Local Hölder continuity of nonnegative weak solutions of inverse variation-inequality problems of non-divergence type

1.   Introduction and financial background

The variational inequality in the following form has received extensive research attention in recent years:

Here, the operator 'min' is used to control the inequality condition, where $Lu$ represents a linear parabolic operator or a degenerate parabolic operator. The existence of solutions to parabolic variational inequalities has been analyzed using the finite element method in the literature ^[1,2,3]. These studies not only provide the discrete schemes but also analyze the convergence between the discrete schemes and the parabolic variational inequalities, thus establishing the existence of solutions to the variational inequality problem. The existence of solutions to parabolic variational inequalities has also been investigated in ^[4,5] using the Banach fixed-point theorem and the surjectivity theorem, respectively. The uniqueness of solutions to parabolic variational inequalities has been analyzed in ^[6,7] through energy estimates of weak solution interpolations in Sobolev spaces. Literature ^[8] estimates the energy of the weak solution's second-order gradient in Sobolev spaces, establishing the higher integrability and regularity of the weak solution. Currently, research on higher integrability and regularity mainly focuses on parabolic initial-boundary value problems ^[9,10], while it is relatively scarce in the field of variational inequality studies. Literature ^[11] investigates the Hölder continuity property of variational inequalities. By utilizing the Poincaré inequality and the Gagliardo-Nirenberg inequality, the Caccioppoli inequality is derived, which is then used to establish the Hölder continuity. The structure of this parabolic variational inequality is relatively simple, consisting of a first-order quasi-linear parabolic operator.

In recent years, research in financial theory has found that the inverse variational inequalities,

such as (2) in the Black-Scholes framework, are more suitable for studying the pricing of American options which enables investors to buy or sell the underlying stock at a predetermined price $K$ at any point within the option's time period $[0, T]$. In the Black-Scholes model, the price of American options also satisfies the variational inequality (2) and the parabolic operator (denoted by ${L_{BS}}$) satisfies ^[12,13]

Here, $S$ represents the underlying stock of the American option, $\sigma$ represents the volatility of the stock and $r$ represents the risk-free interest rate in the stock market. For American options, the initial conditions satisfy the following:

This paper investigates the existence and local Hölder continuity of weak solutions to the variational inequality (2) under the non-divergence degenerate parabolic operator

Additionally, we assume that $u_0$ is nonzero in the interior of $\Omega$, otherwise $u_0 \equiv 0$ in $\Omega$, which easily leads to $u \equiv 0$ in $\Omega_T$ being a solution to (1), rendering the study meaningless.

The motivation behind this research is the lack of documentation on the existence of weak solutions to variational inequality problems in the literature. Specifically, the authors focus on the inverse variational inequality model, where determining the upper bound of weak solutions using the traditional comparison principle is challenging. This motivates the need for new approaches to overcome this difficulty.

This paper introduces two key innovations to address the challenges mentioned above. First, the authors introduce a nonnegative constant ${M_0}$ and construct an integral inequality for ${(u - {M_0})_+}$ based on $Lu \le 0$. This novel approach allows for the determination of an upper bound for weak solutions, which was previously difficult to achieve using the traditional comparison principle. This innovation provides a new perspective on defining weak solutions for variational inequalities. Second, the authors construct an integral inequality using ${(u - k)_ \pm }$ and a nonnegative function $\phi$ on ${W^{1, p}}(\Omega)$. By choosing $\phi$ as a cut-off function, they are able to establish a lower bound for $u$ in a cylindrical subdomain. This lower bound enables the establishment of the Hölder continuity. This contribution is significant as it provides a new method for establishing continuity in variational inequality problems. Overall, this paper makes important contributions to the field by introducing innovative approaches to determine upper bounds for weak solutions and establishing continuity in variational inequality problems. These contributions fill a gap in the existing literature and provide valuable insights for future research in this area.

2.   Structure of weak solutions and preliminaries

In addition to providing several useful lemmas, this section constructs a weak solution to the inverse variational inequality (2) using the global boundedness of $u$. Considering that ${u_0} \ge 0$ in $\Omega$, we can deduce from (2) that

Furthermore, from (1) we also know that $Lu \le 0$ in $\Omega_T$. Therefore, for any nonnegative fixed constant $M_0$ and $t \in (0, T]$, multiplying both sides of $Lu \le 0$ by ${(u - M_0)_ +}$ and integrating over the domain ${\Omega _t}$, we have

