Study of weak solutions of variational inequality systems with degenerate parabolic operators and quasilinear terms arising Americian option pricing problems

1.   Introduction

Let $\Omega \subset \mathbb{R}^N (N \ge 2)$ be a bounded simple connected domain, $0 < T < \infty$ and $Q_T = \Omega \times [0, T]$. We study the following parabolic systems

with quasilinear degenerate parabolic operators, where

The initial boundary value problem of variational inequalities arise in many application in pricing American options and their derivatives. Through the risk neutral strategy, the intrinsic value of many American options can ultimately be attributed to the solution of a variational inequality in the Black-Scholes models. The author refers to ^[1,2,3] and the references their in.

The nonexistence, existence and uniqueness theory for weak solutions of parabolic systems were studied by many existing works, see e.g., ^[4,5,6]. In particular, Hassnaoui and Idrissi ^[6] studied the existence and uniqueness of weak solutions for a nonlinear parabolic system with non-degenerate case of (1.1). Escher, Laurencot and Matioc in ^[7] proved the global existence of nonnegative weak solutions to a degenerate parabolic system without quasilinear terms in (1.1). Furthermore, the authors showed that these weak solutions converge at an exponential rate.

In recent years, there has been tremendous interest in developing existence and uniqueness theory for weak solutions of parabolic variational inequality (see, for example, ^{[3,8,9,10,11,12,13,14]} and the references therein). In 2014, the authors in ^[9] discussed the problem

with second order degenerate elliptic operator

Under the assumptions about $u_0$ and $F$, they proved the existence and uniqueness of the weak solution. When $a(u) = 1$, and $p(x, t) = 2$, the authors in ^[10,11] discussed the existence and numerical algorithm of solution. In ^[12], a new property of variable exponent Lebesgue and Sobolev spaces was examined. Using these properties, the authors proved the existence of the solution of some parabolic variational inequality.

To the best of our knowledge, the existence and uniqueness for multi-variable problem of parabolic variational inequalities (called variational inequality systems) were less studied. We cannot easily put the method in ^[10,11] to the multi-variable case since the systems are coupled with quasilinear terms.

The aim of this paper is to study the existence and uniqueness of solution for parabolic systems with quasilinear degenerate inequalities in a bounded domain. We mainly use comparison theorem and penalty method to construct a sequence of approximation solutions with the help of monotone iteration technique. Then we obtain the existence of solutions to the system (1.1) by a standard limiting process.

The paper is organized as follows. In Section 2, we present our main theorems. Section 3 gives some estimates about penalty problems to prove our main results. Section 4 analyses the existence and uniqueness of solutions to variational inequality system (1.1).

2.   The main results of weak solutions

In spirit of ^[3] and ^[9], we introduce the following maximal monotone graph

The purpose of the paper is to obtain the existence and uniqueness of weak solutions of (1.1), and the weak solution is defined as follows.

Definition 2.1. Function $\{ ({u_1}, {\xi _1}), ({u_2}, {\xi _2})\}$ is called a generalized solution of the systems (1.1) if ${u_i} \in {L^\infty }({Q _T}) \cap L(0, T, W_0^{1, {p_i}}(\Omega))$, ${\partial _t}{u_{_i}} \in {L^2}({\Omega _T})$, ${\xi _i} \in {L^\infty }(0, T;{L^\infty }(\Omega))$, $i = 1, 2$, and satisfies (a) $u_i (x, t) \ge u_{i0}(x)$, (b) $u_i (x, 0) = u_{i0}(x)$, (c) $\xi_i \in G(u_i - u_{i0})$, (d) for every test-function ${\varphi _i} \in C_0^1({\bar Q _T})$

Here $C_0^1({\bar \Omega _T})$ is the space of all continuous and differentiable functions satisfying

Condition (d) of Definition 2.1 implies that

We introduce the constrains to the nonlinear functions ${f_i}, i = 1, 2$ involved in this paper as follows.

Definition 2.2. A function $f = f({u_1}, {u_2})$ is quasimonotone nondecreasing (resp., nonincreasing) if for fixed ${u_1}$ (or ${u_2}$), $f$ is nondecreasing (resp., nonincreasing) in ${u_2}$ (or ${u_1}$).

To study the problem (1.1), we make the following assumptions:

(H1) ${f_i}(x, t, {u_1}, {u_2})$ is quasimonotonically nondecreasing for $u_1$, $u_2$, $i = 1, 2$.

