A general form for precise asymptotics for complete convergence under sublinear expectation

1.   Introduction

The American lookback option is a crucial aspect of option pricing whose value can be determined through a variation-inequality approach. This option allows investors to observe the risk assets' lowest price ${ {S_t}, t \in [0, T]}$ during the time frame $[0, T]$ and purchase them at that price before the transaction time $t$. If the option is exercised on the expiry date, its value is calculated as follows ^[1,2,3]:

where ${J_t} = \mathop {\min }\limits_{t \in [0, t]} {S_t}$. American look-back options allow investors to exercise their options at any point during the time interval $[0, T]$. This means that the value of the option, denoted by $V({S_t}, {J_t}, T)$, is greater than or equal to the difference between the stock price at time $t$, $S_t$ and the minimum stock price within the time interval ($J_t$), which is represented by

According to the literature ^[4], the option value $V(S, J, t)$ at any given time is governed by a variation inequality:

where $r$ represents the risk-free interest rate of the financial market, and $\sigma$ represents the volatility of option-linked stocks.

The theoretical study of variation-inequality has increasingly gained the attention of scholars. In 2022, wu developed a fourth-order $p$-Laplacian Kirchhoff operator, given by

and investigated a variation-inequality problem ^[5]:

where ${\Omega _T} = \Omega \times (0, T)$, ${\Omega}$ is a $N$-dimensional domain with $N \ge 2$, where $\phi_0$ is a given function and $\gamma$ is a positive constant.The Leray-Schauder principle, a penalty function, and inequality amplification techniques were used to establish the existence, stability and uniqueness of the weak solution. Li and Bi ^[6] examined a 2-D variation-inequality system and proved the existence of weak solutions by analyzing upper and lower solutions of the auxiliary problem. The issues of existence for variation-inequality problems have been extensively studied in ^[7,8], with relevant literature reviewed. Additionally, uniqueness and stability of weak solutions for variation-inequality problems have been proven in ^[9,10,11]. Further results on solvability and well-posedness can be found in ^[12,13] and related references. However, research regarding regularity and higher integrability of this type of problem appears to be less explored.

We investigate a kind of variation-inequality problem

defined on $\Omega \times (0, T)$, featuring a non-Newtonian parabolic operator ($4$) in which

Here, ${u_0}:\Omega \to {\rm{R}}$ is measurable, and ${u_0} \in {H^1}(\Omega)$.

We aim to investigate the regularity and higher integrability of weak solutions to problem (1). Our approach involves several steps. First, we establish the existence of $\Delta u$ by utilizing the weak solution of an auxiliary problem. Next, we analyze the first-order spatial gradient estimation and the time gradient estimation of $u$ by employing the weak solution of the variation-inequality. We then combine the results obtained from both these analyses to obtain higher-order integrability using the inequality amplification technique. Final, we construct the weak solution using the difference operator and approximate the estimate of the partial derivative using its estimate. Through this process, we are able to obtain the regularity of the variational inequality problem.

2.   Statement of the problem and some preliminaries

Firstly, we present the weak solution of the variation-inequality. This solution's existence can be found in various literatures ^[5,6]. To do so, we provide a set of maximal monotone maps given by

where $M_0$ is a positive constant.

Using a standard energy method, it can be shown that variation-inequality (3) has a unique solution if and only if,

(a) $u \in {L^\infty }(0, T, {H^1}(\Omega)), \; {\partial _t}u \in {L^\infty }(0, T, {L^2}(\Omega))$ and $\xi \in G\; {\rm{for}}\; {\rm{any}}\; (x, t) \in {\Omega _T}$,

(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 $\varphi \in {C^1}({\bar \Omega _T})$, there admits the equalities

and

To analyze the regularity of variation-inequality (3), we need to introduce the following operators and their corresponding results. Readers can refer to literature ^[14] for proofs of some of these results.

The difference operator of $u(x, t)$ in the $e_i$ direction, denoted by $\Delta^i_h u(x, t)$, is given by

where $e_i$ is the unit vector in the $x_i$ direction.

Lemma 2.1. ^[14] (1) Assume that $\Delta _h^{i*}$ is the conjugate operator of $\Delta _h^i$ satisfies $\Delta _h^{i*} = - \Delta _{ - h}^i$. Then, one gets

(2) $D_j$ and $\Delta _h^i$ are interchangeable, in other words,

(3) Suppose $T_h^i$ represents the displacement operator in the $x_i$ direction. Then,

(4) If $u \in {W^{1, p}}(\Omega)$, then for any $\Omega ' \subset \subset \Omega$, we have $||\Delta _h^iu|{|_{{L^p}(\Omega ')}} \le ||{D_i}u|{|_{{L^p}(\Omega)}}$.