By utilizing the method of integration by parts, we can obtain

Note that $\gamma \in (0, 1)$, $u|\nabla {(u - {M_0})_ + }{|^p}$ and $(1 - \gamma){(u - {M_0})_ + }|\nabla {(u - {M_0})_ + }{|^p}$ are nonnegative. By removing them in (5), we can obtain

which leads to

Also, since ${u_0} \in W_0^{1, p}(\Omega)$, if $M_0$ is sufficiently large, $\int_\Omega {({u_0} - {M_0})_ + ^2{\rm{d}}x} = 0$, which implies $\int_\Omega {(u(\cdot, t) - {M_0})_ + ^2{\rm{d}}x} = 0$. This means that

Next, we define the weak solution of the variational inequality (2). Considering the upper and lower bounds (4) and (7) of the solution to the variational inequality (2) and incorporating the methods from ^[8], we provide a set of maximal monotone maps

where $M_0$ is a positive constant.

Definition 2.1. A pair $(u, \xi)$ is said to be a generalized solution of the inverse variation-inequality (2) if it satisfies the following conditions: $u \in {L^\infty }(0, T, W_0^{1, p}(\Omega))$, ${\partial _t}u \in {L^\infty }(0, T, {L^2}(\Omega))$, and $\xi \in G$ for any $(x, t) \in \Omega_T$, (a) $u(x, t) \ge {u_0}(x), \; u(x, 0) = {u_0}(x)\; {{for}}\; {{any}}\; (x, t) \in {\Omega _T}$, (b) for every test function $\varphi \in {C^1}({\bar \Omega _T})$, there exists an equality that holds:

Finally, we introduce two lemmas that are utilized in the proof of the Hölder continuity of the weak solution to the inverse variation-inequality (2). The detailed proof can be found in ^[14,15].

Lemma 2.1. Suppose that it 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. There exists a positive constant $C$ depending only on $N$ and $p$ such that

Lemma 2.1 is used to obtain a lower bound for the weak solution using the limit method, which is then used to prove the Hölder continuity of the weak solution. Lemma 2.2 is used to obtain the conditions required for Lemma 2.1.

3.   Existence

To characterize the weak solution defined by $\xi \in G(u - u_0)$, we introduce the penalty function

and $\mathop {\lim }\limits_{\varepsilon \to + 0} {\beta _\varepsilon }(z) = \left\{ {\begin{array}{*{20}{c}} 0 & {z > 0, }\\ { - {M_0}} & {z = 0.} \end{array}} \right.$ Consider the following parabolic auxiliary problem

From definition (10), it can be observed that when ${u_\varepsilon } \ge {u_0} + \varepsilon$, $L{u_\varepsilon } = 0$ in ${\Omega _T}$; and at the same time, when ${u_\varepsilon } < {u_0} + \varepsilon$, $L{u_\varepsilon } \le 0$ in ${\Omega _T}$. This is exactly the same as the situation in variational inequality (2), which is also the original intention of constructing the auxiliary problem. Additionally, let $t \to 0$; according to the definition of ${\beta _\varepsilon }(\cdot)$, we have

On the other hand, by utilizing (11), it can be inferred that

which indicates

Furthermore, due to ${u_\varepsilon } = {u_{0, \varepsilon }}(x)\, {\rm{in}}\, \partial {\Omega _T}$, by utilizing the principle of comparison, it can be inferred that

Based on the experience from references ^[4,8], we provide the weak solution of the auxiliary problem without proof and analyze the boundedness and energy estimation of the auxiliary problem (11) on this basis.

Definition 3.1. A function $u$ is considered a generalized solution of variation-inequality (1) if it meets the condition

and for any test-function $\varphi \in {C^1}({\bar \Omega _T})$, the equality

holds.

Next, we will analyze the properties of the weak solution of the parabolic auxiliary problem (11). Let's start by proving ${u_\varepsilon } \le {M_0}\; {\rm{in}}\; {\Omega _T}$. It is important to note that ${\beta _\varepsilon }({u_\varepsilon } - {u_0}) \le 0\, \, {\rm{in}}\, \, {\Omega _T}$. From (11), we can also deduce that $L{u_\varepsilon } \le 0\; {\rm{in}}\; {\Omega _T}$, which allows us to repeat the proof process of (6) and obtain

When $M_0$ is sufficiently large, it follows that $\int_\Omega {({u_0} - {M_0})_ + ^2{\rm{d}}x} = 0$. This indicates

By combining (12) and (15), it can be shown that there exists a sufficiently large positive constant $M_0$ such that

Now, we delve into the estimation of the gradient of $u_\varepsilon$. By selecting $u_\varepsilon$ as the basis function in Definition 3.1, we can derive

It is important to note that ${\beta _\varepsilon }({u_\varepsilon } - {u_0}) \le 0$ and ${u_\varepsilon } \ge 0$ in $\Omega _T$, which enables us to eliminate the nonnegative term $\int {\int_{{\Omega _T}} {{\beta _\varepsilon }({u_\varepsilon } - {u_0}) \cdot {u_\varepsilon }{\rm{d}}x{\rm{d}}t} \le 0}$ and obtain

Building upon this, by utilizing the Hölder and Young inequality, we can derive the following result

For a detailed proof, please refer to ^[8], as it will not be reiterated here.

