This paper analyzes a class of variational inequalities involving a double-phase degenerate parabolic operator with variable exponents, which arises in the valuation of American put options. By establishing energy estimates for the associated penalty problem and applying a limiting argument, the existence of a weak solution is obtained. The uniqueness of the weak solution is also discussed.
Citation: Qingjun Zhao. Existence and uniqueness of solutions to variational inequalities involving a double-phase degenerate parabolic operator with variable exponents arising from American put option valuation analysis[J]. Electronic Research Archive, 2026, 34(2): 1063-1078. doi: 10.3934/era.2026049
This paper analyzes a class of variational inequalities involving a double-phase degenerate parabolic operator with variable exponents, which arises in the valuation of American put options. By establishing energy estimates for the associated penalty problem and applying a limiting argument, the existence of a weak solution is obtained. The uniqueness of the weak solution is also discussed.
| [1] |
L. F. Li, J. Y. Wu, M. Zhu, M. Wang, Analytic solutions for pricing American style options, Finance Res. Lett., 86 (2025), 108467. https://doi.org/10.1016/j.frl.2025.108467 doi: 10.1016/j.frl.2025.108467
|
| [2] |
H. Zhou, D. M. Dang, Numerical analysis of American option pricing in a two-asset jump-diffusion model, Appl. Numer. Math., 216 (2025), 98–126. https://doi.org/10.1016/j.apnum.2025.03.005 doi: 10.1016/j.apnum.2025.03.005
|
| [3] |
W. W. Jiang, J. Li, Local max-norm regularity estimates for gradient solutions of variational inequalities involving a class of doubly degenerate parabolic operators arising from quanto option valuation analysis, AIMS Math., 10 (2025), 21902–21915. https://doi.org/10.3934/math.2025975 doi: 10.3934/math.2025975
|
| [4] |
L. Constantin, J. Giacomoni, G. Warnault, Existence and global behaviour of solutions of a parabolic problem involving the fractional p-Laplacian in porous medium, Nonlinear Anal. Real World Appl., 87 (2026), 104416. https://doi.org/10.1016/j.nonrwa.2025.104416 doi: 10.1016/j.nonrwa.2025.104416
|
| [5] |
S. J. Guo, Existence and stability of positive solutions in a parabolic problem with a nonlinear incoming flux on the boundary, J. Differ. Equations, 436 (2025), 113349. https://doi.org/10.1016/j.jde.2025.113349 doi: 10.1016/j.jde.2025.113349
|
| [6] |
N. Chen, F. S. Li, P. H. Wang, Global existence and blow-up phenomenon for a nonlocal semilinear pseudo-parabolic p-Laplacian equation, J. Differ. Equations, 409 (2024), 395–440. https://doi.org/10.1016/j.jde.2024.07.008 doi: 10.1016/j.jde.2024.07.008
|
| [7] |
C. O. Alves, T. Boudjeriou, Global existence for parabolic p-Laplace equations with supercritical growth in whole RN, J. Math. Anal. Appl., 535 (2024), 128244. https://doi.org/10.1016/j.jmaa.2024.128244 doi: 10.1016/j.jmaa.2024.128244
|
| [8] |
M. Tran, T. Nguyen, Existence of weak solutions to borderline double-phase problems with logarithmic convection terms, J. Math. Anal. Appl., 546 (2025), 129185. https://doi.org/10.1016/j.jmaa.2024.129185 doi: 10.1016/j.jmaa.2024.129185
|
| [9] |
A. Armiti-Juber, C. Rohde, Existence of weak solutions for a nonlocal pseudo-parabolic model for Brinkman two-phase flow in asymptotically flat porous media, J. Math. Anal. Appl., 477 (2019), 592–612. https://doi.org/10.1016/j.jmaa.2019.04.049 doi: 10.1016/j.jmaa.2019.04.049
|
| [10] |
W. Kim, K. Moring, L. Särkiö, Hölder regularity for degenerate parabolic double-phase equations, J. Differ. Equations, 434 (2025), 113231. https://doi.org/10.1016/j.jde.2025.113231 doi: 10.1016/j.jde.2025.113231
|
| [11] |
W. Kim, Calder$\mathrm{\acute{o}}$n-Zygmund type estimate for the singular parabolic double-phase system, J. Math. Anal. Appl., 551 (2025), 129593. https://doi.org/10.1016/j.jmaa.2025.129593 doi: 10.1016/j.jmaa.2025.129593
|
| [12] |
R. A. Mashiyev, O. M. Buhrii, Existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity, J. Math. Anal. Appl., 377 (2011), 450–463. https://doi.org/10.1016/j.jmaa.2010.11.006 doi: 10.1016/j.jmaa.2010.11.006
|
| [13] |
J. W. Hao, J. F. Han, Q. S. Liu, Existence and convergence analysis of second-order delay differential variational-hemivariational inequalities with memory terms, Nonlinear Anal. Real World Appl., 85 (2025), 104373. https://doi.org/10.1016/j.nonrwa.2025.104373 doi: 10.1016/j.nonrwa.2025.104373
|
| [14] |
D. Adak, G. Manzini, S. Natarajan, Virtual element approximation of two-dimensional parabolic variational inequalities, Comput. Math. Appl., 116 (2022), 48–70. https://doi.org/10.1016/j.camwa.2021.09.007 doi: 10.1016/j.camwa.2021.09.007
|
| [15] |
Q. J. Zhao, The ${L^\infty }$ estimate of the spatial gradient of the solution to a variational inequality problem originates from the financial contract problem with advanced implementation clauses, AIMS Math., 9 (2024), 35949–35963. https://doi.org/10.3934/math.20241704 doi: 10.3934/math.20241704
|
| [16] |
X. R. Han, F. H. Yi, An irreversible investment problem with demand on a finite horizon: The optimal investment boundary analysis, Commun. Nonlinear Sci. Numer. Simul., 109 (2022), 106302. https://doi.org/10.1016/j.cnsns.2022.106302 doi: 10.1016/j.cnsns.2022.106302
|
| [17] |
W. S. Zhou, Z. Q. Wu, Some results on a class of degenerate parabolic equations not in divergence form, Nonlinear Anal. Real World Appl., 60 (2005), 863–886. https://doi.org/10.1016/j.na.2004.09.053 doi: 10.1016/j.na.2004.09.053
|
Qingjun Zhao. Existence and uniqueness of solutions to variational inequalities involving a double-phase degenerate parabolic operator with variable exponents arising from American put option valuation analysis[J]. Electronic Research Archive, 2026, 34(2): 1063-1078. doi: 10.3934/era.2026049
DownLoad: