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 Open Access
On some nonlinear boundary value problems related to a BlackScholes model with transaction costs
 Rubén Figueroa^{1}Email author and
 Maria do Rosário Grossinho^{2}
https://doi.org/10.1186/s1366101504109
© Figueroa and Grossinho 2015
 Received: 5 June 2015
 Accepted: 10 August 2015
 Published: 22 August 2015
Abstract
We deal with some generalizations on a BlackScholes model arising in financial mathematics. As a novelty in this paper, we consider a variable volatility and abstract functional boundary conditions, which allow us to treat a very large class of problems involving BlackScholes equation. Our main results involve the existence of extremal solutions in presence of lower and upper solutions. Some examples of applications are provided too.
Keywords
 BlackScholes equation
 functional conditions
 discontinuous ODE’s
1 Introduction
 (1)
Dirichlet conditions: \(B_{1}(V(c),V)=V(c)V_{c}\), \(B_{2}(V(d),V)=V(d)V_{d}\);
 (2)
Initialintegral conditions: \(B_{1}(V(c),V)=V(c) {\int_{c}^{d} k(x) V(x)\,dx}\);
 (3)
Multipoint conditions: \(B_{2}(V(d),V)=V(d) {\sum_{j=1}^{n} V(x_{j})}\),
This paper is organized as follows. In Section 2, we introduce an auxiliary nonlinear boundary value problem and the notions of upper and lower solutions used later. In Section 3, we consider problem (1) with Dirichlet boundary conditions. So, we start from paper [1] and we generalize it by considering \(p(x)\), \(q(x)\) instead of constants p, q, which corresponds to variable volatility in the (BS) model, and by dropping the condition \(V_{c} < V_{d}\). In Section 4, we deal with problem (1) on its full version, that is, with functional boundary conditions. Namely, we provide a result on the existence of extremal solutions between lower and upper solutions by using a generalized iteration with Dirichlet problems. Some examples of application are provided, too.
2 Auxiliary problem and upper and lower solutions
We will use the method of upper and lower solutions for this problem and we will begin by considering the classical notions of \(\mathcal {C}^{2}\)lower and upper solutions. However, afterwards, we will use some weaker notions, namely we will need a notion that allows lower solutions to have ‘angles’. In fact, we will consider a maximum of two classical \(\mathcal{C}^{2}\)lower solutions which is not necessarily differentiable and can exhibit an ‘angle’. So, denoting by \(D_{}f(x)\) and \(D^{+}f(x)\), respectively, the lower lefthand and the upper righthand Diniderivatives of the function f at x, we introduce the following definitions (see [5]).
Definition 2.1
 (1)
\(D_{}\alpha(x_{0}) < D^{+}\alpha(x_{0})\);
 (2)There exists an open interval \(J_{0}\) such that \(x_{0} \in J_{0} \subset(c,d)\), \(\alpha_{J_{0}} \in W^{2,1}(J_{0})\), and for almost all \(x \in J_{0}\) we have$$\alpha''(x)+H\bigl(x,\alpha(x),\alpha'(x) \bigr) \ge0. $$
 (1)
\(D^{}\beta(x_{0}) > D_{+}\beta(x_{0})\);
 (2)There exists an open interval \(J_{0}\) such that \(x_{0} \in J_{0} \subset(c,d)\), \(\beta_{J_{0}} \in W^{2,1}(J_{0})\), and for almost all \(x \in J_{0}\) we have$$\beta''(x)+H\bigl(x,\beta(x),\beta'(x) \bigr) \le0. $$
Notice that if α, β are classical \(\mathcal{C}^{2}\)lower and upper solutions for problem (8), then they are also lower and upper solutions in the sense referred above.
3 Nonlinear problem with Dirichlet conditions
From the study of problem (11), we will deduce later existence and localization results for problem (10). Next proposition establishes adequately the existence of classical \(\mathcal{C}^{2}\)upper and lower solutions for (11).
