We studied the Sobolev estimates and inverse Hölder estimates for a class of variational inequality problems involving divergence-type parabolic operator structures. These problems arise from the valuation analysis of American contingent claim problems. First, we analyzed the uniform continuity of the spatially averaged operator with respect to time in a spherical region and the Sobolev estimates for solutions of the variational inequality. Second, by using spatial and temporal truncation, we obtained the Caccioppoli estimate for the variational inequality and consequently derived the inverse Hölder estimate for the solutions.
Citation: Kaiyu Zhang. Sobolev estimates and inverse Hölder estimates on a class of non-divergence variation-inequality problem arising in American option pricing[J]. Electronic Research Archive, 2024, 32(11): 5975-5987. doi: 10.3934/era.2024277
We studied the Sobolev estimates and inverse Hölder estimates for a class of variational inequality problems involving divergence-type parabolic operator structures. These problems arise from the valuation analysis of American contingent claim problems. First, we analyzed the uniform continuity of the spatially averaged operator with respect to time in a spherical region and the Sobolev estimates for solutions of the variational inequality. Second, by using spatial and temporal truncation, we obtained the Caccioppoli estimate for the variational inequality and consequently derived the inverse Hölder estimate for the solutions.
| [1] |
M. Moradipour, S. A. Yousefi, Using spectral element method to solve variational inequalities with applications in finance, Chaos Solitons Fractals, 81 (2015), 208–217. https://doi.org/10.1016/j.chaos.2015.09.006 doi: 10.1016/j.chaos.2015.09.006
|
| [2] |
J. Shen, W. Huang, J. Ma, An efficient and provable sequential quadratic programming method for American and swing option pricing, Eur. J. Oper. Res., 316 (2024), 19–35. https://doi.org/10.1016/j.ejor.2023.11.012 doi: 10.1016/j.ejor.2023.11.012
|
| [3] |
H. Sultan, A. Hifsa, N. Muhammad, A. A. Pantelous, Pricing American options under Azzalini Ito-McKean skew Brownian motions, Appl. Math. Comput., 451 (2023), 128040. https://doi.org/10.1016/j.amc.2023.128040 doi: 10.1016/j.amc.2023.128040
|
| [4] |
S. B. Boyana, T. Lewis, A. Rapp, Y. Zhang, Convergence analysis of a symmetric dual-wind discontinuous Galerkin method for a parabolic variational inequality, J. Comput. Appl. Math., 422 (2023), 114922. https://doi.org/10.1016/j.cam.2022.114922 doi: 10.1016/j.cam.2022.114922
|
| [5] |
S. Migorski, S. Dudek, Constrained evolutionary variational-hemivariational inequalities with application to fluid flow model, Commun. Nonlinear Sci. Numer. Simul., 127 (2023), 107555. https://doi.org/10.1016/j.cnsns.2023.107555 doi: 10.1016/j.cnsns.2023.107555
|
| [6] |
Z. Wu, W. Li, Q. Zhang, Y. Xiao, New existence and stability results of mild solutions for fuzzy fractional differential variational inequalities, J. Comput. Appl. Math., 448 (2024), 115926. https://doi.org/10.1016/j.cam.2024.115926 doi: 10.1016/j.cam.2024.115926
|
| [7] |
X. Wang, S. Chen, H. Qi, A class of delay differential variational inequalities with boundary conditions, Commun. Nonlinear Sci. Numer. Simul., 129 (2024), 107684. https://doi.org/10.1016/j.cnsns.2023.107684 doi: 10.1016/j.cnsns.2023.107684
|
| [8] |
Y. Bai, N. Costea, S. Zeng, Existence results for variational-hemivariational inequality systems with nonlinear couplings, Commun. Nonlinear Sci. Numer. Simul., 134 (2024), 108026. https://doi.org/10.1016/j.cnsns.2024.108026 doi: 10.1016/j.cnsns.2024.108026
|
| [9] |
J. Zhao, J. Chen, Z. Liu, Second order evolutionary problems driven by mixed quasi-variational-hemivariational inequalities, Commun. Nonlinear Sci. Numer. Simul., 120 (2023), 107192. https://doi.org/10.1016/j.cnsns.2023.107192 doi: 10.1016/j.cnsns.2023.107192
|
| [10] |
P. M. N. Feehan, C. A. Pop, Boundary-degenerate elliptic operators and Hölder continuity for solutions to variational equations and inequalities, Ann. Inst. H. Poincar$\mathrm{\acute{e}}$ C Anal. Non Lineair$\mathrm{\acute{e}}$, 34 (2017), 1075–1129. https://doi.org/10.1016/j.anihpc.2016.07.005 doi: 10.1016/j.anihpc.2016.07.005
|
| [11] |
Y. Sun, T. Wu, Hölder and Schauder estimates for weak solutions of a certain class of non-divergent variation inequality problems in finance, AIMS Math., 8 (2023), 18995–19003. https://doi.org/10.3934/math.2023968 doi: 10.3934/math.2023968
|
| [12] |
J. Li, Z. Tong, Local Hölder continuity of inverse variation-inequality problem constructed by non-Newtonian polytropic operators in finance, AIMS Math., 8 (2023), 28753–28765. https://doi.org/10.3934/math.20231472 doi: 10.3934/math.20231472
|
| [13] |
M. Ptashnyk, Homogenization of some degenerate pseudoparabolic variational inequalities, J. Math. Anal. Appl., 469 (2019), 44–75. https://doi.org/10.1016/j.jmaa.2018.08.047 doi: 10.1016/j.jmaa.2018.08.047
|
| [14] |
J. Kinnunen, J. L. Lewis, Higher integrability for parabolic systems of p-Laplacian type, Duke Math. J., 102 (2000), 253–271. https://doi.org/10.1215/S0012-7094-00-10223-2 doi: 10.1215/S0012-7094-00-10223-2
|
| [15] |
P. Baroni, Marcinkiewicz estimates for degenerate parabolic equations with measure data, J. Funct. Anal., 267 (2014), 3397–3426. https://doi.org/10.1016/j.jfa.2014.08.017 doi: 10.1016/j.jfa.2014.08.017
|
Kaiyu Zhang. Sobolev estimates and inverse Hölder estimates on a class of non-divergence variation-inequality problem arising in American option pricing[J]. Electronic Research Archive, 2024, 32(11): 5975-5987. doi: 10.3934/era.2024277
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