H¨older and Schauder estimates for weak solutions of a certain class of non-divergent variation inequality problems in finance

: This article studies a class of variational inequality problems composed of non-divergence type parabolic operators. In comparison with traditional di ff erential equations, this study focuses on overcoming inequality constraints to obtain H¨older and Schauder estimates for weak solutions. The results indicate that the weak solution of the variational inequality possesses the C α continuity and the Schauder estimate on the W 1 , p space, where α ∈ (0 , 1) and p ≥ 2.


Introduction
Variational inequalities are often used in American-style option valuation analysis, and they provide a good description of early exercise provisions in the presence of uncertain equities. As an application of variational inequality, we investigate the pricing problem of American-style options on two risky assets. Assuming the existence of two risky assets in the financial market, their prices follow: where µ i represents the return rate of the i-th asset, and σ i represents its volatility, i = 1, 2. {W 1 (t), t ≥ 0} and {W 2 (t), t ≥ 0} are two standard 1-D Brownian motions used to describe the background noise of the financial market. Investors who purchase American-style options have the right to choose between risk assets {S 1 (t), t ≥ 0} and {S 2 (t), t ≥ 0} for holding, in terms of value, expected return rate, turnover rate, and asset volatility, with the aim of maximizing their gains while minimizing fluctuations. According to the literature [1][2][3], the value v of American-style option is suitable for the following variational inequality model: where r represents risk-free interest rate and Many documents indicate that when there are costs in stock trading, sigma in Lv is often related to the first spatial gradient of v. The famous Leland model can rewrite sigma as In this equation, the variable σ 2 0 denotes the initial volatility, while Le represents the Leland number. The value of p is greater than or equal to 2.
Taking inspiration from Leland's fee model for American-style options, this paper examines a variational inequality initial-boundary value problem: incorporating the non-divergence parabolic operator Different from (1), we restrict Ω to be a bounded and open subset of R N , and Ω T = Ω × (0, T ). In terms of the parameter γ, we continue to employ the hypothesis conditions from the study in reference [4,5] to verify the presence of weak solutions, which is represented by γ ∈ (0, 1). Theoretical research on variational inequalities has been extensively expanded in many aspects. Some examples include the study of the existence and uniqueness of solutions for 2-D variational inequalities in [4], and with fourth-order p(x)-Kirchhoff operators in [5]. The existence of solutions in whole R N can be found in [6]. Furthermore, the existence and uniqueness of solutions for mixed variational-hemivariational inequality were discovered in [7], without Lipschitz continuity in [8] and with nonlocal fractional operators in [9]. Additionally, stability analysis for variational inequalities, hemivariational inequalities and variational-hemivariational inequalities was analyzed in [10], as well as in reflexive Banach spaces in [11]. Finally, the regularity of weak solutions to a class of fourth-order parabolic variational inequalities was proved in [12].
Inspired by the literature [4][5][6], we studied the Hölder estimate and Schauder estimate for weak solutions of variational inequalities formed by a class of non-divergence parabolic operators. On one hand, starting from the weak solutions constructed in [4][5][6], we obtained several gradient estimates using techniques such as Hölder and Young inequalities, and obtained the Hölder estimate results based on [13]. On the other hand, we constructed test functions for weak solutions using time and space truncation factors, and obtained the Caccioppoli inequality that is suitable for the variational inequality (3), which serves as the cornerstone for proving the Schauder estimate. By varying different parameters in the Caccioppoli inequality, we obtained the Schauder estimate for weak solutions of the variational inequality (3).
In summary, this section provides a definition of the weak solution for the variational inequality based on references [4][5][6], along with a set of maximal monotone maps specified in [1][2][3]5,6]: Here M 0 is a positive constant.
It should be noted by the reader that the above formula implies Note that u 0 ≥ 0 in Ω T . Then, using the second condition of inequality (3) and with ease, we can see that u ≥ 0 in Ω T .