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

Using Holder inequalities and combining with Lemma 2.1 (4), it is clear to verify that

Lemma 2.2. Let $u \in {H^2}(\Omega)$, and $h$ be small enough. If $\int_\Omega {|{D_i}u\Delta _h^iu|{\rm{d}}x} \le C$, then

Proof. Using Taylor expansion method, there is an $\theta \in [0, 1]$, such that

Rearranging the above equation, we have

Since $u \in {H^2}(\Omega)$,

Then, if $h$ is small enough, (9) holds.

Lemma 2.3. Assume $u \in {H^2}(\Omega)$, and $m$ is a positive integer. Let $h$ be small enough. If $|\int_\Omega {{D_i}[{{(\Delta _h^iu)}^m}]\Delta _h^i[{{({D_i}u)}^m}]{\rm{d}}x} | \le C$, then we have

Proof. Since ${D_i}u(x, t) = \Delta _h^iu + {\rm{O}}(h)$, carrying out binomial expansion to ${[\Delta _h^iu + {\rm{O}}(h)]^m}$ gives ${({D_i}u)^m} = {(\Delta _h^iu)^m} + {\rm{O}}(h)$, so

If $h$ is small enough, using Lemma 2.2 obtains

Hence, the first part of Lemma 2.3 is proved from which the second part is an immediate result.□

3.   Exisence of $\Delta u$

First, we consider the existence of $\Delta u$ and use $\varepsilon$ to construct a penalty function ${\beta_\varepsilon}(\cdot)$ to control the variation-inequality (3). The penalty map ${\beta_\varepsilon}: \mathbb{R} \to \mathbb{R}_-$ satisfies

Furthermore, we use the following auxiliary problem to approach the variation-inequality (3):

where

Using a similar method as in ^[5,6], we can find a solution ${u_\varepsilon }$ for problem (11) that satisfies ${u_\varepsilon } \in {L^\infty }(0, T; {W^{1, p}}(\Omega))$, ${\partial _t}{u_\varepsilon } \in {L^\infty }(0, T; {L^2}(\Omega))$, and the identity

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

By choosing $u_\varepsilon$ as the test function in Eq (12), we can follow a similar approach as in ^[5] to obtain the following inequality:

This implies that for any $\varepsilon\in(0, 1)$, there exists a subsequence of $\{u_\varepsilon\}_{0 < \varepsilon < 1}$ (which we still denote by $\{u_\varepsilon\}_{0 < \varepsilon < 1}$) and a function $v$, such that:

Indeed, from ^[13], we know that $\{u_\varepsilon, \varepsilon \in(0, 1) \}$ is a bounded sequence, which allows us to extract a subsequence without loss of generality, denoted again by $\{u_\varepsilon, \varepsilon \in(0, 1) \}$, that converges almost everywhere in $\Omega_T$ to some function $u$:

Now, we verify $v = \Delta u$. By suing integral by part, one gets

Thus, from Eq (14) we have for any $\phi \in C_0^\infty (\Omega)$,

This implies that $v = \Delta u$ a.e. in $\Omega _T$.

4.   Higher integrability of the gradient

This section gives several gradient estimates of $u$. First, choosing $u$ as a test function in (6) gives

Since $\int {\int_{{\Omega _T}} {{\partial _t}u \cdot u{\rm{d}}x{\rm{d}}t} } = \int_0^T {\int_\Omega {{\partial _t}{u^2}{\rm{d}}x{\rm{d}}t} } = \int_\Omega {|u(, T){|^2}{\rm{d}}x} - \int_\Omega {|{u_0}{|^2}{\rm{d}}x}$, one, from (15) can get that

Applying Holder and Young inequalities,

Combining (16) and (17), it is inferred that (note that $p \ge 1$),

Second, letting ${\partial _t}u$ be a test function in (6), we can find that

Using differential transformation technology, one gets

Applying Holder and Young inequalities with parameters $(1/2, 1/2)$,

Combining (19)–(22) and dropping the term $\frac{1}{p}||\nabla u(\cdot, T)|{|_{{L^p}(\Omega)}}$, one can get

Final, we choose $\varphi {\rm{ = div}}(|\nabla u{|^{p - 2}}\nabla u)$ in (7) to arrive at

Applying integral by part and joining with (20),

Applying Holder and Young inequalities to ${\int {\int_{{\Omega _T}} {f{\rm{div}}(|\nabla u{|^{p - 2}}\nabla u){\rm{d}}x{\rm{d}}t} } }$ and $\int {\int_{\Omega_T} {\xi \cdot {\rm{div}}(|\nabla u{|^{p - 2}}\nabla u){\rm{d}}x{\rm{d}}t} }$,

Substituting (26) and (27) to (27) and dropping the term $\frac{1}{p}||\nabla u(\cdot, T)|{|_{{L^p}(\Omega)}}$,

Theorem 4.1. If ${u_0} \in W_0^{1, p}(\Omega)$ \ and \; $f \in {L^1}(0, T; {L^2}(\Omega))$, then

It should be pointed out that the higher-order term ${\rm{div}}(|\nabla u{|^{p - 2}}\nabla u)$ in the non-Newtonian parabolic operator $Lu$ has not been explained before and is substituted into Eq (24). While the final result displays the boundedness of ${\rm{div}}(|\nabla u{|^{p - 2}}\nabla u)$, a more reasonable proof may be required. Thus, we provide such a proof in Section 5.

