Existence and blowup of solutions for non-divergence polytropic variation-inequality in corn option trading

1.   Introduction

First, consider a kind of variation-inequality problem

with the non-Newtonian polytropic operator,

Here, $\Omega \subset {{\mathrm{R}}^N}(N \ge 2)$ is a bounded domain with an appropriately smooth boundary $\partial \Omega$, $p \ge 2$, $m > 0$ and $u_0$ satisfies

The theory of variation-inequality problems has gained significant attention due to its applications in option pricing. These applications are discussed in references ^[1,2,3], where more details on the financial background can be found. In recent years, there has been a growing interest in the study of variation-inequality problems, with a particular emphasis on investigating the existence and uniqueness of solutions.

In 2022, Li and Bi considered a two dimension variation-inequality system ^[4],

involving a degenerate parabolic operator

Using the comparison principle of $L_i u_i$ and norm estimation techniques, the sequence of upper and lower solutions for the auxiliary problem is obtained. The existence and uniqueness of weak solutions are then analyzed. While reference ^[5] considers the initial boundary value problem under a single variational inequality, the author explores more complex non-divergence parabolic operators

Reference ^[5] constructs a more intricate auxiliary problem and proves that the weak solutions are both unique and existent by using progressive integration and various inequality amplification techniques. Readers can refer to references ^[6,7,8] for further information on these interesting results.

In the field of differential equations, there are various literature available on initial boundary value problems that involve the non-Newtonian polytropic operator. In ^[9,10], the authors focused on a specific class of initial boundary value problems that feature the non-Newtonian polytropic operator

To investigate the existence of a weak solution, they made use of topologic degree theory.

Currently, there is no literature on the study of variational inequalities under non-Newtonian polytropic operators (2). Therefore, we attempt to use the results of partial differential equations from literature ^[5,9,10] to investigate the existence and blow-up properties of weak solutions for variational inequalities (1). Additionally, considering the degeneracy of the operator $Lu$ at $u = 0$ and $\nabla {u^m} = 0$, some traditional methods for existence proofs are no longer applicable. Here, we attempt to use the fixed point theorem to solve this issue, and obtain the existence and blow up of generalized solutions.

2.   Statement of the problem and its background

We first consider the case of variation-inequality in corn options. During the harvest season, farmers face the problem of corn storage, while flour manufacturers are concerned about the downtime caused by a lack of raw materials.

In exchange for the farmer storing the raw materials in the warehouse, the flour manufacturer promises the farmer the following contract:

Assuming that the current time is 0, the corn price $S_t$ follows the time interval $[0, \ T]$, and is given by:

where $S_0$ is known, $\mu$ represents the annual growth rate of corn price, and $\sigma$ represents the volatility rate. $\{ {W_t}, t \ge 0\}$ stands for a winner process, representing market noise.

In addition, to avoid significant economic losses for flour manufacturers due to rapid increases in raw material prices, obstacle clauses are often included in the following form: if the price of corn rises more than $B$, the option contract becomes null and void. According to literature ^[1,2,3], the value $V$ of the option contract at any time $t \in [0, T]$ satisfies

where $L_0V = {\partial _t}V + \frac{1}{2}{\sigma ^2}{S^2}{\partial _{SS}}V + rS{\partial _S}V - rV$, $r$ is the risk-free interest rate of the agricultural product market; $B$ is the upper bound of corn prices, which prevents flour manufacturers from incurring significant losses due to rising corn prices. On the one hand, if $x = lnS$, then (3) can be written as

where $L_1V = {\partial _t}V + \frac{1}{2}{\sigma ^2} {\partial _{xx}}V + r {\partial _x}V - rV$. It can be seen that problem (4) is a constant coefficient parabolic variational inequality problem, which has long been studied by scholars (see ^[1,2,3] for details). On the other hand, when there are transaction costs involved in agricultural product trading, the constant $\sigma$ in the operator $LV$ is no longer valid and is often related to ${\partial _S}V$, as well as $V$ itself. For instance, the well-known Leland model ^[5] adjusts volatility $\sigma$ into a non-divergence structure represented by

where $\sigma$ denotes the original volatility and $Le$ corresponds to the Leland number.

Inspired by these findings, we aim to explore more intricate variation-inequality models in (1). When $m = 1$, the non-divergence polytropic structure ${u^m}\nabla (|\nabla {u^m}{|^{p - 2}}\nabla {u^m})$ in model (1) degenerates into a similar n-dimensional structure as model (4). It's worth noting that while model (4) only considers one type of risky asset and is defined in a 1-dimensional space, model (1) studies the problem in an n-dimensional space.