By selecting $u_\varepsilon ^{\gamma - 1}{\partial _t}{u_\varepsilon }$ as the basis function, we have

By utilizing Eq (10) to ${\beta _\varepsilon }({u_\varepsilon } - {u_0})$, we obtain

Given the setting $\mu = \frac{1}{2}(\gamma + 1)$, let us analyze the integration $\int {\int_{{\Omega _T}} {{\partial _t}{u_\varepsilon } \cdot u_\varepsilon ^{\gamma - 1}{\partial _t}{u_\varepsilon }{\rm{d}}x{\rm{d}} t} }$. It is important to note that according to (14), we have $\int_\Omega {u{{(\, \cdot \, , T)}^2}{\rm{d}}x} \le \int_\Omega {u_{0, \varepsilon }^2{\rm{d}}x}$, which consequently leads to

By applying the integral transformation to $\int {\int_{{\Omega _T}} {{u_\varepsilon }|\nabla {u_\varepsilon }{|^{p - 2}}\nabla {u_\varepsilon }\nabla (u_\varepsilon ^{\gamma - 1}{\partial _t}{u_\varepsilon }){\rm{d}}x{\rm{d}}t} }$, we can obtain

Please note that $\gamma \in (0, \, 1)$ and $\mu \in (0, \, 1)$. Substituting Eqs (20)–(22) into (19), we have

Considering the estimation of $\int {\int_{{\Omega _T}} {u_\varepsilon ^\gamma |\nabla {u_\varepsilon }{|^{p - 2}}\nabla {u_\varepsilon }\nabla ({\partial _t}{u_\varepsilon }){\rm{d}}x{\rm{d}}t} }$ in (23), from (16) and (18), we can obtain

Substituting (24) into (23), we obtain

According to the estimates obtained from (16), (18) and (25), it can be concluded that the set $\{ {u_\varepsilon }, \varepsilon \ge {{0\} }}$ possesses a convergent subsequence and a function u such that

It is worth noting that (27) also employs Lebesgue's dominated convergence theorem in its proof.

Lemma 3.1. Let ${u_\varepsilon }$ be a weak solution to the parabolic auxiliary problem (2), then there exists $\xi \in G$ such that

Proof. According to (16), the sequence $\{ {\beta _\varepsilon }({u_\varepsilon } - {u_0}), \varepsilon \ge 0\}$ has a convergent subsequence and, furthermore,

The following demonstrates the validity of $\xi \in G$. It should be noted that when ${u_\varepsilon } \ge {u_0} + \varepsilon$, ${\beta _\varepsilon }({u_\varepsilon } - {u_0}) = 0$ and, as a result,

This implies that when $u > {u_0}$, $\xi = 0$. On the other hand, when ${u_\varepsilon } \le {u_0} + \varepsilon$, $- {M_0} \le {\beta _\varepsilon }({u_\varepsilon } - {u_0}) \le 0$. Based on the boundedness of limits, $- {M_0} \le \xi \le 0$, which indicates $\xi \in G$.

By combining Eqs (26)–(29) and employing the limit method for $\varepsilon$ as described in ^[4], the existence of a weak solution can be obtained.

Theorem 3.1. Assume that $\gamma \in (0, 1)$ and ${u_0} \in W_0^{1, p}(\Omega)$, then (1) admits a solution within the class of Definition 2.1.

4.   Hölder continuity

For any $({t_0}, {x_0}) \in {\Omega _T}$, let $Q = Q(\rho, \theta) = {B_\rho }({x_0}) \times ({t_0} - \theta, {t_0})$, where $\rho$ and $\theta$ are sufficiently small nonnegative constants to ensure $Q \subset {\Omega _T}$. In this section, we consider the Hölder continuity of the weak solution $u$ to the inverse variational inequality (2) on $Q$.

