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

1.   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 $\mu_i$ represents the return rate of the $i$-th asset, and $\sigma_i$ represents its volatility, $i = 1, 2$. $\{ {W_1}(t), t \ge 0\}$ and $\{ {W_2}(t), t \ge 0\}$ 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 $\{ {S_1}(t), t \ge 0\}$ and $\{ {S_2}(t), t \ge 0\}$ 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 $\sigma _0^2$ 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 $\Omega$ to be a bounded and open subset of $\mathrm{R}^N$, and $\Omega_T = \Omega \times (0, T)$. In terms of the parameter $\gamma$, 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 $\gamma \in (0, 1)$.

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 $\mathrm{R}_N$ 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.

Definition 1.1. A pair $(u, \xi)$ is said to be a generalized solution of variation-inequality (1), if $(u, \xi)$ satisfies $u \in {L^\infty }(0, T, {W^{1, p}}(\Omega)), \; {\partial _t}u \in {L^\infty }(0, T, {L^2}(\Omega))$ and $\xi \in G\; {{\rm{for}}}\; {{\rm{any}}}\; (x, t) \in {\Omega _T}$,

(a) $u(x, t) \ge {u_0}(x), \; u(x, 0) = {u_0}(x)\; {{\rm{for}}}\; {{\rm{any}}}\; (x, t) \in {\Omega _T}$,

(b) for every test-function $\varphi \in C_0^\infty ({\Omega _T})$ and $t \in (0, T]$, there admits the equality

It should be noted by the reader that the above formula implies

Note that ${u_0} \geq 0 \ {\rm{in}} \ \Omega_T$. Then, using the second condition of inequality (3) and with ease, we can see that $u \geq 0 \ {\rm{in}} \ \Omega_T$.

2.   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 $t \in (0, T]$, define $\Omega_t = \Omega \times (0, t)$. 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 $\int {\int_{{\Omega _t}} {\xi \cdot u{{\rm{d}}}x{{\rm{d}}}t} }$, we have

from (5). Thus, substituting (9) into (8) and integrating with respect to $\int {\int_{{\Omega _t}} {{\partial _t}{u^2}{{\rm{d}}}x{{\rm{d}}}t} }$, it is not difficult to obtain

On the one hand, it should be noted that $2 - \gamma > 0$. By removing the non-positive term $(2 - \gamma)\int {\int_{{\Omega _t}} {|\nabla u{|^p}u{{\rm{d}}}x{{\rm{d}}}t} }$, we obtain

Using Gronwall's inequality,

On the other hand, by removing the non-negative term $\frac{1}{2}\int_\Omega {{u^2}{{\rm{d}}}x}$ 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, $\int {\int_{{\Omega _t}} {u{\Delta _p}u{{\rm{d}}}x{{\rm{d}}}t} } = - \int {\int_{{\Omega _t}} {|\Delta u{|^p}{{\rm{d}}}x{{\rm{d}}}t} }$, and thus (13) can also be used to obtain $u|{\Delta _p}u| \in {L^\infty }(0, T; {L^2}(\Omega))$.We will now prove $u|{\Delta _p}u| \in {L^\infty }(0, T; {L^2}(\Omega))$. Assuming $\varphi = u{\Delta _p}u$ in (7), it is easy to see that

Here, we attempt to estimate $\int {\int_{{\Omega _t}} {{u^2}|{\Delta _p}u{|^2}{{\rm{d}}}x{{\rm{d}}}t} }$ in (14). With the help of the Hölder and Young inequalities, it is easy to discover that

Combining with (5), we obtain

From (9) we know that $u \in {L^\infty }(0, T; {W^{1, p}}(\Omega))$, such that we use Poincaré insert theory [14, P15] to arrive at $\mathop {\sup }\limits_{{\Omega _t}} u \le ||u|{|_{{L^\infty }(0, T; {W^{1, p}}(\Omega))}}$, such that Using the multivariate mean value theorem and integrating by part gives

Substituting (15)–(17) into (14), we obtain

From reference [13], we have the following result established by Eqs (13) and (18):

Theorem 2.1. For any $({x_1}, {t_1}), ({x_2}, {t_2}) \in {\Omega _T}$ and any $\alpha \in (0, 1)$, there exists

3.   Schauder estimate of solution

This section considers the Schauder estimate of weak solutions. For any $({x_0}, {t_0}) \in {\Omega _T}$, define

where $R$ and $\rho$ are given positive numbers.

