Sobolev estimates and inverse Hölder estimates on a class of non-divergence variation-inequality problem arising in American option pricing

1.   Introduction

This study investigates a specific variational inequality problem described by

with a non-divergence parabolic operator

Here, $\Omega$ denotes a bounded and open subset of $\mathbb{R}^n$. We consider the case where $p \ge 2$, $\gamma \in (0, 1)$ and $T > 0$ are positive constants, $\Omega _ T = \Omega \times (0, T)$, and $u_0$ satisfies

From (1), we can infer that $u > u_0$ and $Lu \le 0$ in $\Omega_T$. It is easily observed that when $u > u_0$, then $Lu = 0$ in $\Omega_T$; conversely, when $u = u_0$, it follows that $Lu \le 0$. Therefore, in some literature ^[1,2,3], the variational inequality (1) is often stated in the following manner:

It is evident that the above formulation is not as concise as the model presented in (1). This paper adopts the statement of the model in (1) for clarity and simplicity.

The study of variational inequality problems of the form (1) originated from the pricing problems of American contingent claims with the inclusion of early exercise provisions ^[1]. The inclusion of early exercise provisions results in a variational inequality model that is characterized by Eq (1). Further studies on this aspect can be found in ^[2,3], and necessary explanations are provided in Section 2. Therefore, we will not repeat them here.

In recent years, there has been an increasing amount of theoretical research on variational inequalities under the framework of linear and quasilinear parabolic operators. In ^[4], the solvability and regularity of quasilinear parabolic obstacle problems were studied using a symmetric dual-wind discontinuous Galerkin (DG) method. Reference ^[5] investigated a new class of constrained abstract evolutionary variational inequalities in three-dimensional space. By utilizing mathematical analysis of the unsteady Oseen model for generalized Newtonian incompressible fluids, sufficient conditions for the existence of weak solutions were obtained. Reference ^[6] focused on studying the existence and stability of weak solutions to variational inequalities under fuzzy parameters. By introducing two parameters into the mappings and constraint sets involved, ^[6] established the existence results for weak solutions of parameter fuzzy fractional differential variational inequalities (PFFDVI) and further analyzed the compactness and continuous dependence on the initial values of PFFDVI. For more results on the existence of solutions, please refer to ^[7,8,9].

There have been some novel results in theoretical research on variational inequalities as well. Reference ^[10] established the local upper bounds, Harnack inequalities, and Hölder continuity up to the boundary for solutions of variational equations defined by degenerate elliptic operators. Studies on the Hölder continuity of solutions to variational inequalities under parabolic operator structures and other regularity results can be found in ^[11,12]. Reference ^[13] applied regularization and penalization operator methods to prove the existence of solutions to nonlinear degenerate pseudo-parabolic variational inequalities defined in regions with microstructures, and derived a priori estimates for solutions to the microscale problem.

Inspired by ^[10,11,12], this study investigates the inverse Hölder estimate for solutions of the variational inequality (1), which has not yet been addressed in the literature. First, we define the integral mean operator $I(t)$ on a spherical region and analyze its uniform continuity with respect to the time variable. Second, using the integral mean operator $I(t)$ and other inequality amplification techniques, we obtain a Sobolev estimate for the variational inequality (1). Then, by combining the Caccioppoli inequality for the variational inequality (1), we derive the inverse Hölder estimate for the gradient of solutions, which allows us to estimate the higher-order norms of the gradient of solutions using lower-order ${L^p}$ norms. Such results play a key role in many regularity studies.

2.   Statement of the problem and its background

The valuation of American options ultimately boils down to a well-posed problem of variational inequality similar to Eq (1). An American call option gives the investor the right to purchase an underlying asset at a predetermined price $K$ at any time within the investment horizon $[0, T]$. It is known that the value of an American option on the maturity date $T$ is given by

American options only grant the investor the right to exercise their option within the investment horizon $[0, T]$ without imposing any obligations. Thus, we have

If the value of the option $C(S, t)$ at time $t$ exceeds $\max \{ S - K, 0\}$, the investor may consider exercising the option, thereby forfeiting the opportunity for higher past returns. In this case, it follows that

as stated in ^[1]. Here, $\sigma$ represents the volatility of the underlying asset linked to the American option, and $r$ denotes the risk-free rate of return in the financial market. On the other hand, if the value $C(S, t)$ of the option at time $t$ is $\max \{ S - K, 0\}$, the investor can retain the option to capture higher returns, leading to

as indicated in ^[1].