(H2) ${f_i}(x, t, {u_1}, {u_2}) \in C(\Omega \times [0, T] \times {R^2})$, and there exists a nonnegative function $g(s) \in {C^1}(\mathrm{R})$ such that

Our main results are present as follows:

Theorem 2.1. Let ($H1$) and ($H2$) be satisfied, and ${u_{i0}} \in {L^\infty }({\Omega _T}) \cap W_0^{1, {p_i}}(\Omega)$, $i = 1, 2$. Then problem (1.1) has a solution $u = ({u_1}, {u_2})$ in the sense of Definition 2.2.

Theorem 2.2. Assume that $f = ({f_1}, {f_2})$ is Lipschitz continuous in $({u_1}, {u_2})$, then the solution of problem (1.1) is unique.

3.   A penalty problem

To prove the theorem, we consider the following penalty problem

where

Here, ${\beta _\varepsilon }(\, \cdot \, )$ is the penalty function satisfying

It is worth noting that when ${u_i} > {u_{i, 0}}$, ${L_i}{u_i} - {f_i}(x, t, {u_1}, {u_2}) = 0$, and when ${u_i} = {u_{i, 0}}$, one gets ${L_i}{u_i} \ge {f_i}(x, t, {u_1}, {u_2})$ in (1.1). In (3.1), ${\beta _\varepsilon }(u_{i\varepsilon }^{} - {u_{i0}})$ plays a similar role. When $u_{i\varepsilon }^{} > {u_{i0}} + \varepsilon$,

and when ${u_{i0}} \le u_{i\varepsilon }^{} \le {u_{i0}} + \varepsilon$, we have

With a similar method as in ^[8], we can prove that regularized problem has a unique weak solution

satisfying the following integral identities

with ${\varphi _i} \in {C^1}({\bar \Omega _T})$ and $t \in (0, T)$.

We start with two preliminary results that will be used several times henceforth.

Lemma 3.1. [^[15], Lemma2.1.] Let $M(s) = {\left| s \right|^{p(x, t) - 2}}s$, then $\forall \xi, \eta {\rm{ }} \in {\rm{ }}{{\rm{R}}^N}$

Lemma 3.2. (Comparison principle) Assume ${u_i}$ and ${v_i}$ are in ${L^{{p_i}}}(0, T;{\rm{ }}{W^{1, {p_i}}}(\Omega))$. If ${L_{i\varepsilon }}{u_i} \ge {L_{i\varepsilon }}{v_i}$ in ${Q_T}$ and ${u_i}(x, t) \le {v_i}(x, t)$ on $\partial {Q_T}$, then ${u_i}(x, t) \le {v_i}(x, t)$ in $Q_T$, $i = 1, 2$.

Proof. We argue by contradiction. Suppose ${u_i}(x, t)$ and ${v_i}(x, t)$ satisfy ${L_{i\varepsilon }}{u_i} \ge {L_{i\varepsilon }}{v_i}$ in $Q_T$, and there is a $\delta > 0$ such that for some $0 < \tau \le T$, ${w_i} = {u_i} - {v_i} > \delta$ on the set

and $|{\Omega _\delta }| > 0$, $i = 1, 2$. Let

where$\delta > \varepsilon > 0$. Since ${F_\varepsilon }(w_i) \le 0$, we multiply ${L_{i\varepsilon }}{u_i} \ge {L_{i\varepsilon }}{v_i}$ by ${F_\varepsilon }({w_i})$ and integrate in ${Q_\tau }$ to have

or equivalently

where ${Q_{\tau, \varepsilon }} = \{ (x, t) \in {Q_T}|w > \varepsilon \}$,

Now let ${t_0} = \inf \{ t \in (0, {\rm{ }}\tau {\rm{ }}]:w{\rm{ }} > \varepsilon \}$, then we estimate $J_1$ as follows

Using $\delta > \varepsilon > 0$ and the function ${F_\varepsilon }(\cdot)$ to (3.5), we have

By the virtue of the first inequality of Lemma 3.1, we use ${w_i} = {u_i} - {v_i} > \delta$ to arrive at

Since ${v_i} \in {L^{{p_i}}}(0, T;{\rm{ }}{W^{1, {p_i}}}(\Omega))$, and we plug the above estimates (3.6) and (3.7) into (3.4) and drop the nonnegative terms, we arrive at

Note that $\mathop {\lim }\limits_{\varepsilon \to 0} (\delta - \varepsilon){\varepsilon ^{ - \frac{1}{2}}}|{\Omega _\delta }| = + \infty$, we obtain a contradiction. This means that $|{\Omega _\delta }| = 0$ and ${w_i} \le 0$ a.e. in ${{\rm{Q}}_\tau }$, $i = 1, 2$.