Variation-inequality (1) degenerates when either $u = 0$ or $\nabla {u^m} = 0$. Classically, there would be no traditional solution. Following a similar way in ^[1,3], we consider generalized solutions and first give a class of maximal monotone maps $G:[0, + \infty) \to [0, + \infty)$ satisfies

Definition 2.1 A pair $(u, \xi)$ is called a generalized solution for variation-inequality, if for any fixed $T > 0$,

(a) $u^m \in {L^\infty }(0, T, W^{1, p}_0 (\Omega))$, ${\partial _t}u \in {L^2}({\Omega _T}),$

(b) $\xi \in G(u - {u_0})\; {\rm{for}}\; {\rm{any}}\; (x, t) \in {\Omega _T}$,

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

(d) for every test-function $\varphi \in {C^1}({\bar \Omega _T})$, there holds

As far as what was mentioned above, ${u^m}\nabla (|\nabla {u^m}{|^{p - 2}}\nabla {u^m})$ degenerates when $u^m = 0$ or $\nabla {u^m} = 0$. We set and use a parameter $\varepsilon \in [0, 1]$ to regularize ${u^m}\nabla (|\nabla {u^m}{|^{p - 2}}\nabla {u^m})$ in operator $Lu$ and the initial boundary condition. Meanwhile, we use $\varepsilon$ to construct a penalty function $\beta _\varepsilon (\cdot)$ and use it to control the inequalities in (1) that the penalty map ${\beta _\varepsilon }:{{\rm{R}}_ + } \to {{\rm{R}}_ - }$ satisfies

In other words, we consider the following regular problem

where

Similar to ^[4,5], problem (8) admits a solution ${u_\varepsilon }$ satisfies $u_\varepsilon ^m \in {L^\infty }(0, T; {W^{1, p}}(\Omega))$, ${\partial _t}u_\varepsilon ^m \in {L^\infty }(0, T; {L^2}(\Omega))$, and the identity

with $\varphi \in {C^1}({\bar \Omega _T})$. Meanwhile, for any $\varepsilon \in (0, \; 1)$,

Indeed, define ${A_\theta }({u_\varepsilon }) = \theta u_\varepsilon ^m + (1 - \theta){u_\varepsilon }$,

One can use a map based on Leray-Schauder fixed point theory

that is,

so that by proving the boundedness, continuity and compactness of operator $M$, the existence result of (6) can be established. For details, see literature ^[11], omitted here.

3.   Existence of solution

In this section, we consider the existence of a generalized solution to variation-inequality (1). Since $u_\varepsilon ^m \in {L^\infty }(0, T; {W^{1, p}}(\Omega))$, ${\partial _t}u_\varepsilon ^m \in {L^\infty }(0, T; {L^2}(\Omega))$, by combining with (9), we may infer that the sequence $\{ u_\varepsilon, \varepsilon \ge 0\}$ contains a subsequence $\{ u_{{\varepsilon _k}}^{}, k = 1, 2, \cdots \}$ and a function $u$, ${\varepsilon _k} \to 0\; {\rm{as}}\; k \to \infty$,

From (9), one can easily show that ${u_{{\varepsilon _k}}} \le u$, $\forall (x, t) \le {\Omega _T}$, $k = 1, 2, 3, \cdots$. So, one can infer that for all $(x, t) \in {\Omega _T}$,

Next, we pass the limit $k \to \infty$. It follows from (13), that for any $(x, t) \in {\Omega _T}$, $k = 1, 2, 3, \cdots$,

so that pass the limit $k \to \infty$,

Choosing $\varphi = {u_{{\varepsilon _k}}} - u$ in (8) and turning $\varepsilon$ into $\varepsilon _k$, one can infer that

Subtracting (18) from (19) and integrating it from 0 to $T$,

From (32), we infer that

Recall that ${u_{{\varepsilon _k}}}(x, 0) = {u_0}(x) + {\varepsilon _k}$ for any $x \in \Omega$. Then

Note that ${\varepsilon _k} \to 0\; {\rm{as}}\; k \to \infty$. So we may infer that $\int {\int_{{\Omega _T}} {({\partial _t}{u_{{\varepsilon _k}}} - {\partial _t}u) \cdot \varphi {\rm{d}}x{\rm{d}}t} } \ge 0$ if $k$ is large enough. Then, removing the non negative term on the left hand-side in (20) and passing the limit $k \to \infty$, it is clear to verify that

Note that $u_{}^m|\nabla {u^m}{|^{p - 2}}\nabla {u^m} = |\nabla {u^{\mu m}}{|^{p - 2}}\nabla {u^{\mu m}}$, $\mu = \frac{p}{{p - 1}}$. As mentioned in ^[12], it follows from (9) that

Thus, by using ${\mathop{\rm sgn}} \left({\nabla u_{{\varepsilon _k}}^{\mu m} - \nabla {u^{\mu m}}} \right) = {\mathop{\rm sgn}} (\nabla \varphi)$, leads to

Subtracting (24) from (25), one can see that

Obviously, if we swap ${u_{{\varepsilon _k}}}$ and ${u }$, one can get another inequality

Combining (26) and (27), we obtain (28) below and give the following Lemma.

