 Research
 Open Access
The uniqueness of the solution for the definite problem of a parabolic variational inequality
 Liping Song^{1}Email author and
 Wanghui Yu^{2}
https://doi.org/10.1186/s136600161267x
© The Author(s) 2016
Received: 15 September 2016
Accepted: 29 November 2016
Published: 13 December 2016
Abstract
The uniqueness of the solution for the definite problem of a parabolic variational inequality is proved. The problem comes from the study of the optimal exercise strategies for the perpetual executive stock options with unrestricted exercise in financial market. Because the variational inequality is degenerate and the obstacle condition contains the partial derivative of an unknown function, it makes the theoretical study of the definite problem of the variational inequality problem very difficult. Firstly, the property which the value function satisfies is derived by applying the Jensen inequality. Then the uniqueness of the solution is proved by using this property and maximum principles.
Keywords
 parabolic variational inequality
 definite problem
 uniqueness
 Jensen inequality
 maximum principles
1 Introduction
The following is a brief introduction of the financial background and related research of the definite solution problem (1)(2).
Executive stock options are called ESOs for short, which are American call options that the company granted to the managers as a compensation. The underlying assets of ESOs are the company’s stocks. The exercise of ESOs can be divided into block exercise and unrestricted exercise. Block exercise is identical to the exercise of the standard American call options, that is, the holders can exercise none of ESOs or exercise all of ESOs at any exercise time. In the unrestricted exercise situation, the holders can exercise any copies of ESOs at arbitrary exercise moment.

a is the number of ESOs held by the manager at time 0 before exercise, \(0< a \leq M\);

\(m_{t}\) is the number of ESOs exercised by the manager until time t;

\(S_{t}\) is the stock price at time t, and s is the stock price at time 0, \(0< s<+\infty\);

K is the strike price of ESOs, and r is the riskfree rate, where r is a constant and \(r>0\);

x is the manager’s wealth at time 0 before exercise, \(0\leq x<+\infty\);