We denote ${(u - k)_ + }$ and ${(u - k)_ - }$ as ${(u - k)_ \pm }$, where $k$ is a positive constant, ${(u - k)_ + } = \max \{ u - k, 0\}$, and ${(u - k)_ - } = \max \{ k - u, 0\}$. Let $\phi \in {W^{1, p}}({\Omega _T})$ be a given function. In the context of (9), we select the test function $w = {\phi ^p} \times {(u - k)_ \pm }$ and set $\phi \ge 0$. It can be readily observed that

It is important to note that $\gamma \in (0, 1)$, $\phi$ and ${(u - k)_ \pm }$ are nonnegative, which leads to the conclusion of

On the other hand, it is easily derived from (8) that

Removing the nonpositive term $(1 - \gamma)\int_{{t_0} - \theta }^{{t_0}} {\int_\Omega {|\nabla u{|^p}{\phi ^p} \times {{(u - k)}_ \pm }{\rm{d}}x{\rm{d}}t} }$ and the nonnegative term $\int_{{t_0} - \theta }^{{t_0}} {\int_\Omega {\xi \cdot {\phi ^p} \times {{(u - k)}_ \pm }{\rm{d}}x{\rm{d}}t} }$, we have

Integrate the temporal gradient term $\int_\Omega {{\partial _t}({\phi ^p} \times (u - k)_ \pm ^2){\rm{d}}x}$ with respect to time, yielding

Upon integrating the spatial gradient term $\int_\Omega {u|\nabla u{|^{p - 2}}\nabla u\nabla ({\phi ^p} \times {{(u - k)}_ \pm }){\rm{d}}x}$, we obtain

Further analysis of $\int_\Omega {u|\nabla u{|^{p - 2}}\nabla u \times {{(u - k)}_ \pm }\nabla {\phi ^p}{\rm{d}}x}$ in (33) reveals that the Hölder and Young inequalities can be employed to obtain

By combining Eqs (33) and (34), and substituting them together with Eq (32) into (31), we obtain the following result, which serves as the cornerstone for proving the weak solution's Hölder continuity.

Theorem 4.1. Let $k \ge 0$ and $\phi \in {W^{1, p}}({\Omega _T})$ be any nonnegative constants. If $({t_0} - \theta, {t_0}) \subset (0, T)$ holds for any nonnegative constant $\theta$, then

For any given $({t_0}, {x_0}) \in {\Omega _T}$, select R to be sufficiently small such that ${Q_n} = Q({R_n}, R_n^p) \in {\Omega _T}$. Furthermore, let us define ${\mu ^ + } = \mathop {ess\sup }\limits_{Q(R, {R^p})} u$, ${\mu ^ - } = \mathop {ess\inf }\limits_{Q(R, {R^p})} u$, $\omega = \mathop {osc}\limits_{Q(R, {R^p})} u = {\mu ^ + } - {\mu ^ - }$ and also utilize the symbol ${R_n} = \frac{1}{2}R + \frac{1}{{{2^{n + 1}}}}R$.

Lemma 4.1. Given the definitions of ${k_n} = {\mu ^ - } + \frac{1}{{{2^{{s_*} + 1}}}}\omega + \frac{1}{{{2^{{s_*} + n + 1}}}}\omega$, $n = 1, 2, 3, \cdots$, it follows that

Proof. According to the definition of ${k_n}$, when $u$ takes ${\mu ^ - }$, ${\left({\frac{{{2^{{s_{_*}}}}}}{\omega }} \right)^{p - 2}}(u - {k_n})_ - ^p$ reaches its maximum; thus,

At this point, $(u - {k_n})_ - ^2$ satisfies

By combining Eqs (37) and (38), the result is proven to hold.

Next, we analyze the weak solution's Hölder continuity. Let ${s_*} > 1$ be set and define the truncation function

Additionally, assume that $\phi _n$ satisfies the condition

Choose $\phi = {\phi _n}(x, t)$ and $k = {k_n}$. Furthermore, due to $\int_{{B_n}} {(\phi _n^p(x, {t_0} - R_n^p) \times (u(x, {t_0} - R_n^p) - {k_n})_ - ^2){\rm{d}}x} = 0$, from (35) we conclude that

After organizing, we have

The validity of the last inequality in the above equation is ensured by utilizing (15). By applying (31) to $\int_{{t_0} - R_n^p}^{{t_0}} {\int_{{B_n}} {(u - {k_n})_ - ^2{\rm{d}}x{\rm{d}}t} } + \frac{1}{{{p^2}}}M_0^{\frac{{p - 1}}{p}}\int_{{t_0} - R_n^p}^{{t_0}} {\int_{{B_n}} {|(u - {k_n})_ - ^p{\rm{|d}}x} {\rm{d}}t}$, the result is

which allows (42) to be rewritten as

Here, it is easy to deduce from (15) that there exists a nonnegative constant $C$ such that