Lemma 3.3. Let be weak solutions of (3.1). Then

where $|{u_0}| = \mathop {\sup }\limits_{x \in \Omega } |{u_0}(x)|$, for details, see ^[16,17].

Proof. First, we prove ${u_{i\varepsilon }} \ge {u_{i0\varepsilon }}$ by contradiction. Assume ${u_{i\varepsilon }} \le {u_{i0\varepsilon }}$ in $Q_T^0$, $i = 1, 2$, $Q_T^0 \subset Q_T$. Noting ${u_{i\varepsilon }} \ge {u_{i0\varepsilon }}$ on $\partial {Q_T}$, we may assume that ${u_{i\varepsilon }} = {\rm{ }}{u_{i0\varepsilon }}$ on $\partial {\rm{Q}}_T^0$. With (3.1) and letting $t = 0$, it is easy to see that

From Lemma 3.2, we have that

Therefore, we obtain a contradiction.

Second, we pay attention to ${u_{i\varepsilon }}(t, x) \le {\left| {{u_{i0}}} \right|_\infty } + \varepsilon$. Applying the definition of ${\beta _\varepsilon }(\cdot)$ gives

Using Lemma 3.2, (3.13) leads to

Thus, combining (3.13) and (3.14) and repeating Lemma 3.2, we have

Third, we aim to prove (3.9). From (3.1), it is easy to see that

It follows by ${\varepsilon _1} \le {\varepsilon _2}$ and the definition of ${\beta _\varepsilon }(\cdot)$ that

$i = 1, 2$. Combining initial and boundary condition in (3.1), we obtain that the inequality (3.9) holds by Lemma 3.2.

Lemma 3.4. For any $(x, t) \in {\Omega _T}$, the solution of problem (3.1) satisfies the estimate

where $C$ is a constant independent of $\varepsilon$.

Proof. Choosing ${\varphi _i} = u_{i\varepsilon }$ in (3.3), we have

It is easy to see that

Then we substitute (3.21) into (3.20) to arrive at

By (3.8) and the property of $f_i$,

Applying (3.2) and (3.8) obtains

Then Lemma 3.4 is proved by combining (3.21)–(3.23).

Lemma 3.5. The solution of problem (3.1) satisfies the estimate

Proof. From (3.3), we have that

where

First, we pay attention to $A_1$. Using some differential transform technique obtains

Since $u_{i0\varepsilon }^{}(x) = {u_{i0}} + \varepsilon$, then

Applying Holder inequalities again, we have that

Using ${{\rm{(a}} + {\rm{b)}}^2} \le 2({a^2} + {b^2})$, the property of $f_i$ and (3.2), we arrive at

Then, we obtain Lemma 3.5 by submitting (3.26)–(3.28) into (3.25).

4.   Proof of the main result

In this section, we are ready to prove that the system (1.1) has a unique generalized solution. By (3.18), (3.19) and (3.24) and the uniqueness of the weak limits, we know that there are functions

such that for some subsequence of $({u_{1\varepsilon }}, {u_{2\varepsilon }})$, denoted again by $({u_{1\varepsilon }}, {u_{2\varepsilon }})$,

where $\mathop \to \limits^w$ stands for weak convergence, $i = 1, 2$.

Lemma 4.1. For any $(x, t) \in {\Omega _T}$, $w_i^{} = |\nabla u_i^{}{|^{{p_i} - 2}}\nabla u_i^{}$, $i = 1, 2$.

Proof. Applying triangle inequality $|a + b| \le |a| + |b|, \ (a, b \in {\rm{R}})$, it is easy to see that

where

By mean of the inequality $|{a^r} - {b^r}| \le |a - b{|^r}, \ (r \in [0, 1], a, b > 0)$, we have

Applying Holder inequality and (4.2), we have

Now we pay our attention to $I_2$. From (4.2), we know that $\nabla u_i^{}\; \in \; {L^{{p_i}}}({Q_{{T_1}}})$. By (4.1), we may conclude that

If not, there exists a measurable domain ${{\rm O}_T}$ satisfying

Then, we obtain a contradiction with (4.2). Applying Holder inequality, we have

Hence Lemma 4.1 is proved.

