Local Hölder continuity of inverse variation-inequality problem constructed by non-Newtonian polytropic operators in finance

1.   Introduction

In recent years, the study of the following variational inequality has attracted the interest of scholars:

where $Lu$ is a linear parabolic operator or a degenerate parabolic operator. Due to the satisfaction of the condition $Lu \geq 0$ in $\Omega_T$, researchers have found it convenient to use the comparison principle to obtain upper bounds for the solutions. This approach has been combined with limit methods ^[1,2], the Leray-Schauder fixed point theorem ^[3,4], or semi-discrete methods ^[5,6] to prove the existence of solutions. Additionally, some scholars have started from weak solutions and obtained integral inequalities for the difference between two weak solutions, analyzing the stability and uniqueness of weak solutions with respect to initial values ^[7,8,9]. The authors from ^[10,11,12] have demonstrated the explosive nature of weak solutions under certain special conditions through energy estimates. The authors have obtained Caccioppoli inequalities that match the variational inequality by analyzing integral inequalities of weak solutions in locally cylindrical regions, and subsequently studied the Schauder estimates for weak solutions ^[13,14].

In recent years, research on the pricing of financial derivative products with embedded early exercise provisions has found that inverse variation-inequalities, such as the one shown below, are more suitable for

For example, researches from ^[15,16] analyzed the pricing problem of American options under the Black-Scholes model, and the value was reduced to the free boundary problem of the variation inequality (2). Therefore, the parabolic operator $Lu$ (denoted as $L_{BS}u$) satisfies

where $\sigma$ represents the volatility of the underlying stock of the option, and $r$ is the risk-free interest rate in the financial market. Based on the aforementioned financial background considerations, the author of this study investigates the inverse variation-inequality problems with the degenerate parabolic operator in non-divergence form

Additionally, we impose the condition that the initial value $u_0$ satisfies ${u_0} \in W_0^{1, p}(\Omega)$. For a recent study on inverse variational inequalities in a different context, please refer to ^[17]. In that study, an inverse quasi-variational inequality is solved using a dynamical system.

In this paper, we provide the weak solution to the variational inequality (1) and prove the existence of weak solutions. We also prove that the weak solution satisfies an energy inequality in a local cylindrical region, and based on this, we establish the Holder continuity and Harnack inequality of the weak solution. The study of such conclusions is usually focused on degenerate parabolic equation initial-boundary value problems, and research on variational inequalities is still rare.

Due to the fact that the inverse variational inequality (2) implies $Lu \le 0{\kern 3pt} {\rm{in}}{\kern 3pt} {\Omega _T}$, it no longer allows us to determine the upper bound of the solution $u$ and establish the existence of weak solutions through the comparison principle, as in the traditional variational inequality (1). First, this difficulty is overcome by analyzing the energy upper bound of ${(u - {M_0})_ +}$. We can select a suitable $M_0$ as an upper bound for $u$, which is also an innovative aspect of this paper. Second, this paper also explores the Harnack inequality and Hölder continuity of weak solutions by analyzing the weak solutions of the reverse variational inequality (2) and combining it with the integral inequality of ${(u - k)_ \pm }$. This analysis is further enhanced by the use of the cut-off factor and the selection of an appropriate $k$, which adds another innovative aspect to this study.

2.   Existence of weak solution

This section is dedicated to addressing our specific problem: We begin by providing a clear definition of a nonnegative weak solution to Eq (1). To begin, it can be inferred from Eq (1) that

In fact, by utilizing inequality (1) once again, we have $Lu \le 0$ in ${\Omega _T}$. Furthermore, since $u(t, x) = {u_0} = 0$ in $\partial \Omega \times (0, T)$, when, using the comparison principle, we can conclude that (5) still holds.