Lemma 3.1 For any $t \in (0, T]$ and $x \in \Omega$,

Proof. One can deduce that (29) is an immediate result of combining (23), (24) and (29).

Following a similar proof showed in (16)–(28), one can infer that

Further, we prove $\xi \in G(u{\rm{ }} - {\rm{ }}{u_0})$. When ${u_{{\varepsilon _k}}} \ge {u_0} + \varepsilon$, we have ${\beta _{{\varepsilon _k}}}({u_{{\varepsilon _k}}} - {u_0}) = 0$, so

If ${u_0} \le {u_{{\varepsilon _k}}} < {u_0} + {\varepsilon _k}$, ${\beta _\varepsilon }({u_\varepsilon } - {u_0}) \le 0$ and ${\beta _\varepsilon }(\cdot) \in {C^2}({\rm{R}})$ imply that

Combining (31) and (32), it can be easily verified that $\xi \in G(u - {u_0})$.

Further, passing the limit $k \to \infty$ in the second line of (44) and the third line of (6),

Combining the equations above, we infer that $(u, \xi)$ satisfies the conditions of Definition 2.1, such that $(u, \xi)$ is a generalized solution of (1).

Theorem 3.1 Assume that $u_0^m \in {W^{1, p}}(\Omega) \cap {L^\infty }(\Omega)$, $\gamma \le 1$. Then variation-inequality (1) admits at least one solution $(u, \xi)$ within the class of Definition 2.1.

4.   Blowup of solution

In this section, we consider the blowup of the generalized solution when $\gamma > 2$ and try to prove it by contradiction. Assume $(u, \xi)$ is a generalized solution of (1). Taking $\varphi = {u^m}$ in Definition 1, it is easy to see that

It follows from (5), (9), and $\xi \in G(u{\rm{ }} - {\rm{ }}{u_0})$ that $\int_\Omega {\xi \cdot {u^m}{\rm{d}}x} \ge 0$. Let

It follows from (c) in Definition 2.1 that $E(t) \ge 0$ for any $t \in (0, T]$, so one can infer that

Using the Poincare inequality gives

Here, $\int_\Omega {|u{|^{(p + 1)m}}{\rm{d}}x}$ need to keep shrinking. By the Holder inequality

so that

Combining (34)–(36), one can find that

Note that $mp > 1$. Using variable separation method, we have that

such that

This means that the generalized solution blows up at ${T^*} = \frac{{E{{(0)}^{\frac{{1 - mp}}{{m + 1}}}}(p + 1)m}}{{p(\gamma - 2)(mp - 1)C(|\Omega |)}}$.

Theorem 4.1 Assume $mp > 1$. if $\gamma > 2$, the generalized solution $(u, \xi)$ of variation-inequality (1) blows up in finite time.

5.   Conclusions

In this study, the existence and blowup of a generalized solution to a class of variation-inequality problems with non-divergence polytropic parabolic operators

We first consider the existence of generalized solution. Due to the use of integration by parts, $-\gamma |\nabla {u^m}{|^p}$ becomes $(1-\gamma)|\nabla {u^m}{|^p}$. In the process of proving $u_\varepsilon ^m \in {L^\infty }(0, T; {W^{1, p}}(\Omega))$ and ${\partial _t}u_\varepsilon ^m \in {L^\infty }(0, T; {L^2}(\Omega))$ in ^[4,5], $(1 - \gamma)|\nabla {u^m}{|^p}$ is required to be greater than 0, therefore eliciting the restriction $\gamma \le 1$. Regarding the restriction of $p$, the condition $p \ge 1$ is required in (24) and the above formula. As what mentioned, we have used the results $u_\varepsilon ^m \in {L^\infty }(0, T; {W^{1, p}}(\Omega))$ and ${\partial _t}u_\varepsilon ^m \in {L^\infty }(0, T; {L^2}(\Omega))$ in literature ^[4,5] where $p \ge 2$ is required, therefore we require the restriction that $p \ge 2$. The results show that variation-inequality (1) admits at least one solution $(u, \xi)$ when $\gamma\le 1$.

Second, we analyzed the blowup phenomenon of a generalized solution. In (38), $mp$ must be big than 1, otherwise (39) is invalid. The results show that the generalized solution $(u, \xi)$ of the variation-inequality (1) blows up in finite time when $\gamma \ge 2$.

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 author declares no conflict of interest.