This proves that any weak convergence subsequence of $|\nabla u_{i\varepsilon }^{}{|^{{p_i} - 2}}{\partial _{{x_l}}}u_{i\varepsilon }^{}$ will have ${\partial _{{x_l}}}w_i$ as its weak limit and hence by a standard argument, and we have that as $k \to \infty$,

Combining the above results, we have, in fact, proved that $u = ({u_1}, {u_2})$ is a generalized solution of (1.1).

Lemma 4.2. For any $(x, t) \in \Omega _T$, it hold that

Proof. Using (3.8) and the definition of ${\beta _\varepsilon }$, we have

Now, we prove $\xi \in G(u - {u_0})$. According to the definition of $G(\cdot)$, we only need to prove that if $u({x_0}, {\rm{ }}{t_0}) > {\rm{ }}{u_0}({x_0})$,

In fact, if $u({x_0}, {\rm{ }}{t_0}) > {u_0}({x_0})$, there are a constant $\lambda > 0$ and a $\delta$-neighborhood ${B_\delta }({x_0}, {\rm{ }}{t_0})$ such that if $\varepsilon$ is small enough, we have

Thus, if $\varepsilon$ is small enough, we have

Furthermore, it follows by $\varepsilon \rightarrow 0$ that

Hence, (4.6) holds, and the proof of Lemma 4.3 completes.

The proof of Theorem 2.1. Applying (3.8), (3.9), and Lemma 4.3, it is clear that

thus (a), (b), and (c) of Definition 1.1 hold. The rest arguments of existence part are the same as those of Theorem 2.1 in ^[8] by a standard limiting process. Thus, we omit the details.

The proof of Theorem 2.2. The following is the uniqueness result to the solution of the system. Assume that $\{ ({u_1}, {\xi _1}), ({u_2}, {\xi _2})\}$ and $\{ ({v_1}, {\zeta _1}), ({v_2}, {\zeta _2})\}$ are two solutions of (1.1). Let ${\varphi _i} = {u_i} - {v_i}$ in Definition 2.1, $i = 1, 2$, then by (2.3),

$i = 1, 2$. Subtracting the 2 equations, we get

Now we prove

On one hand, if ${u_i}(x, t) > {v_i}(x, t)$, then using Lemma2.1 yields

From (2.1) and above inequality, it is easy to see that

Combining (2.1) and (4.8),

On the other hand, if ${u_i}(x, t) < {v_i}(x, t)$, it is easy to have that${\xi _i} \ge 0 = {\zeta _i}$.In this case,

Combining (4.9) and (4.10), (4.7) still holds.

Using the previous inequality and the Lipschitz condition, a simple calculation shows that

Furthermore, it follows by ${{\rm{(a}} + {\rm{b)}}^2} \le 2({a^2} + {b^2})$ and Holder inequalities that

Setting $F(t) = \int_0^{\rm{t}} {\int_\Omega {|{u_1} - {v_1}{|^2}{\rm{ + }}|{u_2} - {v_2}{|^2}{\rm{d}}x} {\rm{d}}t}$, then the above inequality can be written as

A standard argument show that $F(t) = 0$ since $F(0) = 0$, and hence $u_i = v_i, i = 1, 2$. The proof is complete.

5.   Conclusions

In this paper, we study variational inequality systems with quasilinear degenerate parabolic operators in a bounded domain

with quasilinear degenerate parabolic inequalities, where

The existence and uniqueness of the solutions in the weak sense are proved by using the penalty method and the reduction method with assumptions that $p_1$ and $p_2$ are constants satisfying $p_i > 2$. However, there are some problems that have not been solved: when $1 < p_i < 2$, $p_i > 2$ or $p_i$ is $x$-functions, $i = 1, 2$, we cannot use Lemmas 3.1 and 3.2 to prove Lemmas 3.3–3.5. We will continue to study this problem in 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. This work was supported by the Supply and demand docking education employment practice base project of education ministry (No. 20220106223) and the Supply and demand docking education oriented talent training project of education ministry (No. 20220104747).

Conflict of interest

The authors declare no conflict of interest.