Next, we analyze the upper bound of $u$. Let us choose a constant $M_0 > 0$ as a parameter. Multiplying both sides of $Lu \le 0$ by $(u - M_0)_+$ and integrating over $\Omega$ yields (note that $(u - M_0)_+ \ge 0$),

On one hand, when $u \ge {M_0}$ occurs, ${\partial _t}{(u - {M_0})_ + } = {\partial _t}u$, thereby resulting in

On the other hand, when $u < {M_0}$ occurs, ${(u - {M_0})_ + } = 0$ and ${\partial _t}{(u - {M_0})_ + } = 0$, leading to

Combining (6)–(8), we obtain

Note that

when $u < M_0$, while

when $u$ is greater than or equal to $M_0$. Therefore, (9) implies the significance of

Due to $u \ge {u_0} \ge 0$ and $\sigma - \gamma \ge 0$,

is nonnegative. Combining (10), we have

Furthermore, due to ${u_0} \in W_0^{1, p}$, when $M_0$ is sufficiently large,

In this case,

By combining (5) and (12), we can demonstrate that the inverse variational inequality (2) satisfies

Therefore, in ^[12], before providing a weak solution to the inverse variational inequality (2), we first present a set of maximal monotone maps

If $\xi \in G(u - u_0)$, it is easy to see that when $u > u_0$, $\xi = 0$; and in this case $Lu = 0$. When $u = u_0$, $\xi \ge 0$, and in this case we also have $Lu \ge 0$. This inspires us to use $Lu = \xi$ to construct a weak solution for the variational inequality (2).

Definition 2.1. A pair $(u, \xi)$ is considered a generalized solution of the inverse variation-inequality (2) if it satisfies the following conditions:

(a) $u \in {L^\infty }(0, T, {H^1}(\Omega)), \; {\partial _t}u \in {L^\infty }(0, T, {L^2}(\Omega))$.

(b) $\xi \in G\; {\rm{for}}\; {\rm{any}}\; (x, t) \in {\Omega _T}$.

(c) For fixed $\nu = \frac{{\sigma - 1}}{p} + 1$ and for every test-function $\varphi \in {C^1}({\bar \Omega _T})$, there exists an equality

By utilizing (13) and (14), combined with a standard energy method from ^[2,12], we can establish the existence of a weak solution for the inverse variational inequality (2).

Theorem 2.1. Assuming that ${u_0} \in W_0^{1, p}(\Omega)$ in ${\Omega _T}$, the inverse variational inequality (2) has a solution $(u, \xi)$ within the class defined in Definition 2.1.

The final part of this section is dedicated to introducing some notation and presenting several previously established results, which will be used in the subsequent proof of the Hölder continuity. The detailed proof can be found in ^[17].

Lemma 2.1. Assume that $\{ {Y_n}\}, n = 1, 2, 3, \cdots$ is a nonnegative sequence satisfying

If ${Y_0} \le {C^{{{ - 1} / \alpha }}}{b^{{{ - 1} / {{\alpha ^2}}}}}$, then ${Y_n} \to 0, n \to \infty$.

Lemma 2.2. Assuming that $p \ge 2$, there exists a positive constant $C$ such that

where $C$ depends only on $N$ and $p$.

3.   Integral inequality

Along this section, we assume that $u$ is a nonnegative weak solution to Eq (1) with $p \geq 2$. Our objective is to establish an integral inequality, which will be used to determine the Hölder continuity of the weak solution on the domain

where $\rho$ and $\theta$ are positive undetermined constants. Of course, $\rho$ and $\theta$ should be sufficiently small to ensure $Q \subset {\Omega _T}$. Let us define

and introduce the symbol

where ${s_*}$ is a nonnegative undetermined constant. We obtain the Hölder estimate for the weak solution of inequality (2) by using the upper bound estimate of $\mathop {ess\sup }\limits_{Q(\frac{1}{2}R, d{{\left({\frac{1}{2}R} \right)}^p})} u$ that includes $\omega$. In order to estimate $\mathop {ess\inf }\limits_{Q(\frac{1}{2}R, d{ \left(\frac{1}{2}R\right)^p})} u$, we construct ${k_n})_ -$ and simultaneously construct ${k_n})_ +$ to estimate $\mathop {ess\sup }\limits_{Q(\frac{1}{2}R, d{{\left({\frac{1}{2}R} \right)}^p})} u$. In order to prove the Hölder continuity, ${s_*}$ must satisfy the condition ${s_*} > 1$.