This is a backward differential inequality. Let us define $\tau = T - t$ and $x = \ln S$. By combining Eqs (3) and (4), we have

where

It is worth noting for the reader that $x$ is a one-dimensional variable here, as in this financial example, the American option is linked to only one risky asset. Model (5) represents a specific financial case of the main problem studied in model (1), and thus, in model (1), we set $x$ as an n-dimensional variable. Additionally, the variational inequality suitable for American options shows a high degree of structural similarity with Eq (1).

On the other hand, transaction costs are often associated with the exercise of options, which necessitates a modification of the volatility $\sigma$. For instance, Pars and Avellaneda provided a transaction cost model where the volatility $\sigma$ satisfies ^[13]

Here, ${\sigma _0}^2$ represents the long-term volatility level, and the constant $\psi$ is determined by the trading frequency and cost ratio. This adjustment is also consistent with the parabolic operator structure of model (1).

Lastly, the spatial gradient of solutions to the variational inequality (5) applicable to American options not only measures the change in option value with respect to the underlying asset, but it also allows Black and Scholes to construct a risk-free portfolio to hedge against risk.

Based on this, we examine more general cases than variational inequality (5). This article mainly analyzes the Sobolev estimation of the solution to variational inequality (1) and the inverse Hölder estimation of the spatial gradient. Before that, let us give a few useful symbols.

Expanding upon this framework, we broaden our analysis to encompass wider scenarios than those addressed in variational inequality (5). The main objective of this paper is to conduct a thorough investigation into the Sobolev estimation for solving variational inequality (1), along with exploring the inverse Hölder estimation of the spatial gradient. Before delving into these aspects, we introduce a set of relevant mathematical symbols. For a given non-negative constant $\rho$, and any $(x_0, t_0) \in {\Omega _T}$, we define

to denote the ball in space ${{\rm{R}}_n}$ that is also contained within the bounded region $\Omega$. Similarly, we use

to denote the cylinder in space ${{\rm{R}}_{n + 1}}$ that is also contained within ${\Omega _T}$. Lastly, let $|{D_\rho }|$ represent the Lebesgue measure of the set ${D_\rho }$ in space ${{\rm{R}}_n}$, then the averaging operator of $u$ on ${D_\rho }$ is defined as ${I_\rho }(t)$, given by

With the help of the maximal monotone operator,

References ^[8,11,12] analyzed the existence of generalized solutions, whose definition is as follows:

Definition 2.1 A pair $(u, \xi)$ is defined as a generalized solution of the variational inequality (1) if it satisfies the following conditions:

(a) $u \in {L^\infty }(0, T, {W^{1, p}}(\Omega))$ and ${\partial _t}u \in {L^\infty }(0, T, {L^2}(\Omega))$,

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

(c) For every test function $\varphi \in C^1(\bar{\Omega}_T)$ and for each $t \in [0, T]$, the following equality holds:

In order to ensure the solvability of the problem, we still impose the restriction $\gamma \in (0, 1)$ in this study.

Lemma 2.2 The solution $u$ of variational inequality (1) is uniformly bounded in $\Omega _T$. That is, for any $(x, t) \le {\Omega_T}$, there exists a constant ${\vartheta _0}$, independent of $(x, t)$, such that

Proof Note that from (1), we obtain $Lu \le 0$ for any $(x, t) \in {\Omega _T}$. By multiplying both sides of $Lu \le 0$ by $\phi = {(u - {\vartheta _0})_ + }$, we have

Note that when $u \le {\vartheta _0}$, ${\partial _t}{(u - {\vartheta _0})_ + } = 0$ and $\nabla {(u - {\vartheta _0})_ + } = 0$; when $u \le {\vartheta _0}$, ${\partial _t}{(u - {\vartheta _0})_ + } = {\partial _t}u$ and $\nabla {(u - {\vartheta _0})_ + } = \nabla u$, thus