Proposition 3.1
 (1)If \(\frac{V_{d}}{d} \le\frac{V_{c}}{c}\), then the functionis a \(\mathcal{C}^{2}\)lower solution for problem (11).$$ \alpha_{1}(x)=\frac{V_{d}}{d} x $$(12)
The converse is also valid.
 (2)Take \(k >0\) such thatwhere$$ k \ge\sqrt{\frac{Q}{c^{3}}} \sqrt{ \max_{x \in[c,d]} \biggl\vert \frac {k}{2}\bigl(x^{2}cd\bigr)+ \frac{cV_{d} dV_{c} }{dc}\biggr\vert }, $$(13)Then the function$$Q=\max_{x \in[c,d]} q(x). $$is a \(\mathcal{C}^{2}\)lower solution for problem (11).$$ \alpha_{k}(x)=\frac{k}{2} x^{2} + \biggl(\frac{V_{d}V_{c}}{dc}\frac {k}{2}(d+c) \biggr)x + \frac{k}{2} cd  \frac{cV_{d} dV_{c} }{dc} $$(14)
 (3)The functionis a \(\mathcal{C}^{2}\)upper solution for problem (11).$$ \beta(x)=\frac{V_{d}V_{c}}{dc}x + \frac{dV_{c}cV_{d}}{dc},\quad x\in[c,d] $$(15)
Proof
Remark 3.1
Notation 3.1
Next, we state an existence and localization result for problem (11).
Theorem 3.1
 (a)
If \(\frac{V_{d}}{d} \le\frac{V_{c}}{c}\), then problem (11) has extremal \(W^{2,1}\)solutions, that is, the least and the greatest one, in the functional interval \([\alpha_{1},\beta]\).
 (b)
If \(k >0\) satisfies (13), then problem (11) has extremal \(W^{2,1}\)solutions in the functional interval \([\alpha _{k},\beta]\).
Proof
Corollary 3.1
Proof
Proposition 3.2
Proof
(b) Let V be a solution of problem (11) such that \(V\geq\alpha_{1}\) in \([c,d]\). We claim that \(\frac{V_{d}}{d}\geq V'(d)\). In fact, \(\alpha_{1}=\frac{V_{d}}{d}x\leq V(x)\) implies that \(\frac{V_{d}}{d}\geq\frac{V(x)V_{d}}{xd}\) in \([c,d]\), and then, letting \(x\rightarrow d\), we obtain \(\frac{V_{d}}{d}\geq V'(d)\). This inequality together with the fact that the function \(x \in[c,d] \longmapsto xV'(x)V(x)\) is nondecreasing implies that \(xV'(x)V(x) \le 0\) for all \(x \in[c,d]\). □
Next theorem establishes the relations between the convex solutions of (11) and of (10).
Theorem 3.2
Proof
From Theorem 3.1, Theorem 3.2 and Corollary 3.1, it is clear that the following existence and localization result holds.
Theorem 3.3
 (1)
If \(\frac{V_{d}}{d} \le\frac{V_{c}}{c}\), then this problem has the extremal convex \(W^{2,1}\)solutions in the functional interval \([\alpha_{1},\beta]\), where \(\alpha_{1}\) and β are provided, respectively, by (12) and (15);
 (2)If \(\frac{V_{d}}{d} \le\frac{V_{c}}{c}\) and \(k>0\) satisfies (13), then this problem has the extremal convex \(W^{2,1}\)solutions in the functional interval \([\alpha,\beta]\), whereand \(\alpha_{1}\), \(\alpha_{k}\), β are provided, respectively, by (12), (14), (15).$$\alpha(x)=\max\bigl\{ \alpha_{1}(x),\alpha_{k}(x)\bigr\} $$
Remark 3.2
Under the hypotheses of the above theorem, observe that if \(\frac{V_{d}}{d}=\frac{V_{c}}{c}\) then \(\alpha_{1}\) is a solution of problem (10). On the other hand, in the periodic case, \(V_{c}=V_{d}\), the constant function \(V\equiv V_{c}\) is a solution of (10).
Example 3.1
4 Problem with functional boundary conditions
In this section we deal with problem (8) on its full expression and, as said in the introduction, we will use a generalized monotone method. In the construction of this method we will use two technical lemmas. First of them is the following generalization of Bolzano’s theorem.
Lemma 4.1
([7], Lemma 2.3)
The second auxiliary result we need deals with the existence of extremal fixed points for nondecreasing operators defined in the space of absolutely continuous functions.
Lemma 4.2
([8], Proposition 1.4.4)
Now we establish a new result on the existence of extremal convex solutions for problem (8).
Theorem 4.1
 (H_{1}):