Lemma 3.1. Assuming $p \ge 2$ and $\nu = \frac{{\sigma - 1}}{p} + 1$, one can infer

Proof. According to the definition of $k_n^-$, it is easy to obtain

Since $(u - {k_n^-})_ - ^{p\nu }$ reaches its maximum, when $u$ takes the value ${\mu ^ + }$,

holds. At this point, $(u - {k_n^-})_ - ^{\nu + 1}$ satisfies

By combining Eqs (16) and (17), the result is proven to hold.

In order to achieve the desired outcome, a test function $w = {\phi ^p} \times (u - k)_ \pm ^\nu$ is selected, resulting in

Considering that $\int {\int_{{\Omega _T}} {{\partial _t}u \times {\phi ^p} \times (u - k)_ \pm ^\nu {\rm{d}}x{\rm{d}}t} }$ is not suitable for integration calculations, the following transformation is performed:

In $\int_\Omega {{u^\nu }|\nabla {u^\nu }{|^{p - 2}}\nabla {u^\nu }\nabla [{\phi ^p} \times (u - k)_ \pm ^\nu ]{\rm{d}}x}$, a differential transformation is applied to $\nabla [{\phi ^p} \times (u - k)_ \pm ^\nu ]$, resulting in

Further utilizing the Hölder and Young inequalities, we can obtain the expression

Consequently, by combining Eqs (18)–(21), we can obtain the equation

Theorem 3.1 Let $u$ be a weak solution of the inverse variational inequality (2) with $p \ge 2$, then it follows that

In (23), we utilize the condition $\phi (x, {t_0} - \theta) = 0$, which readily yields

Additionally, it is worth noting that by selecting suitable $\phi$ and $(u - k)_ \pm$ in (22), we can obtain local estimates for the weak solution $u$, thereby establishing the Harnack inequality and Hölder continuity.

4.   Results towards local continuity

This section is devoted to analyzing the Hölder continuity of weak solutions of the inverse variational inequality (2). We first examine the local lower bound estimate of weak solutions $u$ of the inverse variational inequality (2), and define a cut-off function ${\phi _n}(x, t)$ on ${Q_n}$ as described in

Additionally, we assume that ${\phi _n}(x, t)$ satisfies the condition

In (23), $(u - k)_ \pm$ is set as ${(u - k)_ - }$, while $k$ is chosen as $k_n^-$, resulting in

Due to the presence of $\sigma - \gamma > 0$ and $\nu = \frac{{\sigma - 1}}{p} + 1 > 1$,

After removing them, we have

Further analysis of $\int_{{t_0} - {{dR} / 2}}^{{t_0}} {\int_{{B_n}} {{\phi ^{p - 1}} \times (u - k_n^-)_ - ^{\nu + 1}{\rm{d}}x{\rm{d}}t} } + \frac{d}{{{p^2}{\nu ^{p - 1}}}}\int_{{t_0} - {{dR} / 2}}^{{t_0}} {\int_{{B_n}} {|(u - k_n^-)_ - ^{p\nu }{\rm{|d}}x} {\rm{d}}t}$ is conducted by applying Lemma 3.1,

By substituting the aforementioned results into Eq (27), we can obtain

For the purpose of facilitating the discussion, let us define ${A_n} = \{ x \in {B_n}|u \le {k_n^-}\}$. Consequently, it can be derived from Eq (28) that

Applying Lemma 2.2 to $||(u - k_n^-)_ - ^{}{\phi _n}||_{{L^p}({Q_n})}^p$, we obtain

Lemma 4.1. If $u$ is a weak solution of the inverse variational inequality (2) with $p > 2$, then

Proof. Due to

it follows that

thereby

Furthermore, due to

it follows that

thereby

By combining Eqs (31) and (32), Lemma 4.1 is proven.

Continuing the analysis of the lower bound for weak solutions by combining (30) and Lemma 4.1 and substituting the obtained result into (29), it can be easily deduced that

Consequently, simplifying (33) yields

This from Lemma 2.2 implies that $\int_{{t_0} - {{dR} / 2}}^{{t_0}} {|{A_n}|{\rm{d}}t} \to 0\; {\rm{as}}\; \to n$, if

It is worth noting that $\sigma \ge 1$ when selecting a sufficiently large $S_*$ such that (34) always holds, thus leading to the following result.

Theorem 4.1. If $\sigma \ge 1$, selecting a sufficiently large ${s_*} > 1$, it holds that

Furthermore, if

then, (35) still holds.