By further removing the non-negative terms $\int_\Omega {u|\nabla (u - {\vartheta _0}){|^p}{\rm{d}}x}$ and $\int_\Omega {u|\nabla (u - {\vartheta _0}){|^p}{{(u - {\vartheta _0})}_ + }{\rm{d}}x}$ from (6), we obtain

Clearly, when ${\vartheta _0}$ is sufficiently large, $\int_\Omega {({u_0} - {\vartheta _0})_ + ^2{\rm{d}}x} = 0$ holds. Therefore, for any $t \in (0, \; T)$,

This demonstrates $u \le {\vartheta _0}\; {\rm{a}}{\rm{.e}}{\rm{.}}\; {\rm{in}}\; {\Omega _T}$ □.

Note that from (1), it is easy to see that $Lu \le 0$ in ${Q_{2\rho, 2s}}$. By choosing the test function $\phi = {\psi ^2}{(u - \lambda)_ + }$, and then integrating $\phi Lu \le 0$ over ${Q_{2\rho, 2s}}$,

From Eq (13) in ^[11] or Theorem 2.1 in ^[12], it is easy to see that when $\gamma \in (0, \; 1)$, for any $t \in (0, T)$,

3.   Sobolev estimates under lower-order $W_1^p$ norm

This section examines the Sobolev estimates for the solution $u$. A Sobolev estimate on a local spherical region ${Q_{\rho, s}}(x, t)$ is constructed using the lower-order $W_1^p$ norm of the solution $u$. Initially, we investigate the time continuity results of an operator ${I_\rho }(t)$.

Lemma 3.1 For any ${Q_{4\rho, s}} \in {\Omega_T}$, there exists a constant $C$, which depends solely on $p$ and $\vartheta _0$, such that

Proof Assume ${t_1} < {t_2}$, choose $\eta$ sufficiently small, and suppose the function ${\psi _1} \in C_0^\infty ({t_1} - \eta, {t_2} + \eta)$ satisfies

By applying integration by parts, it is easy to obtain

Furthermore, due to $|{D_{\bar \rho }}| \times \left({{I_{\bar \rho }}({t_1}) - {I_{\bar \rho }}({t_2})} \right) = \int {\int_{{D_{\bar \rho }} \times ({t_1}, {t_2})} {{\partial _t}u{\rm{d}}x{\rm{d}}t} }$, we have

Considering $(1-\gamma) \int {\int_{{D_{\bar \rho }} \times ({t_1}, {t_2})} {|\nabla u{|^p}{\rm{d}}x{\rm{d}}t} }$ is non-negative, by Lemma 2.2, we obtain

Here, we choose $\bar \rho$ to satisfy $\rho < \bar \rho < 2\rho$. On the other hand, according to [14, p. 5, line 3] and Lemma 4.4 of ^[14], there exists a constant $C$ that depends only on $n$ and $p$ such that

By combining (10) and (11) and substituting the result into (9), the proposition is established. □

Theorem 3.1 Assume $u$ is a solution to the variational inequality (1). If $u \in {L^\alpha }(\tau - 2s, \tau + 2s; \ {W^{1, p}}({D_{2\rho }}(z)))$, then for any $\alpha \in (1, \infty)$, we have

Proof By selecting $\tau - 2s < t < \tau + 2s$ and $1 < \alpha < \infty$, we analyze the Sobolev-type estimates for the solution to the variational inequality problem (1) under the condition

Define

where $\psi (x, t)$ is a cut-off function on ${Q_{2\rho, 2s}}$,

Evidently, when $t \notin (\tau - s, \tau - s)$, for any $x \in {D_{2\rho }}(z)$, we have $\psi (x, \tau) = 0$. When $t \in (\tau - s, \tau - s)$, for any $x \in {D_{2\rho }}(z)$, we have

For convenience, let $\hat \rho = 2\rho$, and define

Thus, by the Hölder inequality,

Further applying the Sobolev inequality to ${\left({\int_{{D_{\hat \rho }}} {{v^{[\alpha + ({2 / n})(\alpha -1)]{n / {(n - 1)}}}}{\rm{d}}x} } \right)^{{{(n - 1)} / n}}}$, we get