Lemma 3.1. (Caccioppoli Type Inequality) If $(u, \xi)$ is a weak solution to the variational inequality problem (3), then for any ${B_R} \subset {\Omega _T}$ and $\lambda \in \mathbb{R}$, there holds the following estimate:

Proof. For spatial variables, use the tangent function $\eta$ on $B_R$ relative to ${B_\rho}$, that is

For a time variable, let $\kappa \in C_0^\infty (\mathbb{R})$ be defined such that $0 \le \kappa \le 1$. If $t\leq t_0-R^2$, we have $\kappa = 0$, and if $t \geq t_0 - \rho^2$, we have $\kappa = 1$. Moreover, $0 \leq \kappa '(t) \leq \frac{C}{(R-\rho)^2}$ in $[0, \ T]$.

Defining $Q_R^s$ as

and selecting test function $\phi$ as ${\eta ^2}{\kappa ^2}(u - \lambda)$, it is easy to obtain

Note that $u(t, x) = \frac{{\partial u}}{{\partial \nu }} = 0$ for any $(x, t) \in \partial \Omega \times (0, T)$. Thus, by applying integration by part on $\int {\int_{Q_R^s} {{\partial _t}[{\eta ^2}{\kappa ^2}{{(u - \lambda)}^2}]{{\rm{d}}}x{{\rm{d}}}t} }$ and $\int {\int_{Q_R^s} {u|\nabla u{|^{p - 2}}\nabla u \cdot \nabla [{\eta ^2}{\kappa ^2}(u - \lambda)]{{\rm{d}}}x{{\rm{d}}}t} }$, we obtain

Substituting (20) and (21) into (19), we can obtain

which means

Using the Hölder and Young inequalities, and applying $|\nabla \eta | \le \frac{C}{{{{(R - \rho)}^2}}}$, we have

Substituting (23) and (24) into (22), we have that

Rearranging the above inequality yields

Since $0 \le \xi \le {M_0}$, using the Hölder inequality to $\int {\int_{Q_R^s} {{\eta ^2}{\kappa ^2}{{(u - \lambda)}^2}{{\rm{d}}}x{{\rm{d}}}t} }$, $\int {\int_{Q_R^s} {{\eta ^2}\kappa \kappa '{{(u - \lambda)}^2}{{\rm{d}}}x{{\rm{d}}}t} }$ and $\int {\int_{Q_R^s} {\xi \cdot {\eta ^2}{\kappa ^2}(u - \lambda){{\rm{d}}}x{{\rm{d}}}t} }$, we have

Removing the non-negative terms $\frac{1}{p}\int {\int_{Q_R^s} {{\eta ^2}{\kappa ^2}|\nabla u{|^p}{{\rm{d}}}x{{\rm{d}}}t} }$ from which implies that for any $s \in {I_\rho }$,

On the other hand, by removing the non-negative term $\int_{{B_R}} {{\eta ^2}{\kappa ^2}{{(u - \lambda)}^2}{{\rm{d}}}x} {|_{t = s}}$ 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 $\rho$ and $R$ to be $\frac{1}{2}R$ and $\frac{3}{4}R$, respectively, in the Lemma 4.1 and setting $\lambda = 0$, we obtain

On the other hand, in Lemma 4.1, we choose $\rho = \frac{3}{4}R$ and $R$, respectively, and set $\lambda = 0$,

Combining (29) and (30), we can obtain the following Schauder estimates.

Theorem 3.2. Let $(u, \xi)$ be a weak solution of variational inequality (3), then

4.   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 ${\Delta _p}u$ 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.

Acknowledgments

The authors sincerely thank the editors and anonymous reviewers for their insightful comments and constructive suggestions, which greatly improved the quality of the paper. This work is supported by the National Social Science Fund of China in 2020 (No.20BMZ043), the Doctoral Project of Guizhou Education University (No.2021BS037) and the Science Research Project of Guizhou Provincial Education Department (Youth Project, No.[2022]158).

Conflict of interest

The authors declare no conflicts of interest.