If not, for any small nonnegative constant $C_1$, we have

This implies that $u \equiv 0$ or $|\nabla {(u - {k_n})_ - }{|^p} \times \phi _n^p \equiv 0$ in ${B_n}$. If $|\nabla {(u - {k_n})_ - }{|^p} \times \phi _n^p \equiv 0$, the continuity result for Hölder holds directly. If $u \equiv 0$, then from (16) it can be seen that this contradicts the fact that $u_0$ is not zero everywhere inside $\Omega$. Combining (43) and (44), we have

To facilitate the discussion, define ${A_n} = \{ x \in {B_n}|u \le {k_n}\}$, and it can be derived from Eqs (43) and (45) that

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

Furthermore, from $(u - {k_n})_ - ^2$ we can also obtain

By combining (47) and (48) and substituting the result into Eq (46), we obtain

Simplifying Eq (49) leads to the inequality of

By utilizing Lemma 2.1, we can obtain $\int_{{t_0} - R_n^p}^{{t_0}} {|{A_n}|{\rm{d}}t} \to 0\; {\rm{as}}\; n \to \infty$ if ${Y_0} \le {\left({\frac{{2p}}{{{R^p}}}} \right)^{\frac{{N + p}}{p}}}{4^{\frac{{{{(N + p)}^2}}}{p}}}$. Consequently, we can draw the following conclusion.

Theorem 4.2. If ${s_*}$ is sufficiently large,

Theorem 4.3. The weak solution of the variational inequality problem (1) possesses Hölder continuity, i.e., there exists a nonnegative constant $\sigma$ such that

Proof. Due to the presence of ${\mu ^ + } = \mathop {ess\sup }\limits_{Q(R, {R^p})} u$ and ${\mu ^ - } = \mathop {ess\inf }\limits_{Q(R, {R^p})} u$,

Substituting (52) into (51) and selecting $\sigma = (1 - \frac{1}{{{2^{{s_{_*}} + 1}}}})$, the theorem proposition holds.

5.   Conclusions

In recent years, numerous scholars have conducted theoretical research on variational inequalities. Variational inequalities of the form (1) are convenient for scholars to use the comparison principle to obtain upper bounds for the solution u, thereby constructing weak solutions through the use of the maximal operator. In the case of the inverse variational inequality (2), since we can only obtain $Lu \le 0$ in $\Omega_T$, the use of the comparison principle can only demonstrate that the solution of the inverse variational inequality (2) has a nonnegative lower bound, thus limiting the study of inverse variational inequalities.

The present study starts by considering $Lu \le 0$ and ${(u - {M_0})_+ }$ and obtains an energy estimate for ${(u - {M_0})_+ }$. It is demonstrated that when $M_0$ is sufficiently large, the upper bound of this estimate is 0, thereby obtaining a global upper bound for the solution of the inverse variation inequality (2). Subsequently, a continuous method is employed to prove the existence of weak solutions for the inverse variation inequality (2). Finally, we analyzed the Hölder continuity of the weak solution to the inverse variation inequality (2). Combining the global upper and lower bounds of the weak solution to the inverse variation inequality (2), we obtained an integral inequality involving ${\phi ^p} \times {(u - k)_ \pm }$ starting from the weak solution, as shown in Theorem 4.1. We then chose $\phi$ as the cut-off factor for the subdomain $Q(\frac{1}{2}R, {(\frac{1}{2}R)^p})$, and by using Hölder and Young inequalities as well as a sequence convergence result (see Lemma 2.1), we established the Hölder continuity of the weak solution.

So far, there are still some limitations in this paper: (i) Regarding the parameter $\gamma$, the proof in (15) relies on the constraint $\gamma \in (0, 1)$, and the proof in (17) relies on the constraint $\gamma < 2$; thus, the paper consistently assumes $\gamma \in (0, 1)$. (ii) Regarding the parameter $p$, the existence of weak solutions and the Hölder continuity are both repeatedly used under the condition $p \ge 2$ using Hölder and Young inequalities. Therefore, the paper also consistently assumes $p \ge 2$. The author intends 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 are very grateful to the five anonymous referees for their insightful comments and constructive suggestions, which considerably improve our manuscript. This work was supported by the Key R & D Projects of Weinan Science and Technology Bureau (No. 2020ZDYF-JCYJ-162) and General Project of Scientific Research Fund of Shaanxi Railway Engineering Vocational and Technical College (NO. 2023KYYB-01).

Conflict of interest

The author declares there is no confict of interest.