Regularity of weak solutions to a class of fourth order parabolic variational inequality problems arising from swap option pricing

1.   Introduction

A swap option is a kind of exotic option involving two stocks. Assume that the stock price $\{ {S_i}(t), t \in [0, T]\}$ follows the following stochastic process:

where $s_i$ is known, $\mu_i$ is the yield rate of stock $\{ {S_i}(t), t \in [0, T]\}$, and $\sigma_i$ is the volatility, $i = 1, 2$. In the financial contract, the swap option allows investors to convert stock $\{ {S_1}(t), t \in [0, T]\}$ to stock $\{ {S_2}(t), t \in [0, T]\}$ within the time interval $[0, T]$. (It is just a right, not an obligation, to make the conversion.) If the yield and turnover of stock $\{ {S_1}(t), t \in [0, T]\}$ are better than those of $\{ {S_2}(t), t \in [0, T]\}$, investors need to compensate the agent for a certain amount of cash $K$, so the value of the swap option on the maturity date $T$ is ^[1,2,3]

Since investors can execute at any time within the interval $[0, T]$,

From ^[4], the value of the swap option with maturity time $T$ at time $0$ satisfies the following parabolic variational inequality:

Let $r$ be the risk-free interest rate in the securities market, and define $q_i$ as the dividend rate of the stock $\{ {S_i}(t), t \in [0, T]\}$, $i = 1, 2$. Parabolic operator $L_0V$ can be written as

If the stock has transaction costs, the operator $Lv$ has a more complex structure in which

Readers can read about the Leland model, the Barles and Soner's model and the Davis model in ^[5].

The author of this study focuses on more complex models, considering a certain kind of variational inequality problem

with the non-Newtonian polytropic operator

Recently, there are many studies about the theoretical research of variational inequality problems. Tao Wu in ^[6] used a fourth-order $p$-Laplacian Kirchhoff operator and considered the following variation-inequality initial-boundary value problem

Based on the Leray-Schauder principle, the existence of solutions to the auxiliary problem is proved. The existence and uniqueness of solution to (3) is then studied in which the parabolic operator $L\phi$ is extended. The 2-D value variational inequality problem is also considered in ^[7] using the limit method. For the existence of weak solutions to variational inequalities, readers can refer to the literature ^[8,9,10]. The stability and uniqueness of weak solutions have also been a hot topic in recent years. Literature ^[7,11] analyzes the upper bound estimate of the difference between two weak solutions and proves the stability and uniqueness of the weak solution about the initial value. However, there is currently no literature on the regularity of weak solutions of such variational inequalities. Literature ^[12] analyzes parametric boundary value problems using optimal variational iteration methods and convergence control techniques, showing that the optimal variational iteration method is an effective method for solving such problems. The authors of ^[13] developed a scheme to examine fractional-order shock wave equations and wave equations occurring in the motion of gases in the Caputo sense with their main finding being the handling of the recurrence relation that produces the series solutions after only a few iterations. Finally, the authors of ^[14] suggested a He-Laplace variational iteration method for the study of some partial differential equations arising in physical phenomena such as chemical kinetics and population dynamics.

In summary, we investigate the regularity of weak solutions for the variational inequality problem (1). We use a variety of techniques, including integral inequalities, partial derivatives, flattening operators, and the finite cover principle, to obtain internal, near edge, and global regularity estimates. Additionally, we provided an ${L^\infty }(0, T;{H^2}(\Omega))$ estimation using the quadratic gradient of the weak solution.

2.   Statement of the problem and its background

We will recall several well-known aspects about the solution to problem (1) and provide a set of maximal monotone maps defined in ^[1,2,3]

where $M_0$ is a positive constant. With a similar method to that used in ^[6,7], variational inequality problem (1) admits a generalized solution $(u, \xi)$ that satisfies

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

(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 $\varphi \in {C^1}({\bar \Omega _T})$,

Indeed, for any generalized solution $u$, (5) can be rewritten as

Following a similar way of ^[6,7], applying the comparison principle shows that

This paper focuses on the internal regularity of weak solutions within a subarea $\Omega ' \subset \subset \Omega$. In doing so, we introduce the difference operator

where $e_i$ is the unit vector in the direction $x_i$. As stated in the literature ^[12], the difference operator has the following results.

Lemma 2.1. (1) Let $\Delta _h^{i*} = - \Delta _{ - h}^i$ be the conjugate operator of $\Delta _h^i$. Then we have

in other words, $\int_{{{\rm{R}}_n}} {f(x)\Delta _h^ig(x){\rm{d}}x} = \int_{{{\rm{R}}_n}} {g(x)\Delta _h^{i*}f(x){\rm{d}}x}$.

(2) Operator $\Delta _h^i$ has the following commutative results

(3) If $u \in {W^{1, p}}(\Omega)$, for any $\Omega ' \subset \subset \Omega$,

(4) Let $h$ be small enough. If $||\Delta _h^iu|{|_{{L^p}(\Omega)}} \le C$, then

where $C$ is independent of $h$.

3.   Internal regularity of solution

In this section, we will investigate the internal regularity of weak solutions. To do so, we require the following auxiliary result.