\(U(\cdot)\) is the utility function of the manager, which is concave and increasing.
Remark 1
The reason of taking \(\alpha< r\) is that stocks with dividends are considered for the American call stock options in general. If the stocks do not have dividends, the American call options are actually European call options. ESOs are American call options, so the case of stocks with dividends is considered. Thus we take \(\alpha< r\).
Because \(0< a \leq M\), \(\gamma> 0\) are both constants, we only consider the case of \(0<\tau\leq A\) for arbitrary positive number A. Let \(Q_{\infty}=(\infty,+\infty)\times(0,A]\), we get the definite solution problem (1)(2).
In [1], a permanent ESOs model with unrestrained exercise is established, and the definite solution problem (1)(2) of a parabolic variational inequality is obtained. The existence and regularity of the solution to the definite solution problem (1)(2) are proved in [2]. In [5], the properties of the free boundary for (1)(2) are studied by using numerical method. However, the uniqueness of the solution to (1)(2) has not yet been proved, which is the main content of this paper.
2 Proof
In order to prove the uniqueness of the solution to (1)(2), we need the following two lemmas.
Lemma 1
Proof
This completes the proof. □
Lemma 2
Proof
By (3) and \(U(y)=e^{\gamma y}\) (\(y\geq0\)), it is clear that \(v(s,x,z)\geqe^{\gamma x}\).
By (4), we have \(u(z,\tau)\geq1\).
This completes the proof. □
Now to prove the uniqueness of the solution to the definite solution problem (1)(2) which satisfies (6).
Remark 2
For the continuoustime optimization problem, the corresponding variational inequality is usually obtained by formal derivation (not strictly derived). Is the solution of the variational inequality really the value function of the original problem? It needs to be further verified, that is, one has to prove a socalled ‘verification theorem’. By a verification theorem, we can prove that when it is sufficiently smooth, the solution of the variational inequality (7)(11) is equal to the value function of the corresponding singular stochastic control problem (obtained by an identical deformation of (3)).
Remark 3
Using the standard method in [4], we can also prove that the value function of the corresponding singular stochastic control problem (obtained by identical deformation of (3)) is a viscosity solution of the variational inequality (7)(11).
Theorem 1
Proof
Suppose there are two solutions \(u_{1}\) and \(u_{2}\) of (7)(11) in \(W^{2,1}_{\infty,\mathrm{loc}}(Q_{\infty})\cap C(\overline{Q}_{\infty})\). Denote \(w=u_{1}u_{2}\). In order to prove \(w=0\), we need to first prove \(w\leq0\) in \(\overline{Q}_{\infty}\).
If \(z\rightarrow+\infty\), i.e. \(s\rightarrow+\infty\), by (3) and \(U(y)=e^{\gamma y}\), we have \(v(s,x,a)\rightarrow0\). By (4), we get \(u(z,\tau)\rightarrow0\), so \(w(z,\tau)\rightarrow0\) holds.
If \(z\rightarrow\infty\), by (11) and the expression of \(V_{\infty }(e^{z})\), we have \(u(z,\tau)\rightarrow1\), then \(w(z,\tau)\rightarrow0\) also holds.
(1) If \((z_{\ast},\tau_{\ast})\in\partial_{p} {\mathcal{O}_{\varepsilon}}\) (\(\partial_{p} {\mathcal{O}_{\varepsilon}}\) is the parabolic boundary of \(\mathcal{O}_{\varepsilon}\)), then by (13)(14), (17) holds.
\({\mathcal{B}}u_{1}(z,\tau)=0\) on \(\partial_{p} {\widehat {\mathcal {O}}_{\varepsilon}}\), so \({\mathcal{B}}u_{1}(z_{0},\tau_{0})=0\). Similar to the proof of (2), \(w(z_{0},\tau_{0})\leq\varepsilon\). Thus \(w\leq\varepsilon\) on \(\partial_{p} {\widehat {\mathcal {O}}_{\varepsilon}}\).
Then by the maximum principle, \(w\leq\varepsilon\) on \(\widehat {\mathcal {O}}_{\varepsilon}\). By \((z_{\ast},\tau_{\ast})\in \widehat {\mathcal{O}}_{\varepsilon}\), (17) holds.
In summary, (17) holds, so \(w\leq\varepsilon\) on \(\mathcal {O}_{\varepsilon}\). Combining with (13), we have \(w\leq\varepsilon\) on \(\overline {Q}_{\infty}\). Let \(\varepsilon\rightarrow0\), we get \(w\leq0\) on \(\overline {Q}_{\infty}\). Similarly, we can prove \(w\geq0\) on \(\overline {Q}_{\infty}\) by exchanging \(u_{1}\) and \(u_{2}\). Thus \(w=0\) on \(\overline {Q}_{\infty}\), i.e., \(u_{1}=u_{2}\) on \(\overline {Q}_{\infty}\).
This completes the proof. □
3 Concluding remarks
The utility function chosen in this paper is an exponential function. However, the conclusion of this paper still holds for other types of utility functions and the proof method is similar.
Declarations
Acknowledgements
The authors wish to thank the anonymous referees for their endeavors and valuable comments. This work is supported by National Natural Science Foundation of China (11471175), Natural Science Foundation of Fujian Province (CN) (2015J05012, 2016J01677, 2016J01678), Educational Scientific Research Project of Young and Middleaged Teachers of Fujian Province (JAT160430), Preresearch Project of National Fund of Putian University (2015079), Breeding Fund Project of Putian University (2014060, 2014061).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Rogers, LCG, Scheinkman, J: Optimal exercise of executive stock options. Finance Stoch. 11(3), 357372 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Song, LP, Yu, WH: A parabolic variational inequality related to the perpetual American executive stock options. Nonlinear Anal., Theory Methods Appl. 74(17), 65836600 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Øksendal, B: Stochastic Differential Equations, 5th edn. Springer, Berlin (2000) Google Scholar
 Pham, H: ContinuousTime Stochastic Control and Optimization with Financial Application. Springer, Berlin (2009) View ArticleMATHGoogle Scholar
 Song, LP: A free boundary problem coming from the perpetual American ESOs. Chin. J. Eng. Math. 31(4), 511520 (2014) MathSciNetMATHGoogle Scholar
 Jaillet, P, Lamberton, D, Lapeyre, B: Variational inequalities and pricing of American options. Acta Appl. Math. 21(3), 263289 (1990) MathSciNetView ArticleMATHGoogle Scholar
 Jiang, LS: The Mathematical Models and Methods in Option Pricing. Higher Education Press, Beijing (2003) Google Scholar
 Song, LP, Yu, WH: The equivalence between block exercise and unrestricted exercise of executive stock options. J. Syst. Sci. Math. Sci. 36(10), 17101720 (2016) Google Scholar