The final inequality sign in the above expression holds because $(\alpha - 1)(1 + \frac{2}{n}) = \alpha + \frac{2(\alpha - 1)}{n} - 1$. Using the Hölder inequality again,

Therefore, by combining inequalities (16)–(18), we obtain

Thus, to estimate $J$, it suffices to analyze ${\left({\int_{{D_{\hat \rho }}} {|\nabla v{|^\alpha }{\rm{d}}x} } \right)^{\frac{1}{\alpha }}}$ and ${\left({\int_{{D_{\hat \rho }}} {{v^2}{\rm{d}}x} } \right)^{{1 / n}}}$ and obtain their upper bounds with respect to $u$. Notice that the cut-off function $\psi (x, t)$ satisfies $0 \le \psi (x, t) \le 1\; in\; {Q_{2\rho, 2s}}$ and $|\nabla \psi (x, t)| \le C\frac{1}{\rho }$, thus

Applying the Minkowski inequality again, we get

Further analyzing $|{I_{\hat \rho }}(t) - {I_\rho }(t)| \times |{D_{\hat \rho }}{|^{\frac{1}{\alpha }}}$ in (21), by ${I_\rho }(t)$, we have

Note that we previously assumed $u \in {L^\alpha }(\tau - 2s, \tau + 2s; {W^{1, p}}({D_{2\rho }}(z)))$. Thus, combining inequalities (21) and (22), and using the Sobolev inequality, we obtain

Here, we have used the conditions $\alpha > 1$ and $\hat \rho > \rho > 0$. Substituting (23) into (20), we obtain an estimate for ${\left({\int_{{D_{\hat \rho }}} {|\nabla v{|^\alpha }{\rm{d}}x} } \right)^{\frac{1}{\alpha }}}$ as

First, we analyze ${\left({\int_{{D_{\hat \rho }}} {{v^2}{\rm{d}}x} } \right)^{\frac{1}{n}}}$. By estimating ${\left({\int_{{D_{\hat \rho }}} {|u - {I_\rho }(t){|^2}{\rm{d}}x} } \right)^{\frac{1}{n}}}$ and applying Minkowski's inequality, we can derive

By utilizing the inequality ${(a + b)^{\frac{1}{{2n}}}} \le {(2n)^{2n}}({a^{\frac{1}{{2n}}}} + {b^{\frac{1}{{2n}}}})$, the estimate for ${\left({\int_{{D_{\hat \rho }}} {|u - {I_\rho }(t){|^2}{\rm{d}}x} } \right)^{\frac{1}{n}}}$ can be reformulated as

Furthermore, by the definition of ${I_\rho }(t)$, we obtain

Thus, by combining inequalities (17) and (18), we derive

The truncation function $\psi (x, t)$ fulfills $0 \le \psi (x, t) \le 1\; {\rm{in}}\; {Q_{2\rho, 2s}}$, thereby yielding an estimate for ${\left({\int_{{D_{\hat \rho }}} {{v^2}{\rm{d}}x} } \right)^{\frac{1}{n}}}$ denoted as

In summary, by substituting the results from (24) and (29) into Eq (19), we obtain

It is readily seen that $\int_{{D_\rho }} {|u(x, t) - I_{\rho }(t){|^{\alpha (1 + {2 / n})}}{\rm{d}}x} \le J$, and thus we obtain the result of Theorem 3.1. □

4.   Inverse Hölder inequality

This section presents an inverse Hölder inequality result. Before that, we introduce a Caccioppoli inequality, which is utilized in the proof.

Lemma 4.1 (Caccioppoli's Inequality) Suppose $u$ is a solution to the variational inequality (1). Then, for any non-negative constant $\lambda$, we have

Proof Note that from (1), it can be easily deduced that $Lu \le 0$ in ${Q_{2\rho, 2s}}$. Let us choose a test function $\phi = {\psi ^2} \times {(u - \lambda)_ + }$, and then integrate $\phi \times Lu \le 0$ over ${Q_{2\rho, 2s}}$, resulting in

First, let us analyze $\int\int_{{Q_{2\rho, 2s}}} {{\partial _t}u \times {\psi ^2}{{(u - \lambda)}_ + }{\rm{d}}x{\rm{d}}t}$ by employing the method of integration by parts, yielding

and

Substituting Eqs (31) and (32) into Eq (30), we obtain

that is

Based on ^[11,12], by applying the Hölder's inequality and Young's inequality to Eq (4), we have