For all \(\gamma\in[\alpha,\beta]\) and all \(y \in E\), we have$$\liminf_{z \to y^{}} B_{i}(z,\gamma) \ge B_{i}(y,\gamma) \ge\limsup_{z \to y^{+}} B_{i}(z, \gamma)\quad (i=1,2); $$
 (H_{2}):

For all \(y \in E\), the functions \(B_{i}(y,\cdot)\) are nonincreasing in \([\alpha,\beta]\) (\(i=1,2\)), that is, if \(\gamma _{1},\gamma_{2} \in[\alpha,\beta]\) are such that \(\gamma_{1}(x) \le \gamma_{2}(x)\) for all \(x \in[c,d]\), then \(B_{i}(y,\gamma_{1}) \ge B_{i}(y,\gamma_{2})\).
In these conditions, problem (8) has the extremal convex solutions in \([\alpha,\beta]\).
Proof
On the other hand, as \(\gamma_{c}\) and \(\gamma_{d}\) are the greatest solutions of equations (21)(22) in, respectively, \([\alpha (c),\beta(c)]\) and \([\alpha(d),\beta(d)]\), this implies that \(\alpha (c) \le\gamma_{c}\), \(\alpha(d) \le\gamma_{d}\), \(\beta(c) \ge\gamma _{c}\) and \(\beta(d) \ge\gamma_{d}\). So, α and β are, respectively, a lower and an upper solution for problem (20). This guarantees that (20) has the greatest convex solution in \([\alpha,\beta]\). (Notice that (16)(17) provides a Nagumotype bound for H between our α and β.)
So, problem (20) with conditions \(V(c)=\gamma_{2c}\), \(V(d)=\gamma_{2d}\) has a solution in \([\hat{\alpha},\beta]\), but this contradicts the fact that \(G\gamma_{2}\) is the greatest solution for this problem in \([\alpha,\beta]\). Then we conclude that \(G\gamma _{1} \le G\gamma_{2}\), and so G is a nondecreasing mapping.
Step 4: \(V^{*}\) is the greatest convex solution of problem ( 8 ) in \([\alpha,\beta]\). First, it is clear, as \(GV^{*}=V^{*}\), that \(V^{*}\) is a solution of problem (8). Now, if V is another solution of (8), then we have that \(V \le GV\) and so (19) implies that \(V \le V^{*}\). So, \(V^{*}\) is the greatest convex solution of problem (8) in \([\alpha,\beta]\).
To obtain the least convex solution of (8) in \([\alpha,\beta ]\), we only have to redefine the mapping G in the obvious way. □
Remark 4.1
Notice that condition (H_{1}) is satisfied, for example, if \(B_{1}(\cdot,\gamma)\) is continuous or if it has only downwards discontinuities.
Remark 4.2
The same argument used in the proof of Theorem 3.3 provides that extremal convex solutions of problem (8) are also extremal convex solutions of problem (1) provided that \(\frac{V(d)}{d} \le\frac{V(c)}{c}\).
Example 4.1
 (B_{1}):

‘The initial value of the solution is one half of its mean value on the whole interval \([c,d]\)’;
 (B_{2}):

‘The final value of the solution has integer part 4’.
 (B_{1}):

\(B_{1}(V(c),V)=V(c)\frac{1}{2} \frac{1}{dc} {\int_{c}^{d} V(s)\,ds}=0\);
 (B_{2}):

\(B_{2}(V(d),V)=[V(d)]+4=0\).
Remark 4.3
Declarations
Acknowledgements
The first author was partially supported by Xunta de Galicia, Consellería de Cultura, Educación e Ordenación Universitaria, through the project EM2014/032 ‘Ecuacións diferenciais non lineares’; and by Ministerio de Economía y Competitividad of Spain under Grant MTM201015314, cofinanced by the European Community fund FEDER. The second author was partially funded by Fundação para a Ciência e Tecnologia through the project UID/Multi/00491/2013 and the Transnational Cooperation FCT PortugalSlovakia ‘Analysis of Nonlinear Partial Differential Equations in Mathematical Finance (20132014)’ and by the EU Grant Program FP7PEOPLE2012ITN STRIKE  ‘Novel Methods in Computational Finance’, No. 304617 (D.S.).
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Authors’ Affiliations
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