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Second-order integro-differential parabolic variational inequalities arising from the valuation of American option
Journal of Inequalities and Applications volume 2014, Article number: 8 (2014)
This article deals with the existence and uniqueness of solution to the -valued parabolic variational inequalities with integro-differential terms which arise from the valuation of American option. The authors use the penalty method to construct a sequence of approximation parabolic problem and hence obtain the existence and uniqueness of solution to the approximation problem by using fixed point theory. Then the solution of parabolic variational inequalities is obtained by showing that the solution of this penalty problem converges to the variational inequalities. The uniqueness of the solution is also proven.
In this paper, we are concerned with the existence and uniqueness of solution to an -valued parabolic variational inequality with the integro-differential terms
where is an open set with a smooth boundary ∂ Ω, is a parabolic domain for some . , and is a divergence form second-order elliptic operator satisfying
Moreover, the integro-differential operator is defined by
which is a continuous integral operator as defined in .
The concrete motivations of studying (1) can be easily found in the literature. If , , and the operator degenerates to
(1) becomes the linear variational inequality based on the famous Black-Scholes equation (see ). Such a variational inequality arises in many applications of American option pricing (see [3, 4]). To deal with this problem, some scholars often introduce a free and moving boundary problem. By adding a certain penalty term to the Black-Scholes equation, the solution of this variational inequality is extended to a fixed domain. Furthermore, this penalty term forces the solution to stay above the payoff function at expiry. Throughout the last decade, a number of papers addressing penalty schemes for American options have been published (see, for instance, [5–9] and references therein).
However, several Black-Scholes models proposed in recent years, such as the model found in [8, 9], have allowed the risk assets driven by levy processes. In this model, the risk asset S as in the Black-Scholes model, follows the stochastic process (assume )
from which we obtain the following PIDE:
where is a kind of levy process ( for details, see [10, 11]). The authors in  generalize (2) and prove the existence and uniqueness of a classical solution to a more general problem in the parabolic domain , where Ω is an open, unbounded subset of , with a smooth boundary ∂ Ω. A related work in the context of quantum mechanics has been studied in [12, 13].
Therefore, the authors of this paper intend to study a more complex variational inequality involved in American option based on the more complicated PIDE type Black-Scholes equation than (2). And we will consider -valued parabolic variational inequality (1) using the penalty method. The rest of this article is as follows. In Section 2, we state the main result. Section 3 discusses the penalty problem which will be used to prove our main result. In Section 4, we show the proof of our main result.
2 Main result
In this section, we present some notations and lemmas which are important to be used to prove our main result. Define as the space of all -value functions satisfying
Let and . By , we mean that the ℝ-value functions space satisfies
which is Banach spaces with the norm
Further, we use the following Sobolev spaces which are modified from ℝ-value version to -value version
with the norm
As mentioned in , we shall use to denote the Holder space which satisfies
For , we also use some following common notation from PDEs without mentioning:
Moreover, we use the following lemmas to show the existence and uniqueness of the solution for the penalty approximation of our main problem.
Definition 2.1 Assume that X is a real Banach space, the space consists of all continuous functions with
Then is a Banach space endowed with the norm .
Definition 2.2 A mapping is called to be compact if and only if the sequence is precompact for each bounded sequence , that is, there exists a subsequence such that converges in X.
Lemma 2.3 (Schaefer’s fixed point theorem)
Suppose that is a continuous and compact mapping. Further, assume that the set
is bounded. Then A has a fixed point.
The following two lemmas from the linear theory of parabolic partial differential equations can be found in .
Lemma 2.4 (Energy estimates)
Consider the problem
with . Then there exists a unique solution to problem (9) that satisfies
where C is a positive constant depending only on Ω, T and the operator L.
Lemma 2.5 (Improved regularity)
There exists a unique weak solution
to problem (3), with . Moreover,
We also have the estimate
where C is a positive constant depending only on Ω, T and the operator L.
Throughout this section, we impose the following assumptions:
(A1) The coefficients , , belong to the Holder space .
(A2) The operator is (uniformly) parabolic, that is, there exists a constant such that
(A3) There exists a positive constant B such that for all and , we have
(A4) and belong to the Holder spaces and , respectively. Moreover, for all , we have that , .
(A5) The following two consistency conditions
hold for all , .
(A6) is nonnegative and belongs to , , and there exists satisfying
(A7) For some , satisfies the estimate
for all , where is a positive constant independent of .
(A8) If , then . If in , then in .
Using these assumptions, we will elaborate our main result.
Theorem 2.6 Under hypotheses (A1)-(A8), there exists a unique solution to variational inequality (1) satisfying
The proof of Theorem 2.6 will be given in Section 4.