Moreover, by selecting $\phi = {\psi ^2}{(u - \lambda)_ - }$ and repeating the proof process of Eqs (30)–(34), we can easily obtain

By combining Eqs (34) and (35), the theorem is proven.□

Theorem 4.1 Define $q = \max \{ p - 1, {{pn} / {(n + 2)}}\}$ and let $u$ be a solution to the variational inequality (1). Then, we have

Proof It is important to note that ${I_{2\rho }}(t)$ is not a constant over ${Q_{2\rho, 2s}}$, and thus, we choose

in the Caccioppoli inequality, resulting in

By utilizing the Hölder and Young inequalities, we can obtain

Combining Eqs (36) and (37), we can estimate $\int {\int_{{Q_{\rho, s}}} {u|\nabla u{|^p}{\rm{d}}x{\rm{d}}t} }$ by analyzing only $\int \int_{{Q_{2\rho, 2s}}} {|u - a({Q_{2\rho, 2s}}){|^p}{\rm{d}}x{\rm{d}}t}$. Using the Minkowski inequality, we have

Next, we analyze the $\int\int_{{Q_{2\rho, 2s}}} {|u - {I_{2\rho }}(t){|^p}{\rm{d}}x{\rm{d}}t}$ and $|{I_{2\rho }}(t) - a({Q_{2\rho, 2s}}){|^p}$ in Eq (38). By the definition of ${I_\rho }(t)$ and Lemma 3.1, we have

Therefore, by utilizing the Hölder and Young inequalities, we can estimate the second term on the right-hand side of Eq (38) as follows:

Now, let us estimate the first term on the right-hand side of Eq (38). By considering Theorem 3.1, we have

Furthermore, by utilizing the Sobolev inequality on ${D_{4\rho }}$, we obtain

By substituting (42) into (41) and combining it with (40), we obtain

Here, we make use of the fundamental result $\nabla u \in {L^p}(\Omega)$, which is detailed in (8). With this, the proof of Theorem 4.1 is complete. □

5.   Discussion and conclusions

This study considers a type of variational inequality problem involving a non-divergence parabolic operator, as shown in (1) and (2). In other words, the Sobolev estimates and inverse Hölder estimates are examined for the solutions of variational inequality (1). First, we define the averaging operator of the variational inequality (1) on the local spatial region ${D_\rho }$ as ${I_\rho }(t)$ and prove the uniform continuity of the mean inequality ${I_\rho }(t)$ with respect to time $t$. Second, we establish a Sobolev inequality for the averaging operator ${I_\rho }(t)$, which serves as the cornerstone for proving the inverse Hölder estimates, as stated in Theorem 3.1. Finally, we examine the inverse Hölder estimate problem in local cylindrical regions. The proof relies on Lemma 4.1 and Theorem 3.1, as well as commonly used amplification techniques such as Minkowski inequality, Young's inequality, and Hölder's inequality.

There are still some areas for improvement in the proofs presented in this paper. It is important to note that we make use of the condition $\gamma \in (0, \, 1)$, as when $\gamma > 1$ holds, Eqs (6), (9), and (30) cannot be used as they are in this paper. In such cases, $1 - \gamma < 0$, and we cannot eliminate the non-negative terms containing $1 - \gamma$. Furthermore, in ^[8], the existence of weak solutions to similar problems is discussed under the condition $\gamma \in (0, 1)$, which we also continue to adopt here. On the other hand, in order to employ Young's inequality and Hölder's inequality, we also restrict $p \ge 2$. It is worth noting that in the study of regularity theory for parabolic equations, the literature has considered the case $p \in (1, 2)$, and we also aim to explore this limitation in our future research.

Use of AI tools declaration

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

Acknowledgments

The authors are very grateful to the anonymous referees for their insightful comments and constructive suggestions, which considerably improve our manuscript. This work is supported by Growth Project of Young Scientific and Technological Talents in Colleges and Universities of Guizhou Province (NO. KY[2022]191).

Conflict of interest

The author declares there is no conflict of interest.