5.   Regularity of solution

This section considers the regularity estimate of weak solutions. We draw inspiration from literature ^[14] and introduce the difference operator $\Delta _h^i$ and its conjugate $\Delta _h^{*i}$ into the test function. Since $u \in {L^\infty }(0, T; {W^{1, p}}(\Omega))$, we have

By choosing $\varphi$ as a test function in (7), we get:

We first consider $\int {\int_{{\Omega _T}} {{\partial _t}u \cdot \Delta _h^{i*}(|\Delta _h^iu{|^{p - 2}}\Delta _h^iu){\rm{d}}x{\rm{d}}t} }$. Using Lemma 2.1 (1) and some differential transformation technologies

Substituting (30) into (29) and removing the non negative term $\frac{1}{p}\int_\Omega {|\Delta _h^iu(x, T){|^2}{\rm{d}}x}$, one gets

Using the commutative properties of $\Delta _h^{i*}$ and $\nabla$

It follows from Lemma 2.4 that if $h$ is small enough,

So, combining (31) and (32) and applying Lemma 2.1 (5), inequality (31) can be rewritten as

Applying Holder and Young inequalities,

Inserting (34) and (35) into (33), it is inferred that

From Lemma 2.1 (4), we have $\int_\Omega {|\Delta _h^i{u_0}{|^2}{\rm{d}}x} \le \int_\Omega {|{D_i}{u_0}{|^2}{\rm{d}}x}$, so one can use lemma 2.1 (5) to arrive at

Adding the above formula from 1 to N, we summarize the following result.

Theorem 5.1. If ${u_0} \in W_0^{1, p}(\Omega)\; {\rm{and}}\; f \in {L^1}(0, T; {L^2}(\Omega))$, then

Using Poincare inequality twice, it can be easily verified that

where ${C_{{\rm{poincare}}}}$ is the poincare parameter, such that we obtain the following theorem.

Theorem 5.2. For ${u_0} \in W_0^{1, p}(\Omega)\; {\rm{and}}\; f \in {L^1}(0, T; {L^2}(\Omega))$, we have $u \in L(0, T; {W^{1, \; 2p - 2}}(\Omega))$.

6.   Conclusions

In this paper, we investigate the variation-inequality problem (3) featuring a non-Newtonian operator. First, we establish the norm boundedness of the gradient $\nabla {u_\varepsilon }$ based on the weak solution of the auxiliary problem. We then utilize weak limit to prove the existence of the gradient of the solution to the variation-inequality (3). Second, we analyze the higher order integrability of the solutions of variation-inequality (3). To achieve this, we use the weak Eq (6) of variation-inequality (3), which is fundamental in the study of higher order integrability, to obtain the gradient estimate of the solutions (18) and (23). Further, we select the higher order term of the non-Newtonian parabolic operator as the test function and combine the gradient estimations (18) and (23) to establish the higher order integrability of the weak solution. Last, from the perspective of regularity, we obtain high-order gradient estimates for the variational inequality (3).

It is important to note that using second-order spatial partial derivatives as test functions to analyze regularity and higher-order integrability may not meet the conditions for weak solutions. To avoid discussing the existence and rationality of the test function, we construct it using the difference function. This leads to the derivation of integral inequality (31), which is crucial for proving regularity. The regularity of weak solutions is estimated using Holder and Young inequalities.

There are still some points worth discussing in this article. If we introduce the cutoff factor $\eta$ in formula (29) to construct the test function as follows:

Here, $\eta \in C_0^\infty (\Omega)$ is a cutoff factor on $\Omega ' \subset \subset \Omega$ satisfying:

By following the proof in Section 5, we obtain the following estimate:

If $p = 2$, we can easily obtain the following:

In this case, it is easy to deduce that $u \in L(0, T, {H^k}(\Omega))$, where $k$ is a positive integer satisfying $k \ge 2$. However, when $p \ge 2$, it is impossible to obtain a similar result as in the case of $p = 2$ because we cannot prove $||\Delta u|||_{{L^{2p - 2}}({\Omega _T})}^2 \le ||{\rm{div}}(|\nabla u{|^{p - 2}}\nabla u)||_{{L^2}({\Omega _T})}^2$.

Acknowledgments

Z. Sun was partially supported by the Humanities and Social Sciences Research Project of the Ministry of Education of China (No.21XJC910001). The author is grateful to the anonymous referees for their valuable comments and suggestions.

Conflict of interest

The author declares no conflict of interest.