doi: 10.1098/rspa.2014.0943
  • [P5] D.-C. Chang* and Y. Li: Heat kernels for a family of Grushin operators, Method. Anal. Appl. 21 (2014), 291-312. doi: 10.4310/MAA.2014.v21.n3.a2
  • [P4] O. Calin, D.C. Chang*, and Y. Li: On the heat kernel of a left invariant elliptic operator, in: Excursions in Harmonic Analysis Vol. 2, 197-209, edited by T. D. Andrews et al., Birkhauser/Springer, New York, 2013. doi: 10.1007/978-0-8176-8379-5_10
  • [P3] O. Calin, D.C. Chang*, J. Hu, and Y. Li: Heat kernels of a class of degenerate elliptic operators using stochastic method, Complex Variables and Elliptic Equations 57 (2012), 155-168. doi: 10.1080/17476933.2011.581756
  • [P2] O. Calin, D.C. Chang*, J. Hu, and Y. Li: On heat kernels of a class of degenerate elliptic operators, Journal of Nonlinear and Convex Analysis 12 (2011), 309-340.
  • [P1] D.C. Chang* and Y. Li: SubRiemannian geodesics in Grushin plane, Journal of Geometric Analysis 22 (2012), 800-826. doi: 10.1007/s12220-011-9215-y

Scientific Computing

  • [S6] Liyuan Chen, Yutian Li, and Tieyong Zeng*: Variational image restoration and segmentation with Rician noise, Journal of Scientific Computing to appear. doi: 10.1007/s10915-018-0826-3
  • [S5] Xu Guo, Yutian Li*, and Hong Wang: A high order finite difference method for tempered fractional diffusion equations with applications to the CGMY model, SIAM Journal on Scientific Computing 40 (2018), A3322-A3343. doi: 10.1137/18M1172739
  • [S4] Xu Guo, Yutian Li*, and Hong Wang, A fourth-order scheme for space fractional diffusion equations, Journal of Computational Physics 373 (2018), 410-424. doi: 10.1016/j.jcp.2018.03.032    Description and Supplementary
  • [S3] Xu Guo, Yutian Li*, and Hong Wang, A fast finite difference method for tempered fractional diffusion equations, Communications in Computational Physics 24 (2018), 531-556. doi: 10.4208/cicp.OA-2018-0001
  • [S2] J. Zhu* and Y. Li: A new approach of eigenmodes for varying refractive-index profile's waveguides, IEEE Transactions on Microwave Theory and Techniques 64 (2016), 3131-3138. doi: 10.1109/TMTT.2016.2600325
  • [S1] Y. Li and J. Zhu*: Efficient approximations of dispersion relations in optical waveguides with varying refractive-index profiles, Optics Express 23 (2015), 11952-11964. doi: 10.1364/OE.23.011952

Quantitative Finance

  • [F4] Xu Guo, Yutian Li*, and Hong Wang: Multi-asset options under CGMY processes, Computer and Mathematics Methods with Applications 76 (2018), 1500-1514. doi: 10.1016/j.camwa.2018.07.002
  • [F3] Haiming Song, Kai Zhang*, and Yutian Li: Finite element and discontinuous Galerkin methods with perfectly matched layers for American options, Numerical Methods: Theory Methods and Applications 10 (2017), 829-851. doi: 10.4208/nmtma.2017.0020
  • [F2] X. Guo and Y. Li*: Valuation of American options under the CGMY model, Quantitative Finance 16 (2016), 1529-1540. doi: 10.1080/14697688.2016.1158854
  • [F1] Jiguang Han, Ming Gao, Qiang Zhang*, and Yutian Li: Option prices under stochastic volatility, Applied Mathematics Letters 26 (2013), 1-4. doi: 10.1016/j.aml.2012.07.014


Research Grants

No. Title Sourse Duration Amount Role
GRF 201513 Heat Kernals for High Step Subelliptic Operators Hong Kong RGC Jan. 2014 – Dec. 2016 HK$ 592,987 PI
GRF 12303515 Heat Kernel Asymptotics for Hörmander Type Operators Hong Kong RGC Sep. 2015 – Aug. 2018 * HK$ 451,255 PI
GRF 12328416 The index theorem for subelliptic operators on contact manifolds – A heat kernel approach Hong Kong RGC Sep. 2016 – Aug. 2019 * HK$ 488,501 PI
Start-up Start-up grant Hong Kong Baptist University Jan. 2013 – Dec. 2015 HK$ 120,000 PI
FRG2/14-15/015 Optimal Trajectories in Nonholonomic Systems Hong Kong Baptist University Jun. 2015 – May 2016 HK$ 133,200 PI
PF01 000861 Numerical Methods for Fractional Diffusion Equations The Chinese University of Hong Kong, Shenzhen Mar. 2018 – Feb. 2021 RMB 1,650,000 PI
11801480 On a Class of Subelliptic Operators National Natural Science Foundation of China Jan. 2019 – Dec. 2021 RMB 250,000 PI

A * indicates the grant is terminated since I left Hong Kong.



Working Projects


1. Connection formulas for Heun's equation


2. Heat kernel for subelliptic operators induced by a pseudo-convex domain


3. Index theory for subelliptic operators


4. Spectral methods for fractional diffusion equations


5. Analytic study of Heston's stochastic volatility model


6. Model selection and model average in option pricing