Next, we analyze the upper bound of the weak solution. By applying Lemma 4.1, it is easy to obtain

Consequently, in Eq (23) we set $(u - k)_ \pm$ as ${(u - k_n^+)_ + }$ and eliminate the nonpositive term $\frac{{\sigma - \gamma }}{{{\nu ^p}}}\int {\int_{{\Omega _T}} {|\nabla {u^\nu }{|^p} \times {\phi ^p} \times (u - k_n^+)_ + ^\nu {\rm{d}}x{\rm{d}}t} }$, resulting in

Note that

By applying Lemma 3.1, we can obtain

which implies that

For the sake of convenience in the discussion, we will continue to use the symbol ${A_n} = \{ x \in {B_n}|u \le {k_n^+}\}$, hence

By utilizing Lemma 2.2, we can obtain the following estimation

Following the same approach as in Lemma 4.1, we can deduce that

Combining (41) and (42) and substituting the result into (40), we can simplify and deduce that

Clearly, the equation above and Lemma 2.2 imply that $\int_{{t_0} - {{dR} / 2}}^{{t_0}} {|{A_n}|{\rm{d}}t} \to 0\; {\rm{as}}\; \to n$, if

Thus we have

Due to the fact that

combining Eqs (35) and (45), we obtain

Theorem 4.2. (Hölder continuity) For any $(x, t) \in Q(\frac{1}{2}R, d{(\frac{1}{2}R)^p})$, if $\sigma > 1$, there exists a nonnegative constant $C$ such that

Furthermore, if (34) and (44) hold, the above inequality still holds.

In fact, by choosing

in (46), the conclusion of Theorem 3.3 is evident. Furthermore, by selecting

we have the following result.

Theorem 4.3. (Harnack's inequality) Assuming $\sigma \ge 1$, there exists a nonnegative constant $C$ such that

The Harnack inequality implies the following Hölder modulus estimate, as indicated by the literature in ^[17].

Theorem 4.4. (Hölder's modulus estimate) Let $u \in {L^\infty }(0, T;W_0^{1, p}(\Omega))$ be a weak solution of the inverse variation-inequality (2) and $\sigma \ge 1$. Then, there exists a constant $C$ and $\beta \in (0, 1)$, for any $\Omega ' \subset \Omega$, such that ${[u]_{\beta, \frac{1}{2}\beta; {{\Omega '}_T}}} \le C$.

5.   Conclusions

This paper aimed to explore a specific type of inverse variation inequality problem

which was formulated using degenerate parabolic operators in non-divergence form

First, by incorporating ${(u - {M_0})_ + }$ into $Lu \le 0$ in ${\Omega _T}$, we obtained the following integral inequality:

Subsequently, we derived the upper and lower bounds for the inverse variation inequality problem (2) and utilized them to construct a weak solution for the inverse variation-inequality problem (2).

Next, in the weak solution, the test function $w = {\phi ^p} \times (u - k)_ \pm ^\nu$ was chosen and an integral inequality was obtained using the Hölder and Young inequalities, as shown in Theorem 3.1. Finally, the incorporation of a cut-off factor in Theorem 3.1 yields the Hölder continuity of the weak solution to problem (2), the Harnack inequality and the Hölder modulus estimate.

There are still some areas in this paper that can be improved. The current study only considered the case where $\sigma > \gamma$, and the existence of weak solutions cannot be proven if $\sigma < \gamma$. Additionally, in the proof process of the Hölder continuity in Section 4, the condition $\sigma > \gamma$ was also used to ensure that $\frac{{\sigma - \gamma }}{{{\nu ^p}}}\int {\int_{{\Omega _T}} {|\nabla {u^\nu }{|^p} \times {\phi ^p} \times (u - k)_ + ^\nu {\rm{d}}x{\rm{d}}t} }$ was nonnegative. In Lemma 2.2, the parameter $p$ was restricted to be greater than two. In future research, we will attempt to analyze the impact of these restrictive conditions on the results.

Use of AI tools declaration

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

Acknowledgments

The author sincerely thanks the editors and anonymous reviewers for their insightful comments and constructive suggestions, which greatly improved the quality of the paper.

Conflict of interest

The authors declare no conflicts of interest.