Hölder estimates of solution
This section considers the Hölder estimate of weak solutions, and first provides several gradient energy estimates for weak solutions. For any t ∈ (0, T ], define Ω t = Ω × (0, t). By choosing u as the test function in (6), we can obtain After simplification, the equation can be written as follows: Now using the Hölder and Young inequalities to analyze Ω t ξ · udxdt, we have from (5). Thus, substituting (9) into (8) and integrating with respect to Ω t ∂ t u 2 dxdt, it is not difficult to obtain 1 2 On the one hand, it should be noted that 2 − γ > 0. By removing the non-positive term (2 − γ) Ω t |∇u| p udxdt, we obtain Using Gronwall's inequality, On the other hand, by removing the non-negative term 1 2 Ω u 2 dx in (10), it is easy to obtain Combining (11) and (12), it is easy to see that It is worth noting that using integration by parts yields, Ω t u∆ p udxdt = − Ω t |∆u| p dxdt, and thus (13) can also be used to obtain u|∆ p u| ∈ L ∞ (0, T ; L 2 (Ω)).We will now prove u|∆ p u| ∈ L ∞ (0, T ; L 2 (Ω)). Assuming φ = u∆ p u in (7), it is easy to see that Ω t ∂ t u · u∆ p udxdt + Ω t u 2 |∆ p u| 2 dxdt − γ Ω t |∇u| p u∆ p udxdt = Ω t ξ · u∆ p udxdt.

Schauder estimate of solution
This section considers the Schauder estimate of weak solutions. For any (x 0 , t 0 ) ∈ Ω T , define where R and ρ are given positive numbers.  (3), then for any B R ⊂ Ω T and λ ∈ R, there holds the following estimate: Proof. For spatial variables, use the tangent function η on B R relative to B ρ , that is For a time variable, let κ ∈ C ∞ 0 (R) be defined such that 0 ≤ κ ≤ 1. If t ≤ t 0 − R 2 , we have κ = 0, and if t ≥ t 0 − ρ 2 , we have κ = 1.
and selecting test function ϕ as η 2 κ 2 (u − λ), it is easy to obtain Note that u(t, x) = ∂u ∂ν = 0 for any (x, t) ∈ ∂Ω × (0, T ). Thus, by applying integration by part on Substituting (20) and (21) into (19), we can obtain Using the Hölder and Young inequalities, and applying |∇η| ≤ C (R−ρ) 2 , we have Substituting (23) and (24) into (22), we have that Rearranging the above inequality yields Since 0 ≤ ξ ≤ M 0 , using the Hölder inequality to Removing the non-negative terms 1 p Q s R η 2 κ 2 |∇u| p dxdt from which implies that for any s ∈ I ρ , On the other hand, by removing the non-negative term B R η 2 κ 2 (u − λ) 2 dx| t=s in (26), we can easily obtain Combining (27) and (28), the desired conclusion follows and the proof is completed. □ Now, we analyze the Schauder estimates for weak solutions of the variational inequality (3). On the one hand, choosing ρ and R to be 1 2 R and 3 4 R, respectively, in the Lemma 4.1 and setting λ = 0, we obtain sup On the other hand, in Lemma 4.1, we choose ρ = 3 4 R and R, respectively, and set λ = 0, Combining (29) and (30), we can obtain the following Schauder estimates.

Conclusions
This paper studies the variational inequality problem associated with non-divergence type parabolic operators as follows: First, the Hölder estimate for the weak solution of the variational inequality (3) was analyzed. By using the maximal operator G to overcome the inequality constraints in model (1), combined with the Hölder inequality, Young's inequality, Gronwall's inequality, etc., the energy estimates for the spatial gradient and spatial second-order gradient of the weak solution of the variational inequality (3) were obtained, thus obtaining the Hölder estimate for the weak solution of the variational inequality (3).
Second, the Shauder estimate problem for weak solutions of the variational inequality (3) was studied. By utilizing weak solutions constructed with maximal operators, and combining spatial cutoff factors and time cutoff factors, the Caccioppoli inequality for the variational inequality (3) was obtained. Based on this, the Shauder estimate for the weak solutions of the variational inequality (3) was obtained by selecting different parameters.
There are still some areas for improvement in the current paper. The non-linear structure ∆ p u present in Lu ( see (4), for details) restricts the possibility of obtaining higher-order Hölder and Schauder estimates through spatial partial derivatives. In addition, the inequality constraints in model (3) make it impossible to obtain higher-order Hölder and Schauder estimates through partial derivative operations. We will attempt to overcome these limitations in future research.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.