Lemma 3.1. Let $\eta \in C_0^\infty (\Omega)$ be the cutoff factor on $\Omega ' \subset \subset \Omega$ that satisfies

where $d = {\rm{dist}}(\Omega ', \Omega)$; then,

Proof. It follows from Lemma 2.1 (3) that

such that

Using ${(a + b)^2} \le 2{a^2} + 2{b^2}$ gives

Therefore, the proof of Lemma 2.2 is finished (note that $0 \le \eta \le 1$).

It is important to note that the quadratic gradient $\triangle u$ does not satisfy the condition for weak solutions. Therefore, we plan to construct a test function using the spatial difference operator $\Delta _h^{i}$. Let $h < D$ and choose $\varphi = \Delta _h^{i*}({\eta ^2}\Delta _h^iu)$ as a test function in (5). Since $u \in H_0^1(\Omega)$, we have $\varphi \in H_0^1(\Omega)$, so that

Now we prove that

It follows from Lemma 2.1 (2) that

such that

Thus (10) follows. Combining (9) and (10) and using Lemma 2.1 (2) to $\int {\int_{{\Omega _T}} {\nabla u\nabla \Delta _h^{i*}({\eta ^2}\Delta _h^iu){\rm{d}}x{\rm{d}}t} }$ gives

Because

(11) can be written as

Applying Holder and Young inequalities as well as Lemma 3.1,

Substituting (13), (14) and (15) (note that $0 \le \eta \le 1$), it is clear to verify

It follows from Lemma 2.1 (3) that

such that

This, from Lemma 2.1 (4), implies that

Theorem 3.1. If $u \in {L^\infty }(0, T;{H^1}(\Omega))$, ${u_0} \in {H^1}(\Omega)$, then for any $\Omega ' \subset \subset \Omega$,

4.   Global regularity of solution

In this section, we examine the near-edge regularity of weak solutions. Suppose $U$ is a neighborhood containing ${x_0} \in \partial \Omega$. Drawing inspiration from reference ^[12], we introduce a flattening operator $\Psi$, which belongs to $C^2$ to transform the proof process into an internal regularity problem in Section 3.

Theorem 4.1. Assume $u \in {L^\infty }(0, T;{H^1}(\Omega))$ and ${u_0} \in {H^1}(\Omega)$. For any ${x_0} \in \partial \Omega$, if $x_0$ belongs to $U$, then

According to the finite-covers theorem, there are finite neighborhoods ${U_1}, \; {U_2}, \cdots, {U_N}$, satisfies

such that

Combining Theorem 4.1 and (17), we give the following global regular estimation.

Theorem 4.2. If $u \in {L^\infty }(0, T;{H^1}(\Omega))$ and ${u_0} \in {H^1}(\Omega)$, then

Indeed, from the perspective of ${L^\infty }(0, T;{H^2}(\Omega))$ norm estimation, we have a better result. Choose $\Delta u$ as a test function in (6), such that

Note that $\int_0^T {\int_\Omega {|{\partial _t}\nabla u{|^2}{\rm{d}}x{\rm{d}}t} } = \int_\Omega {|\nabla u(, T){|^2}{\rm{d}}x} - \int_\Omega {|\nabla {u_0}{|^2}{\rm{d}}x}$. Thus,

Applying Holder and Young inequalities and combining with (8), we have

Combining with (4) also gives

Substituting (20) and (21) in (19), one can get

Rearranging the above inequality, we prepare the following theorem which is better than the global regular estimation in (18).

Theorem 4.3. If ${u_0} \in {H^1}(\Omega)$, then

where $C$ depends on $p, \gamma, \int_\Omega {|\nabla {u_0}{|^2}{\rm{d}}x}, |{u_0}{|_\infty }, T$ and $|\Omega |$.

5.   Conclusions

This paper discusses the regularity of weak solutions for a class of variational inequality problems involving the fourth order parabolic operator. The existence of weak solutions is typically established through first order energy estimates, which can be proven by selecting an appropriate test function in the weak solution equation. As weak solutions exhibit higher-order norm estimates, scholars have paid much attention to their regularity. However, constructing test functions using second-order partial derivatives, which do not satisfy the conditions for weak solutions, is a challenge when proving the regularity of these solutions. Nonetheless, the spatial difference operator retains the differential order of the weak solution $u$, leading us to employ spatial difference operators and cutoff factors in test function construction. In this paper, we have constructed a test function

using these operators, which satisfies the condition for weak solutions. Subsequently, we have used the test function $\varphi$ in (5) and the cutoff factor $\eta$ to obtain inequality (11) which is a cornerstone for general energy estimates. As variational inequalities are more complex than equal parabolic equations, we have introduced a maximum monotone operator in weak solutions (5) and (11) based on ^[6,7]. We have combined the Holder and Young inequalities to obtain an estimate of the internal regularity of the weak solution without imposing any further existence conditions on the second partial derivative of the weak solution $u$.

At present, this paper gives the limiting condition $\gamma \ge 0$. Although this condition is not directly used in this paper, according to literature ^[6,7], this condition ensures that $u \in {L^\infty }(0, T, {H^1}(\Omega)) \cap {L^\infty }(0, T, {L^p}(\Omega)),$ ${\partial _t}u \in {L^\infty }(0, T, {L^2}(\Omega))$. So this paper continues to use this restriction and we will try to weaken it in the future.

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. The research was partially supported by the Basic Research Program of Shanxi Province (No. 202103021224310).

Conflict of interest

The author declares no conflict of interest.