3 The penalty problem
In order to prove the existence and uniqueness of the solution, we consider the following penalty approximation of problem (1)
where is the penalty function satisfying
Next, we introduce a change of variables
to transform problem (5) into the zero boundary condition of the form
Here is the unique solution to the problem
For further details, see Theorem 10.4.1 in .
Definition 3.1 v is said to be a weak solution to problem (7) if
for all .
Lemma 3.2 If and , then
The mapping is absolutely continuous with
for any . The proof is quite standard and can be found in .
Lemma 3.3 If v is a weak solution to problem (7), then there exists a positive constant C such that
where C is independent of v.
Proof Choosing as the test function in (9), we obtain
From (1), we easily have that
This and (6) lead to
By (A7), so that we get
where is a positive constant which depends only on the region Ω. It follows by (A2) and (A3) that
Now, we use Theorem 10.4.1 in  and substitute (11)-(15) into (10) to arrive at
It follows, by using the Cauchy inequality with , that
Next, we choose to arrive at
where and are positive constants and
Note that . Therefore we choose a positive constant to obtain
On the one hand, letting , (17) gives
Using the differential form of Gronwall inequality with , we get
where is a positive constant independent of v.
On the other hand, an integration of (17) from 0 to T with yields
Using inequality (18), we obtain
Thus, the proof is ended by letting . □
Lemma 3.4 The solution to problem (7) satisfies
Proof Using (A8), there exists a positive constant M satisfying
Here we plan to finish the proof by using contradiction. Assume that is not empty, that is, at least there exists one k satisfying . Let , from (16), one gets
Using the standard maximum principle and (A4), one gets
This is obviously contradictory. Therefore, we conclude that for any . □
Lemma 3.5 If (A1)-(A8) are satisfied, there exists a unique weak solution to problem (7) satisfying
where N is a positive constant independent of v.
Proof Given , set
The use of (A8) leads to
By Lemma 2.4, there exists a unique solution to
Define the mapping
where v is derived from w via (21).
Here we plan to prove the existence and uniqueness by Schaefer’s fixed point theorem. So that we need to present the continuity and compactness of the mapping M. In this proof we only prove the continuity of the mapping M. The compactness can be obtained by following similar arguments, so we omit it here.
Let be a sequence such that
By the improved regularity (4), we obtain
By using (A8) and Lemma 3.4 with , , we have
The two inequalities above lead to
Next, we pay our attention to the sequence . Since in , from (A8) we have
This and (20) lead to the fact that the sequence is bounded, that is,
Combing (23) with (24), the sequence is bounded uniformly in . In a similar way, is uniformly bounded in . By using Rellich’s theorem (see ), there exist a subsequence and a function which satisfies
We combine (22) with (25) to arrive at
Further, by Lemma 3.3, we have that is bounded. Hence the existence and uniqueness of this theorem are proven by using Lemma 2.3 with .
Finally, we pay our attention to the estimate (19). Letting in (23), we obtain
It follows by that
Therefore, the proof is complete. □
4 The proof of the main result
In this section, we prove that the solution to problem (5) converges to that of problem (1) when . From Lemma 3.3 and Lemma 3.5, we conclude that in (7) exists a unique solution satisfying
Thus, there exists a subsequence of still denoted by itself for convenience, and such that
Letting in (5) and using the maximum principle, we arrive at
Comparing (1) with (26), we only need to prove that
holds. Note that derived from Lemma 3.3. Thus, we can end the proof by showing
In fact, there exists a positive constant such that for all ,
holds when ε is sufficiently small. Further, from (15) we have that
Therefore, we conclude that
Now, we prove the uniqueness by contradiction. Assume that and are the solutions of (1) and . That is, there exists at least one k satisfying . For simplicity, we assume that is not empty (if not, we assume is not empty), so that we have
This and (27) lead to
By the maximum principle and (A8), we have that
This is obviously contradictory. Therefore, we conclude that problem (1) has a unique solution. Moreover the estimate can be easily obtained by Lemma 3.5 and (28) with .
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This work was supported by the National Nature Science Foundation of China (Grant No. 71171164) and the Doctorate Foundation of Northwestern Polytechnical University (Grant No. CX201235). The authors are sincerely grateful to the referees and the associate editor handling the paper for their valuable comments.
The authors declare that they have no competing interests.
YS and YS carried out the proof of the main part of this article, MW corrected the manuscript and participated in its design and coordination. All authors read and approved the final manuscript.
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Sun, Y., Shi, Y. & Wu, M. Second-order integro-differential parabolic variational inequalities arising from the valuation of American option. J Inequal Appl 2014, 8 (2014). https://doi.org/10.1186/1029-242X-2014-8
- American option
- variational inequalities